Financial Engineering
Zvi Wiener
mswiener@mscc.huji.ac.il
02-588-3049
FE-W
http://pluto.mscc.huji.ac.il/~mswiener/zvi.html
EMBAF
Elementary Stochastic Calculus
Following
Paul Wilmott, Introduces Quantitative Finance
Chapter 7
FE-W
http://pluto.mscc.huji.ac.il/~mswiener/zvi.html
EMBAF
Coin Tossing
Ri = -1 or 1 with probability 50%
E[Ri] = 0
E[Ri2] = 1
E[Ri Rj] = 0
Define
j
Sj 
R
i
i 1
Zvi Wiener
FE-Wilmott-IntroQF Ch7
slide 3
Coin Tossing
 
E Sj 0
   E R
E S
2
j
2
1

 R1 R 2    j
E S 6 R1 ,  , R 5   S 5
Zvi Wiener
FE-Wilmott-IntroQF Ch7
slide 4
Markov Property
No memory except of the current state.
Transition matrix defines the whole dynamic.
Zvi Wiener
FE-Wilmott-IntroQF Ch7
slide 5
The Martingale Property


E Si S j , j  i  S j
Some technical conditions are required as well.
Zvi Wiener
FE-Wilmott-IntroQF Ch7
slide 6
Quadratic Variation
i
 S
 S j 1 
2
j
j 1
For example of a fair coin toss it is = i
Zvi Wiener
FE-Wilmott-IntroQF Ch7
slide 7
Brownian Motion
E S ( t )   0


E S (t )   t
Zvi Wiener
2
FE-Wilmott-IntroQF Ch7
slide 8
Brownian Motion
Finiteness – does not diverge
Continuity
Markov
Martingale
Quadratic variation is t
Normality: X(ti) – X(ti-1) ~ N(0, ti-ti-1)
Zvi Wiener
FE-Wilmott-IntroQF Ch7
slide 9
Stochastic Integration
t
W (t ) 

n
f ( ) dX ( )  lim
n 

0
tj  j
Zvi Wiener
f ( t j 1 )  X ( t j )  X ( t j 1 ) 
j 1
t
n
FE-Wilmott-IntroQF Ch7
slide 10
Stochastic Differential Equations
t
W (t ) 

f ( ) dX ( )
0
dw  f ( t ) dX
dX has 0 mean and standard deviation
Zvi Wiener
FE-Wilmott-IntroQF Ch7
t
slide 11
Stochastic Differential Equations
dw  g ( t ) dt  f ( t ) dX
t
W (t ) 

0
Zvi Wiener
t
g ( ) d  

f ( ) dX ( )
0
FE-Wilmott-IntroQF Ch7
slide 12
Simulating Markov Process
The Wiener process
 X  N (0,  t )
The Generalized Wiener process
S  at  bX
The Ito process
S  a (S , t )t  b(S , t )X
Zvi Wiener
FE-Wilmott-IntroQF Ch7
slide 13
value
time
Zvi Wiener
FE-Wilmott-IntroQF Ch7
slide 14
Ito’s Lemma
dt
dX
dt
0
0
dX
0
dt
F ( X ,t)
dF 
dF
dX
Zvi Wiener
FE-Wilmott-IntroQF Ch7
2
dX 
1 d F
2 dX
2
dt
slide 15
Arithmetic Brownian Motion
dS    dt    dX
S (t )  S ( 0 )    t     X (t )  X ( 0 ) 
At time 0 we know that S(t) is distributed
normally with mean S(0)+t and variance 2t.
Zvi Wiener
FE-Wilmott-IntroQF Ch7
slide 16
Arithmetic BM
dS =  dt +  dX
S


time
Zvi Wiener
FE-Wilmott-IntroQF Ch7
slide 17
The Geometric Brownian Motion
S  St   SX
Used for stock prices, exchange rates.
 is the expected price appreciation:
 = total - q.
S follows a lognormal distribution.
Zvi Wiener
FE-Wilmott-IntroQF Ch7
slide 18
The Geometric Brownian Motion
dS   S  dt   S  dX
F  Log  S 

 
 dt    dX
dF    

2 

2
Zvi Wiener
FE-Wilmott-IntroQF Ch7
slide 19
The Geometric Brownian Motion
2


 
 t    X ( t )  X ( 0 ) 
S ( t )  S ( 0 )  Exp    

2



V (S , t)
dV 
Zvi Wiener
V
t
dt 
V
S
dS 
1
FE-Wilmott-IntroQF Ch7
2
 V
2
 S
2
2
S
2
dt
slide 20
Geometric BM
dS = Sdt + SdX
S
time
Zvi Wiener
FE-Wilmott-IntroQF Ch7
slide 21
The Geometric Brownian Motion
2

 
  
ln  S T   ln  S t     

2


S

~ N t, t
S

2
S t 1  S t  S t   t  
Zvi Wiener

N ