Probability for finance

PATRICK ROGER PROBABILITY FOR FINANCE DOWNLOAD FREE TEXTBOOKS AT BOOKBOON.COM Probability for Finance Patrick Roger Strasbourg University, EM Strasbourg Business School May 2010 Download free books at BookBooN.com 2 Probability for Finance © 2010 Patrick Roger & Ventus Publishing ApS ISBN 978-87-7681-589-9 Download free books at BookBooN.com 3 Probability for Finance Contents Contents Introduction 8 1. 1.1 1.1.1 1.1.2 1.1.3 1.2 1.2.1 1.2.2 1.2.3 1.3 1.3.1 1.3.2 1.3.3 1.3.4 1.3.5 Probability spaces and random variables Measurable spaces and probability measures σ algebra (or tribe) on a set Ω Sub-tribes of A Probability measures Conditional probability and Bayes theorem Independant events and independant tribes Conditional probability measures Bayes theorem Random variables and probability distributions Random variables and generated tribes Independant random variables Probability distributions and cumulative distributions Discrete and continuous random variables Transformations of random variables 10 10 11 13 16 18 19 21 24 25 25 29 30 34 35 2. 2.1 Moments of a random variable Mathematical expectation 37 37 Please click the advert what‘s missing in this equation? 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Moller - Maersk. www.maersk.com/mitas Download free books at BookBooN.com 4 Probability for Finance Contents Expectations of discrete and continous random variables Expectation: the general case Illustration: Jensen’s inequality and Saint-Peterburg paradox Variance and higher moments Second-order moments Skewness and kurtosis The vector space of random variables Almost surely equal random variables The space L1 (Ω, A, P) The space L2 (Ω, A, P) Covariance and correlation Equivalent probabilities and Radon-Nikodym derivatives Intuition Radon Nikodym derivatives Random vectors Definitions Application to portfolio choice 39 40 43 46 46 48 50 51 53 54 59 63 63 67 69 69 71 3. 3.1 3.1.1 3.1.2 3.1.3 Usual probability distributions in financial models Discrete distributions Bernoulli distribution Binomial distribution Poisson distribution 73 73 73 76 78 Please click the advert 2.1.1 2.1.2 2.1.3 2.2 2.2.1 2.2.2 2.3 2.3.1 2.3.2 2.3.3 2.3.4 2.4 2.4.1 2.4.2 2.5 2.5.1 2.5.2 Download free books at BookBooN.com 5 Probability for Finance Contents 3.2 3.2.1 3.2.2 3.2.3 3.3 3.3.1 3.3.2 3.3.3 Continuous distributions Uniform distribution Gaussian (normal) distribution Log-normal distribution Some other useful distributions 2 The X distribution The Student-t distribution The Fisher-Snedecor distribution 81 81 82 86 91 91 92 93 4. 4.1 4.1.1 4.1.2 4.1.3 4.1.4 4.1.5 4.2 4.2.1 4.2.2 4.3 4.3.1 4.4 4.4.1 Conditional expectations and Limit theorems Conditional expectations Introductive example Conditional distributions Conditional expectation with respect to an event Conditional expectation with respect to a random variable Conditional expectation with respect to a substribe Geometric interpretation in L2 (Ω, A, P) Introductive example Conditional expectation as a projection in L2 Properties of conditional expectations The Gaussian vector case The law of large numbers and the central limit theorem Stochastic Covergences 94 94 94 96 97 98 100 101 101 102 104 105 108 108 www.job.oticon.dk Download free books at BookBooN.com 6 Probability for Finance Law of large numbers Central limit theorem 109 112 Bibliography 114 Please click the advert 4.4.2 4.4.3 Contents Download free books at BookBooN.com 7 Probability for Finance Introduction                                                                                                                                                                                                                                                                                                                                                                                                                    Download free books at BookBooN.com 8                                    Probability for Finance                                                                                                                  Introduction                                                                                                                                                                                     Turning a challenge into a learning curve. 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Visit accenture.com/bootcamp Download free books at BookBooN.com 9 Probability for Finance Probability spaces and random variables                                   t = 0    T = 1.                                                                                            .                                   T                                                          P                                                    P            Download free books at BookBooN.com 10 Probability for Finance Probability spaces and random variables               ,                             σ   σ                   P()        σ          A  P()    ∈ A  ∀ B ∈ A, B c ∈ A  B c     B   B c = {ω ∈ /ω ∈ / B} . A           (Bn , n ∈ N)    A, +∞ n=1 Bn ∈ A.    A        (, A)          A                   T = 1         ω    A       ω ∈ A  A    ω ∈ / A.     ,         .           = {ω1 , ω 2 , ω 3 , ω 4 } ,    A = {∅, }             A′ = {∅, {ω 1 , ω 2 } , {ω 3 , ω 4 } , }  A = P(),              A          (Bn , n ∈ N)    A,   ∩+∞ n=1 Bn ∈ A A       ∅ ∈ A.    σ                                σ             Download free books at BookBooN.com 11 Probability for Finance Probability spaces and random variables                                                  Γ = {B1 , ..., BK }        Bi ∩ Bj = ∅  i = j  ∪K i=1 Bi = .                 A                 A.          A                  Γ,    ∅,         Γ       .       Bj                 Bj )       Γ  Bj           Γ     Bj       ∅         it’s an interesting world Please click the advert Get under the skin of it. Graduate opportunities Cheltenham | £24,945 + benefits One of the UK’s intelligence services, GCHQ’s role is two-fold: to gather and analyse intelligence which helps shape Britain’s response to global events, and, to provide technical advice for the protection of Government communication and information systems. In doing so, our specialists – in IT, internet, engineering, languages, information assurance, mathematics and intelligence – get well beneath the surface of global affairs. If you thought the world was an interesting place, you really ought to explore our world of work. www.careersinbritishintelligence.co.uk TOP GOVERNMENT EMPLOYER Applicants must be British citizens. GCHQ values diversity and welcomes applicants from all sections of the community. We want our workforce to reflect the diversity of our work. Download free books at BookBooN.com 12 Probability for Finance Probability spaces and random variables            Γ = {B1 , ..., BK }            Γ,   BΓ ,          Γ.        BΓ        BΓ  ∅, ,         Γ.               BΓ  2K      A        T > 1                   T  P)           t < T,            P).     A′  P)     A  A′     A             A    A′ .    , A′ )       A′            = {ω 1 , ω 2 , ω 3 , ω 4 } ,   A′ = {∅, {ω 1 , ω 2 } , {ω 3 , ω 4 } , }     P).                 ∈ A′      B  A′  B c   A′  {ω 1 , ω 2 } = {ω 3 , ω 4 }c .       A′     A′  {ω 1 , ω 2 } ∪ {ω 3 , ω 4 } = .                             A  A′   A′ ⊂ A    Γ  Γ′         Card(   Card(         Card( < Card(P(.                        P(    Download free books at BookBooN.com 13 Probability for Finance Probability spaces and random variables            Γ      Γ′     Γ′       Γ.   Γ       Γ′ .    A′     A   Γ  A       Γ′  A′ .                       2K     K K                A′     A;                                                       u d),        = {uu; ud; du; dd}                A′ = {∅; {uu; ud} ; {du; dd} ; }          P).   {du; dd} = {uu; ud}c           {uu; ud} {du; dd} =  ∈ A.                         1 ր ց ր u ց d ր ց uu = u2 ud du  dd = d2      {uu; ud}          .                              {uu; ud} .                            ud  du     ud            Γ′     Γ. Download free books at BookBooN.com 14 Probability for Finance Probability spaces and random variables                   du.                 R,                 BR .           R     R.      BR                           Brain power Please click the advert By 2020, wind could provide one-tenth of our planet’s electricity needs. Already today, SKF’s innovative knowhow is crucial to running a large proportion of the world’s wind turbines. Up to 25 % of the generating costs relate to maintenance. 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Visit us at www.skf.com/knowledge Download free books at BookBooN.com 15 Probability for Finance Probability spaces and random variables                                                                                                                    (, A)         A     A  [0; 1]   P () = 1     (Bn , n ∈ N)     A +∞  +∞   P Bn = P (Bn ) n=1 n=1   (, A, P )               ∅                                     B    B c ,    P (B) + P (B c ) = P () = 1     P (B c ) = 1−P (B).          B       B c            σ                       Download free books at BookBooN.com 16 Probability for Finance Probability spaces and random variables            (, A, P )      P (∅) = 0  ∀ (B1 , B2 ) ∈ A × A, B1 ⊆ B2 ⇒ P (B1 ) ≤ P (B2 )   (Bn , n ∈ N)     Bn ⊂ Bn+1     A    lim P (Bn ) = P Bn n→+∞ n∈N   (Bn , n ∈ N)     Bn ⊃ Bn+1     A    Bn lim P (Bn ) = P n→+∞ n∈N  ∀ B ∈ A, P (B c ) = 1 − P (B)      ∅    P ( ∅) = P () + P (∅) = P () = 1.   P (∅) = 0     B1 ⊆ B2 ⇒ P (B2 ) = P (B1 (B2 B1c )) = P (B1 ) + P (B2 B1c ) ≥ P (B1 )   n   (Bn , n ∈ N)     un = P   p=1 Bp           P () = 1       (Bn , n ∈N)           P n∈N Bn .   n   (Bn , n ∈ N)     vn = P B   p p=1           P (∅) = 0       (Bn , n ∈N)           P n∈N Bn .        P (B B c ) = P (B) + P (B c )  B   B c     B B c = ,   P (B B c ) = P () = 1   P (B c ) = 1 − P (B)  Download free books at BookBooN.com 17 Probability for Finance Probability spaces and random variables           Card() = N  A = P() ;      A                1 ∀ω ∈ , P (ω) = N                       [0; 1] × [0; 1]       R2 ;         σ                        A         , P (A)       A P   P () = 1; P                      [0; 1] × [0; 1]                                           B = [a; b] × [c; d]    (d − c)(b − a) ≤ 1.                                            B    (d − c)(b − a).                  (, A, P )                                  B ⊂             A      P (ω)               P ({ω})                                                         Download free books at BookBooN.com 18 Probability for Finance Probability spaces and random variables                               B1 , B2  A    P (B1 P (B1 ) × P (B2 ).  B2 ) =   B2 ∈ A   P (B2 ) = 0     B1  B2    P (B1 |B2 ),     P (B1 B2 ) P (B1 |B2 ) = P (B2 )                B2             B2 .        B1         B2 ,      .    B1 B2 = ∅,      B1         B1         B1  B2      B2        B1 .        B1  B2               P (B1 B2 ) P (B1 ) × P (B2 ) P (B1 |B2 ) = = = P (B1 ) P (B2 ) P (B2 )                = [0; 1] × [0; 1]           (x, y)      B1 = 0; 12 ×       1 ; 1  B2 = 0; 13 × 0; 12 ;    3 1 2 1 × = 2 3 3 1 1 1 P (B2 ) = × = 3 2 6 P (B1 ) = Download free books at BookBooN.com 19 Probability for Finance Probability spaces and random variables           B2       (x, y) ∈ B1     x ∈ 0; 13  y      1/3      13 ; 12 .   (x, y) ∈ B2   y ≤ 12 .  (x, y)   B1      y ≥ 13,    1/3   y ∈ 13 ; 12 .    y ∈ 0; 12          P (B1 |B2 ) = 13   B1 B2 = 0; 13 × 13 ; 12 ,        1 1 1 1 P (B1 B2 ) = −0 × − = 3 2 3 18    P (B1 |B2 ) = 1 18 1 6 = 1 = P (B1 ) 3 B1     B2 . Trust and responsibility Please click the advert NNE and Pharmaplan have joined forces to create NNE Pharmaplan, the world’s leading engineering and consultancy company focused entirely on the pharma and biotech industries. – You have to be proactive and open-minded as a newcomer and make it clear to your colleagues what you are able to cope. The pharmaceutical field is new to me. But busy as they are, most of my colleagues find the time to teach me, and they also trust me. Even though it was a bit hard at first, I can feel over time that I am beginning to be taken seriously and that my contribution is appreciated. Inés Aréizaga Esteva (Spain), 25 years old Education: Chemical Engineer NNE Pharmaplan is the world’s leading engineering and consultancy company focused entirely on the pharma and biotech industries. We employ more than 1500 people worldwide and offer global reach and local knowledge along with our all-encompassing list of services. nnepharmaplan.com Download free books at BookBooN.com 20 Probability for Finance Probability spaces and random variables                                B1  B2                          B1 ,          B2     B1              σ          G  G ′  A     ∀B ∈ G, ∀B ′ ∈ G ′ , P (B ∩ B ′ ) = P (B) × P (B ′ )           G× G ′             P()                                                  (, A).                         ∅.                                  B ∈ A   AB     AB = A B  A ∈ A ;  AB     B.    (B, AB )                                Download free books at BookBooN.com 21 Probability for Finance Probability spaces and random variables          B ∈ AB    B = B   (Cn , n ∈ N)       Cn = An B            Cn = An An B = B n∈N n∈N n∈N   B ∈ AB  An ∈ A   n∈N An    C = A B ∈ AB  CBc    C  B.          c  B = Ac B Bc B CBc = A B  = Ac B ∈ AB  A     n∈N    B ∈    P (B) = 0    P (. |B ),   P (B1 |B )    B1 ,       (B, AB ) .     P (B |B ) = 1.  (Cn , n ∈ N)        AB             P P B B) n∈N Cn n∈N (Cn P Cn |B = =  P (B) P (B) n∈N   n, Cn ⊂ B,                    P C B)  n n∈N n∈N P (Cn ) n∈N P (Cn = = = P (Cn |B ) P (B) P (B) P (B) n∈N                         t        B.                                         Download free books at BookBooN.com 22 Probability for Finance Probability spaces and random variables         Please click the advert                                                                        (B, AB , P (. |B )). Download free books at BookBooN.com 23 Probability for Finance Probability spaces and random variables                                                                                                                                                                                                                                 (B1 , B2 , ..., Bn )       C ∈ A,          P (C |Bj )P (Bj ) P (Bj |C ) = n i=1 P (C |Bi )P (Bi )    Bj     n     C= C Bi i=1    P (C) = n  i=1  n     P C Bi = P (C |Bi )P (Bi )  i=1    P C Bj = P (C |Bj )P (Bj ) = P (Bj |C )P (C)  Download free books at BookBooN.com 24 Probability for Finance Probability spaces and random variables               P (C)                                               C         B1       B2 = B1c      P (B1 |C ) = P (C |B1 )P (B1 ) P (C |B1 )P (B1 ) + P (C |B2 )P (B2 )    P (B1 ) = 10−4 P (C |B1 ) = 0.99 P (C |B2 ) = 0.01     P (B1 |C ) =   0.99 × 10−4 ≃ 0.01 0.99 × 10−4 + 0.01 × (1 − 10−4 )                                                T = 1                                                  R+                R                             (S − S )/S ,                ln(S /S ),     −∞.  Download free books at BookBooN.com 25 Probability for Finance Probability spaces and random variables                                                      [−2%; 2%]                                    (, A)  (E, B)                E X   → E   ∀B ∈ B, X −1 (B) ∈ A    X −1 (B)    X −1 (B) = {ω ∈  / X(ω) ∈ B} . X     A    X        E = R                   A.              A)      PX  BR  PX (B) = P (X −1 (B)) .                             X            B        PX (B)              E            R  Rn       R+    N     E ⊆ R         E = Rn                        A .    X          X   B = [−2%; +2%] ,      X −1 (B)                 A                   X       (, A)     (E, B) .     X  BX      A     BX = A ∈ A / ∃B ∈ B , A = X −1 (B) Download free books at BookBooN.com 26 Probability for Finance Probability spaces and random variables               BX     A,      A            A          X                                                  t      Ft           s > t.                  Ft ⊂ Fs  Please click the advert                   uu  ud                             A = P,      R                        A. Download free books at BookBooN.com 27 Probability for Finance Probability spaces and random variables                   Card() = 4  X, Y                               X               X(ω)         Y (ω) = 2.         ω 2  ω 4          Y        X.  ω1 ω2 ω3 ω4 X     Y         X  Y       BX = P() BY = {∅, , {ω 1 } , {ω 2 , ω 4 } , {ω 3 } , {ω 1 , ω 3 } , {ω 1 , ω 2 , ω 4 } , {ω 3 , ω 2 , ω 4 }}    BY          Y                        St  t      uSt−1   p St = dSt−1   1 − p          = {uu; ud; du; dd}                           S0  B0 = {∅, } .             ∅  .   1,   S1            {du; dd}  {uu; ud}               B1       B1 = {∅, , {du; dd} , {uu; ud}}                          Download free books at BookBooN.com 28 Probability for Finance Probability spaces and random variables              B0  B0 ⊂ B1 .          B1                             S2 = udS0 .           S1   .                             B2          (S1 , S2 )    S2 .      BS              S1  S2 .         A  B    P (A ∩ B) = P (A) × P (B)        X  Y   (, A, P )     (E, B)       (A, B) ∈ B 2 ,   X −1 (A)  Y −1 (B)       = {ω 1 , ω 2 , ω 3 , ω 4 } A = P()            X  Y         1 1  −1 2   (X, Y ) =   1 2  −1 1                                     Y = 1, X           ω 1  ω 4    Y = 2,         X  ω 2  ω 3 .   Y         BX                   X  Y                 X  Y                  X  Y         aX  bY  a              Download free books at BookBooN.com 29 Probability for Finance Probability spaces and random variables        b                          X  Y           BX  BY  .            X             (, A)    (R, BR )    X         BR      P  A    X       (, A)    (E, B) ;      X     PX  B,     ∀B ∈ B, PX (B) = P X −1 (B) = P ({ω ∈  / X(ω) ∈ B})   (E, B) = (R, BR ) ,      X  FX ,     R  [0; 1]   FX (x) = P ({ω ∈  / X(ω) ≤ x}) = PX ((−∞; x]) Download free books at BookBooN.com 30 Probability for Finance Probability spaces and random variables          ()   = () X(ω) 35 120   3×() 3×() () = 63 120 ()  = 21 120 1 120                             A                     PX               P                    X                                       n!      k    nk = k!(n−k)!   10!     n       10 = 3!(10−3)! = 120 . 3 1        A = P()  P (ω) = 120 .    PX        X                  {X = k}  k = 0, ..., 3           P (X = 3) =  {X = 2} ,          1 . 120              P (X = 2) = 3×7 .      {X = 1}    120           72 = 21  63 7 10     P (X = 1) = 120 .  P (X = 0) = / 3 = 35/120.     3  63 + 35 + 21 + 1 = 120    {X = k}     {ω ∈    X(ω) = k}. Download free books at BookBooN.com 31 Probability for Finance Probability spaces and random variables                  (, A, P )          PX   BX      A   BX  A.                                                FX     X       lim FX (x) = 0 x→−∞  lim FX (x) = 1 x→+∞  FX          B1 ⊂ B2 ⇒ P (B1 ) ≤ P (B2 ))  x ≤ y,   (−∞; x] ⊆ (−∞; y]   PX ((−∞; x]) ≤ PX ((−∞; y]) .        FX (x)        (−∞; x] ,        (xn , n ∈ N)       x   Bn = (−∞; xn ]      B = (−∞; x] .                             −∞  +∞  n                                                      x   P (X ≥ x) = 1 − FX (x) = 0.99  X              X       V aR(99%) = FX−1 (0.01)                   |x| .         Download free books at BookBooN.com 32  x     X    Y.       Probability for Finance Probability spaces and random variables                                 X       Y,        ∀x ∈ R, FX (x) ≤ FY (x)  FX  FY     X  Y.       X  Y            X  Y,                 P ({X ≥ x}) ≥ P ({Y ≥ x})     x,         x     X    Y.                  Please click the advert  Download free books at BookBooN.com 33 Probability for Finance Probability spaces and random variables                    X        (xn , n ∈ N)     P ({ω ∈  / X(ω) = xn }) = P (X = xn ) = 1 n∈N n∈N (xn , n ∈ N)      X.     Y         fY                 x FY (x) = fY (y)dy −∞  FY     Y. fY       Y     .         +∞ fY (y)dy = 1 −∞     X                       X.                    B ∈ A        B,  B    B (ω) = 1  ω ∈ B = 0                 ω    ω     B.                                                       K = 1000   XT          T.          100 × {XT ≥1000}  {XT ≥ 1000} = {ω ∈  / XT (ω) ≥ 1000} .    Download free books at BookBooN.com 34 Probability for Finance Probability spaces and random variables                                  X        {x1 , ..., xn } ,  xi = xj  i = j     Γ = {B1 , ..., Bn }     n  xi Bi X= i=1       Bi = {ω ∈  / X(ω) = xi } , i = 1, 2, ..., n.  Card() = N  X(ω i ) = xi    X= N  i=1  xi {ω } i     Card() < +∞           {ω }      i                                                    X,           Y = g(X)  g                    •            g                         K   T,      YT = g(XT ) = max(XT − K; 0)  XT   T     Download free books at BookBooN.com 35 Probability for Finance Probability spaces and random variables        •                                Xt  t.       [0; t]      Xt = ln(Xt )  Yt = ln 1                     •                                            fX  fY                              X      fX  g        R  R   fY  Y = g(X)    fX (g −1 (x))  x ∈ Y () |g ′ (g −1 (x))| = 0  fY (x) =  Y () = {y ∈ R / y = Y (ω)  ω ∈ } . Download free books at BookBooN.com 36 Probability for Finance Moments of a random variable                                                                                                                                                                                                                                                                                 Download free books at BookBooN.com 37 Probability for Finance  Moments of a random variable                                                                                             40       21       40        63                                $40 × 1 40 21 40 63 +$ × +$ × = $1 120 21 120 63 120                                                             Sharp Minds - Bright Ideas! 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Dedicated Analytical Solutions FOSS Slangerupgade 69 3400 Hillerød Tel. +45 70103370 www.foss.dk Download free books at BookBooN.com 38 Probability for Finance Moments of a random variable                 X        {x1 , ..., xn } , xi ∈ R   i.  pi = P (X = xi )  i = 1, ..., n;    X   P            E(X),    E(X) = n  xi pi i=1   X       fX   FX     X   P        E(X),    +∞  +∞ xfX (x)dt = xdFX (x) E(X) = −∞ −∞  n   xi              X = B  E(X) = E(B ) = P (B).   X ni=1 xi Bi  Bi = {X = xi } ,     n  n n    E(X) = E xi Bi = xi E (Bi ) = xi pi i=1 i=1 i=1        EP   E         P.  E                                            P,       Q.            EP  EQ        .                           X    Y  X + Y      Download free books at BookBooN.com 39 Probability for Finance  Moments of a random variable                                                     V               (, A, P )  X          X  E(X)   XdP,           E(X) = XdP = sup {E(Y ), Y ≤ X}  Y ∈V      E(Y )   Y ∈ V  Y                  X                 X.                                                                   XdP,     E(X)           x      FX            X    P.         .            X            X      X = X+ − X−  X + = max(X; 0)  X − = max(−X; 0).    E(X)           V        f       sup∈ f (x)          f(x)  x ∈ A.  Download free books at BookBooN.com 40 Probability for Finance Moments of a random variable        X         X  E(X),         E(X) = E(X + ) − E(X − )         X +  X −            E(X)       X       P,           P     E(X)     E (|X|)    |X| = X + + X −                X, Z       A, B    A   X = A ⇒ E(X) = P (A)  0 ≤ X ≤ Z ⇒ 0 ≤ E(X) ≤ E(Z)  {X ≥ 0  A ⊂ B} ⇒ E (XA ) ≤ E (XB ) ∀c ∈ R, E(cX) = cE(X)  E(X + Z) = E(X) + E(Z)  |E(X)| ≤ E (|X|)   X = A       P (A)       1 − P (A)          E(X) = P (A)      Y = 0     V    E(X) ≥ E(Y ) = 0.     Z ≥ X,   sup {E(Y ), Y ≤ X} ≤ sup {E(Y ), Y ≤ Z} Y ∈V  Y ∈V    E(Z) ≥ E(X).     A ⊂ B  X ≥ 0,  XA ≤ XB          Download free books at BookBooN.com 41 Probability for Finance  Moments of a random variable          X ∈ V            X    c > 0.    X  cX  X + −X −  (cX)+ −(cX)− .        c   (cX)− = −cX +  (cX)+ = −cX −       E(cX) = E (cX)+ − E (cX)−     = −cE X − + cE X + = −c (−E(X)) = cE(X)                   X  X + − X −   |X| = X + +X −  E (|X|) = E(X + )+E(X − ) ≥ |E(X + ) − E(X − )|       x  y    x + y > x − y  x + y > y − x.      (x1 , x2 , ..., xn )     X   n      E(X)    x = n1 xi . Please click the advert i=1 Download free books at BookBooN.com 42 Probability for Finance Moments of a random variable                                                                                     X     E [u(X)]      u                                          50 = 12 (0 + 100)              E(X)         X.        E [u(X)] ≤ u [E(X)]  X  u(X)                        X       u      R  R   u(X)       E [u(X)] ≤ u [E(X)]             x1  x2   p  1 − p.    pu(x1 ) + (1 − p)u(x2 ) ≤ u(px1 + (1 − p)x2 )       u(x)        (x1 , u(x1 ))  (x2 , u(x2 )).    f      (x, y)   λ ∈ [0; 1] , f(λx + (1 − λ)y) ≥ λf (x) + (1 − λ)f(y) Download free books at BookBooN.com 43 Probability for Finance  Moments of a random variable            u         u           X        X                          u′ > 0  u′′ < 0                                                                                                                          2n            n                 N                          1/2.  P (N = n) = 21n       n − 1                 2n.         X       E(X) = +∞  n=1 n 2 × P (N = n) = +∞  n=1 2n × 1 = +∞ 2n                                                    Download free books at BookBooN.com 44 Probability for Finance Moments of a random variable                                                         E(ln(X)) = +∞  n=1     ln(2n ) × +∞  1 n = ln(2) n 2 2n n=1 +∞ +∞  +∞   n 1 = =2 n 2 2k n=1 n=1 k=n    E(ln(X)) = 2 ln(2) = ln(4)                                                                Please click the advert Student Discounts + Student Events + Money Saving Advice = Happy Days! 2009 Download free books at BookBooN.com 45 Probability for Finance  Moments of a random variable                                    X.                                                                          X,    2 (X)         2 (X) = E(X 2 )  2 (X)  X       X         X,  V (X)  σ 2 (X)       V (X) = σ 2 (X) = E (X − E(X))2 V (X)           Y = X − E(X),   V (X) = 2 (Y ). Y        E(Y ) = 0    X        V (X) = E (X − E(X))2 = E(X 2 ) − E(X)2     E (X − E(X))2 = = = =   E X 2 − 2XE(X) + E(X)2   E X 2 − 2E [XE(X)] + E(X)2 E(X 2 ) − 2E(X)2 + E(X)2 E(X 2 ) − E(X)2       σ      σ                    Download free books at BookBooN.com 46 Probability for Finance Moments of a random variable          card() =  P (ω) = 0.25   ω,  X     2  3   X=  −1  0 E(X) = 1      Y = X − E(X)      1  2   Y =  −2  −1    X  Y        V (X) = V (Y ) = 0, 25 × 12 + 22 + (−2)2 + (−1)2 = 2.5              V (X) = V (X + c)      c.      (x1 , x2 , ...., xn)     X               n 1  2 s = (xi − x)2  n − 1 i=1   n − 1   n         X         x.    X          X,  σ(X)      V (X)  σ(X) = V (X)                                                                                                  Download free books at BookBooN.com 47 Probability for Finance  Moments of a random variable                                                                                                                                                                                                                   Please click the advert what‘s missing in this equation? 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Moller - Maersk. www.maersk.com/mitas Download free books at BookBooN.com 48 Probability for Finance Moments of a random variable             n     X,   n (X)         n (X) = E(X n )    X           3.    X,  Sk(X)    Sk(X) = 3 (X − E(X)) σ(X)3        (x1 , x2 , ...., xn )     X, Sk(X)    = Sk   xi − x 3 n (n − 1)(n − 2) s                        = −0.73           Sk                                            Download free books at BookBooN.com 49 Probability for Finance  Moments of a random variable                X    4 (X − E(X)) σ4        X    κ(X) = eκ(X) = κ(X) − 3        κ = 3.            κ(X)                 κ(X) = 8.93                                                                                             L0 (, A)          (, A).               ∀ω ∈ , (X + Y ) (ω) = X(ω) + Y (ω) ∀ω ∈ , ∀c ∈ R, (cX)(ω) = cX(ω)      L0 (, A)                                                                                   P   .        P                  Download free books at BookBooN.com 50 Probability for Finance         Moments of a random variable            n   Rn   x  y,            d(x, y)  x  y        d     Rn       Rn ,   n  x=y⇔ (xi − yi )2 = 0  i=1   xi = yi  i = 1, ..., n.             [a; b] .          d(f, g) =  b a |f(x) − g(x)| dx              d(f, g) = 0  f = g  f(x) = 0  [a; b]  g(x) = 0  [a; b[  g(b) = 1.     d(f, g) = 0  f  g               f  g                                 R   f Rg  f  g            R        f Rf   f Rg ⇔ gRf)   f Rg  gRh ⇒ f Rh)        d          R       [a; b] .  fˆ  ĝ        f  g,   d(fˆ, ĝ)           (f, g)     fˆ × ĝ.                      d    S     S × S  R    d(x, y) = 0  x = y  d(x, y) = d(y, x)   d(x, z) ≤ d(x, y) + d(y, z)     Download free books at BookBooN.com 51                     Probability for Finance Moments of a random variable  L1 (, A, P )             P     (, A, P ).      X  Y   (, A, P )   P  P    L1 (, A, P )       R   P (ωXRY ∈  /⇔X(ω) X == Y YP(ω))  = 1        X = Y a.s ⇔ P (X = Y ) = 1                (, A, P )    A ∈ A  P   P (A) = 0.                     L1 (, A, P )      P      (, A, P ).      R   L1 (, A, P )  XRY ⇔ X = Y P       Please click the advert    P   P                       P   P                   Download free books at BookBooN.com 52 Probability for Finance Moments of a random variable            L1 (, A, P )   L1 (, A, P )     R      L1 (, A, P ).     L0 (, A, P )     R      L0 (, A).    L1 (, A, P )      L0 (, A, P ).      L1 (, A, P )  R+ ,  X → X1       X → X1 = E(|X|)     L1  R   X  E(X),  X → E(X),           L1 (, A, P )      L0 (, A, P )                  X → X1    X1 = 0 ⇔ X = 0 P         X + Y 1 ≤ X1 + Y 1    ω ∈ ,   |X(ω) + Y (ω)| ≤ |X(ω)| + |Y (ω)| ,  E(|X + Y |) ≤ E(|X|) + E(|Y |)        αX1 = |α| X1                                                             L                 S     S  R ,  .   x = 0     x = 0  ∀x ∈ S, ∀c ∈ R, cx = |c| x  ∀(x, y) ∈ S × S, x + y ≤ x + y  Download free books at BookBooN.com 53 Probability for Finance  Moments of a random variable                                                                                             L1 (, A, P ))        d1 (X, Y ) = X − Y 1 , L1 (, A, P )                 d1   L1             (Xn , n ∈ N∗ )   L1    X ∈ L1     lim E (|Xn − X|) = 0 n→+∞ L    Xn → X.   L1                  Rn      L1 .                                  L2(, A, P )   L2 (, A, P )          L2 (, A, P )              L2 (, A, P )         L2 (, A, P )      L1 (, A, P )   X  Y     L2 (, A, P )    XY   L1 (, A, P ). Download free books at BookBooN.com 54 Probability for Finance        Moments of a random variable                                  E(XY )2 ≤ E(X 2 )E(Y 2 )   Z = X + tY  t ∈ R      E Z 2 = E X 2 + 2tXY + t2 Y 2 ≥ 0     = E X 2 + 2tE (XY ) + t2 E Y 2           t.             ∆′        ∆′        ∆′ = E (XY )2 − E X 2 E Y 2             X  Y   L2 .            XY            L2 . www.job.oticon.dk Download free books at BookBooN.com 55 Probability for Finance  Moments of a random variable             L2 ×L2  R   ., .    (X, Y ) → X, Y  = E(XY )       L2 .         X2 = X, X = E(X 2 )      d2    d2 (X, Y ) = X − Y 2 .    ., .    X, X = E(X 2 ) > 0  X   P                    L1 ,       L2   L2           (Xn , n ∈ N∗ )   L2    X ∈ L2        lim E (Xn − X)2 = 0 n→+∞ L2                             L2           Rn ,           L2 .                   R2 ,    f : R2 → R    ∀x ∈ R2 , f (x) = a1 x1 + a2 x2   a1  a2     x′ = (x1 , x2 ).      (a1 , a2 )    f.    a′ = (a1 , a2 )        x ∈ R2  f(x)           H                      H          H   Download free books at BookBooN.com 56 Probability for Finance        Moments of a random variable    a  x.         f  R2  R      a ∈ R2 .                  L2 (, A, P ).    f       L2  R   Yf ∈ L2     X ∈ L2  f (X) = X, Yf  = E(XYf )   X         f (X)          X → f (X)              Card() = N           Yf   f (X) = X, Yf  = E(XYf ) = N  X(ω i )Yf (ω i )P (ω i )  i=1      X = ei = {ωi } ,      ω i . f (ei ) = ei , Yf  = P (ω i )Yf (ω i )  f (ei )                        ω i .          P (ω i )  Yf (ω i ).   Yf (ω i )               f(ei )              Yf (ω i )         Yf (ω i )                  Yf (ω i )           ω i       X,            X= N  xi ei i=1  X(ω i ) = xi .    f (X) = X, Yf  =  N  i=1 xi f (ei ) = N  xi Yf (ω i )P (ω i )  i=1      R ,        x  y    < x, y >= x y .  Download free books at BookBooN.com 57                C          C    z) Probability for Finance  Moments of a random variable                L2               x  R2     C ⊂ R2 .    C     z     C     x.    z      x  C.    y     y.          x − z  y − z    90◦  270◦ .                                < x − z, y − z > ≤ 0                C          C    z)  Please click the advert   A        ∀λ ∈ [0; 1] , ∀(x, y) ∈ A × A, λx + (1 − λ)y ∈ A.   R ,      x  y    < x, y > / x . y .    A        ∀λ ∈ [0; 1] , ∀(x, y) ∈ A × A, λx + (1 − λ)y ∈ A.   R ,      x  y    < x, y > / x . y . Download free books at BookBooN.com 58 Probability for Finance        Moments of a random variable       R2                   C       L2  X ∈ L2 .   Z ∈ C   X − Z, Y − Z  0   Y ∈ C Z      X  C.                                                                            Download free books at BookBooN.com 59 Probability for Finance  Moments of a random variable                         X  Y      L2 (, A, P )    X  Y,  Cov(X, Y )   σ XY )    cov(X, Y ) = E [(X − E(X)) (Y − E(Y ))]    X  Y            X(ω) Y (ω) ω1   ω2   ω3   ω4       X  Y          E(X) = E(Y ) = 2.           X(ω) − E(X) Y (ω) − E(Y ) ω1   ω2   ω3   ω4                 cov(X, Y ) = 1 (−1 × 1 + (−2) × (−1) + 1 × (−1) + 2 × 1) = 0.5 4            P             X  Y.           a, b, c, d            X, Y, Z, W   Cov(aX +bY, cZ +dW ) = ac×σ XZ +ad×σ XW +bc×σ Y Z +bd×σ Y W       Cov(aX, Y ) = aCov(X, Y ). Download free books at BookBooN.com 60 Probability for Finance        Moments of a random variable          V (X + Y ) = V (X) + V (Y ) + 2Cov (X, Y )  Cov (X, X) = V (X).    Cov (X, Y )         X  Y.                                            X  Y     L2 ;     X  Y    ρXY ,    ρXY = Cov(X, Y ) σ(X)σ(Y )  σ(X)  σ(Y )      X  Y. X Y ρXY     Cov( σ(X) , σ(Y ),       )                   X  Y       X, Y  ρXY = X2 Y 2  ρXY               X  Y.                                                                          √ 1 σ(X) = ((−1)2 + (−2)2 + (1)2 + (2)2 ) = 2.5 = 1.58 4  1 σ(Y ) = ((1)2 + (−1)2 + (−1)2 + (1)2 ) = 1 4 0.5 ρXY = = 0.316 1.58 Download free books at BookBooN.com 61 Probability for Finance  Moments of a random variable                      X  Y                     ρ          X  Z      L2  a, b, c, d    Cov(aX + b, cZ + d) = ac × Cov(X, Z) ρaX+b,cZ+d = sign(ac) × ρXZ            Y  W             σ(aX + b) = |a| σ(X)  σ(cY + d) = |c| σ(Y )              Turning a challenge into a learning curve. 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Visit accenture.com/bootcamp Download free books at BookBooN.com 62 Probability for Finance Moments of a random variable            X  Y  L2                                                                                                                 X1        X1                   X0 = 90       E (X1 ) = 1 [0 + 200] = 100 2                                  X0 = E (X1 ) = 90 1 + Riskpremium                X0                                                                 Download free books at BookBooN.com 63 Probability for Finance  Moments of a random variable                     Q   P     X0 = EQ (X1 )                  = {ω 1 , ω 2 } , X1 (ω 1 ) = 200  X1 (ω 2 ) = 0.  Q    Q(ω 1 ) = q1 = 0.45 Q(ω 2 ) = q2 = 1 − q1 = 0.55          EQ (X1 ) = 90 = X0 .   Q                    q1 × 200 + q2 × 0 = 90 q1 + q2 = 1                           Q                                                      X1 (ω 1 ) = 200 Y1 (ω 1 ) = 150 X1 (ω 2 ) = 100 X0 = 130 Y1 (ω 2 ) = 110 Y0 = 120         Q   X0 = EQ (X1 ).     130 = 200Q(ω1 ) + 100 (1 − Q (ω1 ))    Q(ω 1 ) = 0.3.      Q′   Y0 = EQ′ (Y1 ).      150Q′ (ω 1 ) + 110 (1 − Q′ (ω 1 )) = 120 Download free books at BookBooN.com 64 Probability for Finance Moments of a random variable           Q′ (ω 1 ) = 0.25. Q  Q′                                                                                         Q  Q′            (X0 , Y0 )              θ           200 150 1 0 θX + θY + θZ = 100 110 1 0  θZ                                                          θX = −2; θY = 5; θZ = −350            200 150 1 0 −2 +5 − 350 = 100 110 1 0         −2 × 130 + 5 × 120 − 350 = −10                              (X0 , Y0 )                 X1       Y1 ,   X0     Y0               −2X0 + 5Y0 = 350.                              Y1       Y0 = 122. Download free books at BookBooN.com 65 Probability for Finance  Moments of a random variable            Q”     122 = 150Q”(ω 1 ) + 110 (1 − Q”(ω 1 ))  Q”(ω 1 ) = 12 40 = 0.3.         Q”      Q                                Q                                    r     1          1+r ,      X1   1 X0 = EQ (X1 )  1+r it’s an interesting world Please click the advert Get under the skin of it. Graduate opportunities Cheltenham | £24,945 + benefits One of the UK’s intelligence services, GCHQ’s role is two-fold: to gather and analyse intelligence which helps shape Britain’s response to global events, and, to provide technical advice for the protection of Government communication and information systems. In doing so, our specialists – in IT, internet, engineering, languages, information assurance, mathematics and intelligence – get well beneath the surface of global affairs. If you thought the world was an interesting place, you really ought to explore our world of work. www.careersinbritishintelligence.co.uk TOP GOVERNMENT EMPLOYER Applicants must be British citizens. GCHQ values diversity and welcomes applicants from all sections of the community. We want our workforce to reflect the diversity of our work. Download free books at BookBooN.com 66 Probability for Finance Moments of a random variable                                                      Q                                         ω           ω                {ω}          ω 1              P (ω 1 ) > 0    A10           P (ω 1 ),           A10 = EQ (A11 ) ,    EQ (A11 ) = Q(ω 1 ).                     P        Q.                             A11 ,             Q         P  ∀B ∈ A, P (B) = 0 ⇒ Q(B) = 0    Q << P.     P  Q    ∀B ∈ A, P (B) = 0 ⇔ Q(B) = 0      Q << P  P << Q.                    Download free books at BookBooN.com 67 Probability for Finance  Moments of a random variable          Q << P         A  φ    ∀B ∈ A, Q(B) = φdP B       P (B) = 0 ⇒ Q(B) = 0       φ,              Q(B) = B φdP,     φ = dQ     dP   φ        Q dP    P.   P  Q   dQ  dQ   dP dQ dP = 1/ dP dQ    P  Q      (, A)  φ = dQ .      dP EQ (X1 ) = E (φX1 )          X1  EQ (X1 )     X1 .          P     φX1      E (φX1 ) = φ, X1           L2 (, A, P ) . φ                  Card() = N  A = P ()  P (ω) > 0   ω            φdP = φ(ω)P (ω) Q({ω}) = {ω} φ     φ(ω) = Q(ω) P (ω)        φ                        Download free books at BookBooN.com 68          Probability for Finance Moments of a random variable                                               n          (, A, P )     (Rn , BRn ) .    X = (X1 , ...., Xn )′   Xi                                       Brain power Please click the advert By 2020, wind could provide one-tenth of our planet’s electricity needs. 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Visit us at www.skf.com/knowledge Download free books at BookBooN.com 69 Probability for Finance  Moments of a random variable                X = (X1 , ...., Xn )′    FX  Rn  [0; 1]   FX (x) = P (∩ni=1 {Xi ≤ xi })  x ∈ Rn    (x1 , x2 .., xn )′ .    Xi         X     fX  Rn  R    x  x  xn FX (x) = ... fX (x)dx1 ...dxn −∞ −∞ −∞                                 E(X)        Xi  X          V (X1 ) ... Cov(X1 , Xj ) Cov(X1 , Xn )    X =   Cov(Xj , X1 )  V (Xj ) Cov(Xn , X1 ) V (Xn )       X     2  σ 1 ... σ 1j σ 1n    X =  2  σ j1  σj σ n1 σ 2n                           X    n    U, W   n   Rn .  E(U ′ X) = U ′ E(X) E (U ′ X, W ′ X) = U ′ E(XX ′ )W V (U ′ X) = U ′ X U CoV (U ′ X, W ′ X) = U ′ X W Download free books at BookBooN.com 70 Probability for Finance Moments of a random variable          U ′ X = n     ′ i=1 Ui Xi      V (U X)     X   (n, n) ′   XX   n × n   E(XX ′ )    n × n     E(Xi Xj )                                  n   X       U ∈ Rn          n       U,  R,     ′ R=UX= n  Ui Xi i=1               E(R) = U ′ E(X) V (R) = U ′ X U   E(X)                               U         n  Ui = 1 i=1     U ′  =1       Rn                              e. X                                        (n, n)  M        ∀x ∈ R , x = 0 ⇔ x′ Mx > 0. Download free books at BookBooN.com 71 Probability for Finance  Moments of a random variable              1 min U ′ X U 2    U ′ E(X) = e U ′  =1   12                             1 L (U, λ, ) = U ′ X U + λ (e − U ′ E(X)) +  (1 − U ′ ) 2          X =   E(X) =         ∂L = U − λ− =  ∂U ∂L = e − U ′ =  ∂λ ∂L = 1 − U ′ =  ∂         U = λ−1 +−1         e = λ′ −1 +′ −1  1 = λ′ −1 +′ −1         1  U= (eC − A)−1 +(b − eA)−1  D  A B C D = = = = ′ −1  ′ −1  −1  BC − A2 √           x → x′ −1 x     Rn      x, y = x′ −1 y.      D    Download free books at BookBooN.com 72 Probability for Finance Usual probability distributions in financial models                                                                                                                                                 χ2 ,  t                                      X       p  X        p  1 − p.  Download free books at BookBooN.com 73 Probability for Finance Usual probability distributions in financial models              B ∈ A  P (B) = p,     B       p.      B ∼ B(p)                    X   a  b (a > b)  1  p  1 − p,   Y = a−b (X − b)        p  1 − p. Y   B(p).                                             ln(u)  ln(d) u  up  d  down).        S0 ,     S1 ,     uS0  dS0 .     ln(S1 ) = ln(S0 ) + X  X       ln(u)  ln(d).                                                       B = {SPT ≥ K}  SPT               T  K                                   P (B).       X ∼ B(p),  E(X) = p  σ 2 (X) = p(1 − p)      X        p,   E(X)     E(X) = p × 1 + (1 − p) × 0 = p    X,   σ 2 (X)            σ 2 (X) = E(X 2 ) − E(X)2 = p − p2 = p(1 − p)               X = X  . Download free books at BookBooN.com 74 Probability for Finance Usual probability distributions in financial models            Y    y1  y2   p  (1 − p).    σ 2 (Y ) = p(1 − p)(y1 − y2 )2  1       X = y −y (Y − y2 )  B(p)   Y = (y1 − y2 )X + y2     E(Y ) = (y1 − y2 )E(X) + y2 = py1 + (1 − p)y2 σ 2 (Y ) = (y1 − y2 )2 σ 2 (X) = p(1 − p)(y1 − y2 )2                      Y          y1 = ln(u)  y2 = ln(d).          u 2  σ 2 (Y ) = p(1 − p) ln d Trust and responsibility Please click the advert NNE and Pharmaplan have joined forces to create NNE Pharmaplan, the world’s leading engineering and consultancy company focused entirely on the pharma and biotech industries. – You have to be proactive and open-minded as a newcomer and make it clear to your colleagues what you are able to cope. The pharmaceutical field is new to me. But busy as they are, most of my colleagues find the time to teach me, and they also trust me. Even though it was a bit hard at first, I can feel over time that I am beginning to be taken seriously and that my contribution is appreciated. Inés Aréizaga Esteva (Spain), 25 years old Education: Chemical Engineer NNE Pharmaplan is the world’s leading engineering and consultancy company focused entirely on the pharma and biotech industries. We employ more than 1500 people worldwide and offer global reach and local knowledge along with our all-encompassing list of services. nnepharmaplan.com Download free books at BookBooN.com 75 Probability for Finance Usual probability distributions in financial models                                                                         u  d                          X        n  p  X       n    Xi , i = 1, ..., n,     B(p).      n k P (X = k) = p (1 − p)n−k k   n!  nk = k!(n−k)!       k   n.   X   B(n, p).                                  S    St   t  ,  (t + 1)     St+1 = St × Xt+1  Xt+1   u  d   p  1 − p.   Xt    St       St = S0 ×      ln  St S0  = t  Xs s=1 t  ln(Xs ) s=1           s = 0  s = t           t      ln(u)  ln(d)   p  1 − p. Download free books at BookBooN.com 76 Probability for Finance Usual probability distributions in financial models         ln(St )  B(n, p)      n k P (ln(St ) = ln(S0 ) + k × u) = p (1 − p)t−k k    nk            k     t − k                          X ∼ B(n, p)  E(X) = np  σ 2 (X) = np(1 − p)     B(n, p)       n        B(p)).      X ∼ B(n, p) E(X) = np  σ 2 (X) = np(1 − p)       n                         t        St E ln = t (p ln (u) + (1 − p) ln(d)) S0     u 2 St 2 σ ln = tp(1 − p) ln S0 d                                                           StSt−St .        Download free books at BookBooN.com 77 Probability for Finance Usual probability distributions in financial models                                               X        λ  X         λk ∀k ∈ N, P (X = k)= exp(−λ) k!    X ∼ P(λ). Please click the advert       P(2).            Download free books at BookBooN.com 78 Probability for Finance Usual probability distributions in financial models         P(2)       X ∼ P(λ)  E(X) = λ  σ 2 (X) = λ            xk         ex = +∞ k=0 k! . +∞  +∞  +∞  λk λk E(X) = kP (X = k) = k exp(−λ) = exp(−λ) k k! k! k=0 k=0 k=1 +∞ +∞ k   λk−1 λ = λ exp(−λ) = λ exp(−λ) = λ exp(−λ) exp(λ) = λ (k − 1)! k! k=1 k=0             σ 2 (X) = E(X 2 ) − E(X)2 = exp(−λ) +∞  k=0 k2 λk − λ2 k!  Download free books at BookBooN.com 79 Probability for Finance Usual probability distributions in financial models             +∞  k=0 k k 2λ k! +∞  k 2λ +∞  λk−1 = k =λ k k! (k − 1)! k=1 k=1 +∞ +∞   λk−1 λk−1 = λ +λ (k − 1) (k − 1)! (k − 1)! k=1 k=1 2 = λ +∞ k  λ +∞ k  λ +λ k! k! k=0   = λ + λ exp(λ) k=0 2       σ 2 (X) = λ.    P(λ)                                      B(n, p)  n    p                                                         n        p                    np    np(1 − p)          np(1 − p) ≃ np  p                            λ = np.                       P(λ),                                 Download free books at BookBooN.com 80 Probability for Finance Usual probability distributions in financial models              X        [a; b] , a < b,    fX     1  x ∈ [a; b] b−a fX (x) = 0     X ∼ U([a; b]).   FX )  X        x−a  b−a  x ∈ [a; b] FX (x) = 0  x < a  1  x > b             [0; 1] .          [c; d]   [a; b] d−c = FX (d) − FX (c) PX ([c; d]) = PX (]c; d]) = b−a                [a; b]                              a  b.       X ∼ U([a; b])  E(X) = b+a 2  σ 2 (X) = (b−a) 12   X      [a; b] ,    X     +∞ 1 E(X) = xfX (x)dx = b−a −∞ 2 2 1 (b − a ) b+a = = 2 b−a 2  b a  2 b 1 x xdx = b−a 2 a Download free books at BookBooN.com 81 Probability for Finance Usual probability distributions in financial models                 [0; 1]          2 σ (X) =  +∞ −∞ 3 2 x fX (x)dx −  b+a 2 2  3 b  2 1 x b+a = − b−a 3 a 2 1 (b − a3 ) 1 2 − (a + 2ab + b2 ) 3 b−a 4 1 2 1 = (a + ab + b2 ) − (a2 + 2ab + b2 ) 3 4 (b − a)2 = 12 =                                         Download free books at BookBooN.com 82 Probability for Finance Usual probability distributions in financial models                                                       Please click the advert                                                                                                           Download free books at BookBooN.com 83 Probability for Finance Usual probability distributions in financial models           X       m  σ   X ∼ N (m, σ))    fX      2  1 1 x−m fX (x) = √ exp − 2 σ σ 2π fX                   x = m            2/3            [m − σ; m + σ]          [m − 2σ; m + 2σ] .          N (0, 1)           N (0, 1)                       χ2 ,             Download free books at BookBooN.com 84 Probability for Finance Usual probability distributions in financial models           X ∼ N (m, σ 2 ), E(X) = m  σ 2 (X) = σ 2              2   +∞ 1 1 x−m E(X) = √ dx x exp − 2 σ σ 2π −∞  y = x−m , σ       +∞ 1 1 2 E(X) = √ (σy + m) exp − y dy 2 2π −∞      +∞  +∞ m σ 1 2 1 2 = √ y exp − y dy + √ exp − y dy 2 2 2π −∞ 2π −∞   +∞ σ 1 = − √ exp − y 2 +m=m 2 2π −∞    E(X) = m.        exp − 12 y 2            y = x−m        σ    +∞ 1 1 2 2 2 σ (X) = √ (σy + m) exp − y dy − m2 2 2π −∞      2  +∞ σ 1 2 2mσ +∞ 1 2 2 = √ y exp − y dx + √ y exp − y dx 2 2 2π −∞ 2π −∞          0           2mσ) ;           +∞ σ2 1 2 √ y × y exp − y dx 2 2π −∞        +∞ +∞ σ2 1 2 1 2 = √ y exp − y − − exp − y dx 2 2 2π −∞ −∞   +∞     +∞ 1 1 1 1 = σ 2 √ y exp − y 2 +√ exp − y 2 dx 2 2 2π 2π −∞ −∞               1.    σ 2 (X) = σ 2 . Download free books at BookBooN.com 85 Probability for Finance Usual probability distributions in financial models                                                                           Please click the advert            0  t    r = ln SSt  St   t  (t > 0).                St = S0 er                  Download free books at BookBooN.com 86 Probability for Finance Usual probability distributions in financial models       X       m  σ 2  ln(X) ∼ N (m, σ 2 ).    X       2   √ 1 1 ln(x)−m exp − 2  x > 0 σ xσ 2π fX (x) =  0    X ∼ LN (m, σ 2 ).             m = 0  σ = 1                                                                                  Download free books at BookBooN.com 87 Probability for Finance Usual probability distributions in financial models                X ∼ LN (m, σ 2 ), E(X) = exp m + exp (2m + σ 2 ) (exp(σ 2 ) − 1)) σ 2   σ 2 (X) =         1 E(X) = √ σ 2π +∞ 0  1 exp − 2  ln(x) − m σ 2  dx   y = ln(x),      1 E(X) = √ σ 2π  +∞ −∞  1 exp(y) exp − 2  y−m σ 2  dy               +∞ 1 (y − (m + σ 2 ))2 σ2 1 √ dy exp − exp m + E(X) = 2 σ2 2 σ 2π −∞   σ2 = exp m + 2                      (m + σ 2 )   σ 2 .        V (X)     E(X 2 ) = exp (2(m + σ 2 ))     V (X) = exp (2m + σ 2 ) (exp(σ 2 ) − 1))    Y ∼ N (0, 1)  X         σ2 X = exp m− + σY 2  m  σ    σ > 0. X   1           m  σ              X    K   1      max(X − K; 0)              Download free books at BookBooN.com 88 Probability for Finance Usual probability distributions in financial models           E (X − K)+  (x)+ = x  x > 0  (x)+ = 0   fX    X,  :   E (X − K)+ = +∞ +∞ fX (x) max(x − K; 0)dx = fX (x)(x − K)dx 0 K +∞ +∞ xfX (x)dx − K fX (x)dx =  xfX (x)dx − KP (X ≥ K)  = = K +∞ K K +∞ xfX (x)dx − KP (ln(X) ≥ ln(K))  K      X     σ2 P (ln(X) ≥ ln(K)) = P m− + σY ≥ ln(K) 2    σ ln(K) − m − 2  = P Y ≥ σ      N (x)              ln(K) − m − σ2   P (X ≥ K) = 1 − N  σ    σ − ln(K) + m − 2  = N  σ                                    +∞ − ln(K) + m + σ2  xfX (x)dx = em N   σ K Download free books at BookBooN.com 89 Probability for Finance Usual probability distributions in financial models                                                   m = 3%  σ = 20%.                                      [900; 1000] .            [1000; 1100]?        Sharp Minds - Bright Ideas! 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Dedicated Analytical Solutions FOSS Slangerupgade 69 3400 Hillerød Tel. +45 70103370 www.foss.dk Download free books at BookBooN.com 90 Probability for Finance Usual probability distributions in financial models                                                                                               χ2   t       χ2       Y   χ2   n      Y     Y = n  Xi2  i=1   Xi        ∀i, Xi ∼ N (0, 1).                σ 2      (X1 , ....Xn )         (m, σ 20 ),   Y   Y = 2 n   Xi − m  σ0 j=1   χ2   n        n σ 20 Y 1 = (Xi − m)2 n n j=1              n  1 Xi ,     m      X = n i=1 Y ∗ ,    m  X      χ2  n   2 1  n − 1       n−1 Xi − X     j=1          Download free books at BookBooN.com 91 Probability for Finance Usual probability distributions in financial models                       χ2       χ2                          t       Y   −t   n     Y    Z Y =  X n  Z       X   χ2   n                                                                  Download free books at BookBooN.com 92 Probability for Finance      Usual probability distributions in financial models                                             n)      n/(n − 2)       3(n−2)/(n−4).      n > 4.     n = 6,                                       F                  Y         Y = X n X n   X1 (X2 )   χ2   n1 (n2 )          F (n1 , n2 )      F (n2 , n1 )          F                           n1  n2                                                 F               Download free books at BookBooN.com 93 Probability for Finance Conditional expectations and Limit theorems                                                                        t           t + 1,        t.                                                                                          (, A, P )   = {ω 1 , ω 2 , ω 3 , ω 4 } , A = P()  P (ω i ) = 0.25   i = 1, .., 4.    X  Y                                       Download free books at BookBooN.com 94 Probability for Finance Conditional expectations and Limit theorems         ω1 ω2 ω3 ω4 X 1 2 3 4 Y 1 1 2 2     X  Y    1 (1 + 2 + 3 + 4) = 2.5 4 1 E(Y ) = (1 + 1 + 2 + 2) = 1.5 4  E(X) =        Y       X  .  Y (ω) = 1,    ω    ω1  ω 2 .      {ω 1 , ω 2 }         {Y = 1}            {Y = 1} .           (P (ω i |{Y = 1}), i = 1, ..., 4) =  1 1 ; ; 0; 0 2 2             X        E (X |{Y = 1})  E (X |{Y = 1}) =  X(ω i )P (ω i |{Y = 1}) = 1 (1 + 2) = 1.5 2      E(X)        {Y = 1}                               Y          Y.     Y,                         X    1.5.                Download free books at BookBooN.com 95 Probability for Finance Conditional expectations and Limit theorems                 X  Y     (xi , i = 1, ..., n)  (yj , j = 1, ..., p) .         X  {Y = yi }      PX|Y (. |yi )    PX|Y (x |yi ) = P (X = x |Y = yi ) = P ({X = x} ∩ {Y = yi }) P ({Y = yi }) Please click the advert        P ({Y = yi }) = 0              Y. PX|Y (. |yi )          X. Download free books at BookBooN.com 96 Probability for Finance Conditional expectations and Limit theorems           fXY          , fX  fY     X  Y.     y  fY (y) > 0,     X  {Y = y}    fX|Y (. |y )    fX|Y (x |y ) = fXY (x, y) fY (y)    X     fX  B      P (B) = 0    X   B    fX (x |B ) =  fX (x) P (B)  x ∈ X(B) 0                                                  A.           X    x1 , ..., xN ,    B  A,    E(X |B )     N  E(X |B ) = xi P ({X = xi } |B ) i=1         X   fX    B  A,    E(X |B )    1 E(X |B ) = P (B)  X(B)  xfX (x)dx =  +∞ −∞ xfX (x |B )dx      Download free books at BookBooN.com 97 Probability for Finance Conditional expectations and Limit theorems                {Y = 2}   E(X |{Y = 2}) = N  i=1  xi P ({ω i } |{Y = 2}) = 3 × P (ω 3 |{Y = 2}) + 4 × P (ω 4 |{Y = 2})  1 (3 + 4) = 3.5  = 2      P (ω 1 |{Y = 2}) = P (ω 2 |{Y = 2}) = 0.                               X       Y.      Y,                                               X,   x1 , ..., xN ,      Y,    y1 , ..., yM ,   E(X |Y ),       ∀ω ∈ {Y = yj } , E(X |Y )(ω) = N  i=1 xi P ({X = xi } |{Y = yj })          X  Y      ω1 ω2 ω3 ω4 E(X |Y ) 1.5 1.5 3.5 3.5      X    Y Download free books at BookBooN.com 98 Probability for Finance Conditional expectations and Limit theorems             X  Y     fX  fY       fX|Y (x |y )  .      X  {Y = y}    +∞ xfX|Y (x |y )dx E (X |Y = y ) = −∞       X  Y        +∞ ∀ω ∈ {Y = y} , E(X |Y )(ω) = xfX|Y (x |y )dx −∞       Y     {Y = yj }     .        E(X |Y )               Y,          Y         E(X |Y ).       E(X |Y )  BY .          Please click the advert Student Discounts + Student Events + Money Saving Advice = Happy Days! 2009 Download free books at BookBooN.com 99 Probability for Finance Conditional expectations and Limit theorems                                                                                        (  X ∈ L1 (, A, P )),    B  A,   B   Z,  ∀B ∈ B, E (ZB ) = E (XB )       •  Z         Z  Z ′                             E(X |B ). •        X      E(X |B )           B.                  •       X  B, E(X |B ) = X.    Card() = , P (ω i ) = pi   ω i   B   B = {∅, {ω1 , ω 2 } , {ω 3 , ω 4 } , }  B1 = {ω 1 , ω 2 } B2 = {ω 3 , ω 4 }   X    X = (x1 ; x2 ; x3 ; x4 ) .     p1 x1 + p2 x2 = p1 z1 + p2 z2 p3 x3 + p4 x4 = p3 z3 + p4 z4     Card  , X                Download free books at BookBooN.com 100 Probability for Finance Conditional expectations and Limit theorems            Z   (z1 ; z2 ; z3 ; z4 ) .      B1      B2 .  Z  B         B1   B2 .    z1 = z2 z3 = z4    1 [p1 x1 + p2 x2 ] = E (X |B1 ) p1 + p2 1 = z4 = [p3 x3 + p4 x4 ] = E (X |B2 ) p3 + p4 z1 = z2 = z3         B1 (B2 )            X    B1 (B2 ).       X   B,  E (X |B )     X.     L2 , A, P )                           L2 (, A, P ) .                                R2 ,       d(x, y) = (x1 − y1 )2 + (x2 − y2 )2  x′ = (x1 , x2 )  y ′ = (y1 , y2 ) .    x ∈ R2 ,         z = (z1 , z1 )             x.     minz (x1 − z1 )2 + (x2 − z1 )2          z1 = z2 . Download free books at BookBooN.com 101 Probability for Finance     L2 (, A, P ) Conditional expectations and Limit theorems      z1 = x +x .    z     2  2   x ∈ R             z − x    z. < z − x, z >= (z1 − x1 )z1 + (z1 − x2 )z1 x2 − x1 x1 − x2 = z1 + z1 = 0 2 2      R2       ∗ d (x, y) = p(x1 − y1 )2 + q (x2 − y2 )2  p + q = 1, p > 0, q > 0.                   z1 = px1 + qx2 z1                 x        L2                   X     L2 (, A, P ) ,    E(X |B )  B        B   L2 (, B, P ) .    L2 (, A, P )     R4  L2 (, B, P )  R2                  E(X |B )      X  L2 (, B, P ) .    E (X |B )         minZ∈L ,B,P ) E (X − Z)2 = minZ∈L ,B,P ) d(X, Z)2 = E (X − E (X |B ))2                 E (X |B )  B      z1 = z2 z3 = z4                         P   PB        B  L , B, P ).        P       B.  Download free books at BookBooN.com 102 Probability for Finance Conditional expectations and Limit theorems           E (X − Z)2 = p1 (x1 −z1 )2 +p2 (x2 −z1 )2 +p3 (x3 −z3 )2 +p4 (x4 −z3 )2        z1  z3          ∂E (X − Z)2 = −2 [p1 (x1 − z1 ) + p2 (x2 − z1 )] = 0  ∂z1   ∂E (X − Z)2 = −2 [p3 (x3 − z3 ) + p4 (x4 − z3 )] = 0  ∂z3      1 (p1 x1 + p2 x2 ) = E (X |B ) (ω 1 ) = E (X |B ) (ω2 ) p1 + p2 1 = z4 = (p3 x3 + p4 x4 ) = E (X |B ) (ω 3 ) = E (X |B ) (ω4 ) p3 + p4 z1 = z2 = z3                                                Please click the advert what‘s missing in this equation? 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Moller - Maersk. www.maersk.com/mitas Download free books at BookBooN.com 103 Probability for Finance Conditional expectations and Limit theorems               (X, Y )      L2 (, A, P )  B, B ′    A  B ⊂ B′    X    c ∈ R, E (X |B ) = c  ∀(a, b) ∈ R2 , E (aX + bY |B ) = aE (X |B ) + bE (Y |B )   X ≤ Y, E (X |B ) ≤ E (Y |B )  E (E (X |B′ ) |B ) = E (X |B )   X  B E (XY |B ) = X E (Y |B )   X    B, E (X |B ) = E(X)                            c     c .               B       c      L2 (, B, P ) .   L2 (, B, P )      L2 (, A, P ) ,                                         E (X |B′ )      X  L2 (, B′ , P ) . E (E (X |B′ ) |B )     L2 (, B, P )  E (X |B ′ )         L2 (, B ′ , P )    L (, B, P )            L2 (, B, P ) .                   B = {∅, } E (X |B ) = E(X)   E (E (X |B ′ )) = E (X)  B′       E (X − E(X) |B ) = 0  E(X)    2       X − E(X)     Y  L2 (, B, P )  E((X − E(X)) Y ) = E (X − E(X)) E(Y ) = 0  Download free books at BookBooN.com 104 Probability for Finance Conditional expectations and Limit theorems                       X −E(X)  Y.                                                                                                 X = (X1 , ...., Xn )      n    ai Xi     i=1  m′ = (E(X1 ), ..., E(Xn ))      X     X.   fX  X     n   1 1 1 n ′ −1  ∀x ∈ R , f(x) = √ exp − (x − m) X (x − m) 2 2π Det(X )   Det(X )                X = (X1 , ...., Xn )        m  X ;  p < n  Y1 = (X1 , ...., Xp )  Y2 = (Xp+1 , ...., Xn ) .  X       Σ11 Σ12 X = Σ21 Σ22  Σii      Yi  Σij          Yi  Yj  i, j = 1, 2, i = j.     Y1   Y2 = y2 ∈ Rn−p         E (Y1 |Y2 = y2 ) = E(Y1 ) + Σ12 Σ−1 22 (y2 − E(Y2 ))  Y |Y =y = Σ11 − Σ12 Σ−1 22 Σ21                        Download free books at BookBooN.com 105 Probability for Finance Conditional expectations and Limit theorems        p = 1  n = 2      p = 1  n = 2  σ 12 (y2 − m2 ) σ 22 σ2 = σ 21 − 12 σ 22 E (X1 |X2 = x2 ) = m1 + X |X =x  ρ12           X |X =x = σ 21 (1 − ρ212 )               x′ = (x1 , x2 ))   −1 √1 exp − 12 (x − m)′ X (x − m) (2π) |Det(X )| fX (x1 , x2 )   fX |X (x1 |x2 ) = = 2  fX (x2 ) 1 1 x −m   √ exp − 2 σ σ  2π   −1 exp − 12 (x − m)′ X (x − m) σ2   = √  2 2 2  2π σ 1 σ 2 − σ 212 1 x −m exp − 2 σ    2  σ2 − m 1 x 2 2 −1 = √  2 2 exp − (x − m)′ X (x − m) − 2 2 σ 2π σ 1 σ 2 − σ 12 2       −1 X 1 = 2 2 σ 1 σ 2 − σ 212  −σ 12 σ 22 −σ 12 σ 21  −1  A = (x − m)′ X (x − m),   A = σ 22 x21 − 2σ 22 x1 m1 − 2x1 σ 12 x2 + 2x1 σ 12 m2 + σ 21 σ 22 − σ 212 σ 22 m21 + 2m1 σ 12 x2 − 2m1 σ 12 m2 + σ 21 x22 − 2σ 21 x2 m2 + σ 21 m22 σ 21 σ 22 − σ 212           2 fX (x1 , x2 ) σ2 1 (−σ 22 x1 + σ 22 m1 + σ 12 x2 − σ 12 m2 ) =√  2 2 exp − fX (x2 ) 2 σ 22 (σ 21 σ 22 − σ 212 ) 2π (σ 1 σ 2 − σ 212 ) Download free books at BookBooN.com 106 Probability for Finance   expectations and Limit theorems   Conditional              σ 12 (x2 − m2 ) σ 22 σ 212 2 = σ1 − 2 σ2 E (X1 |X2 = x2 ) = m1 + X |X =x    g    2  σ  x − m − (x − m ) 1 1 2 2 1 1 σ     g(x1 ) =  exp −   √ σ  2 2 σ σ 1 − σ σ 2 −  2π 1 σ   1 (−σ 22 x1 + σ 22 m1 + σ 12 x2 − σ 12 m2 ) σ2 exp − = √  2 2 2 σ 22 (σ 21 σ 22 − σ 212 ) 2π (σ 1 σ 2 − σ 212 ) 2         g(x1 ) = fX |X (x1 |x2 ). Please click the advert     X |X =x = σ 21 (1 − ρ212 )  X2 = x2              X1                    X1       ρ12                  Download free books at BookBooN.com 107 Probability for Finance Conditional expectations and Limit theorems                                                                                                                                                                                                   β                                                                                                   L1  L2 .                                                            (Xn , n ∈ N)        X         (, A, P ) ; P  (Xn , n ∈ N)   X     Xn → X    ε > 0 lim P (|Xn − X| > ε) = 0 n→+∞ Download free books at BookBooN.com 108 Probability for Finance Conditional expectations and Limit theorems        a.s  (Xn , n ∈ N)   X     Xn → X      0 ⊂   P (0 ) = 1   ∀ω ∈ 0 , lim Xn (ω) = X(ω) n→+∞   PXn PX       Xn X  (Xn , n ∈ N) L   X     Xn → X)       f   f(x).dPXn (x) = f(x).dPX (x) lim n→+∞ R R                                                                                                                                         X           E(X) =    A > 0      P (X ≥ A) ≤ 1 A        A > 1                X.                                      X                              Download free books at BookBooN.com 109 Probability for Finance Conditional expectations and Limit theorems                 X ∈ L2 (, A, P )   E(X) = m  V (X) = σ 2 ;   B > 0   σ2 P (|X − | ≥ B) ≤ 2 B                                                                 P (|X − | ≥ Aσ) ≤ 1 A2  A      X       1 2A2  1           A = 2×0.01 = 7.0711.          A = 2.32,             P (X −   −Aσ) ≤ www.job.oticon.dk Download free books at BookBooN.com 110 Probability for Finance Conditional expectations and Limit theorems                (Xn , n ∈ N)                  σ),     Zn = n1 ni=1 Xi  (Zn , n ∈ N)             ε > 0 P (|Zn − | ≥ ε) ≤ σ2 nε2                                     (Xn , n ∈ N)         Xn   X   X     L2 )         limn→+∞ E(Xn ) = E(X)  limn→+∞ V (Xn − X) = 0         (Xn ,  n ∈ N)           Zn = n1 ni=1 Xi  (Zn , n ∈ N)          E(|Xn |) = +∞,   Zn                                                                                        K  ri = E(ri ) + β ik Fk + εi  k=1 Download free books at BookBooN.com 111 Probability for Finance Conditional expectations and Limit theorems             ri       i, F1 , ..., FK       β ik      i      k   εi           i.       Cov(Fk , Fj ) = 0  j = k)      Cov(Fk , εi ) = 0).      Cov(εi , εm ) = 0  i = m).                                      N            N N N N K 1  1  1  1  ri = E(ri ) + β ik Fk + εi  N i=1 N i=1 N i=1 k=1 N i=1   N K N N  1  1  1  = E(ri ) + β Fk + εi  N i=1 N i=1 ik N i=1 k=1             N  1           N εi                i=1                                                                   (Xn , n ∈ N)           p;   Tn   n Xi − np Tn = i=1 np(1 − p)                               p  Download free books at BookBooN.com 112 Probability for Finance Conditional expectations and Limit theorems          n      u  d                         up                   n , n ≥ 1         Y = Y1n , ..., Yk(n)   k(n) n 2      n,  sn = V . Y   i=1 Yi   n ,n ≥ 1      ε > 0,   U = U1n , ..., Uk(n)   Uin = Yin  |Yin | ≤ εsn = 0    V lim n→+∞  k(n) i=1 s2n Yin  =1                            n    Y = Y1n , ..., Yk(n) , n ≥ 1          n n n n         Y1 − E (Y1 ) , ...., Yk(n) − E Yk(n) , n ≥ 1 k(n) n        n ≥ 1,  Zn = i=1 Yi   E (Zn ) →   V (Zn ) → σ 2 = 0   Zn        Z                    u  d       u  d                                     Download free books at BookBooN.com 113 Probability for Finance Bibliography                                                                                                              ◦                                                                                     Download free books at BookBooN.com 114 Probability for Finance Bibliography                                                                                                                                                                                                                                                                      Download free books at BookBooN.com 115

Probability for finance

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Probability for finance

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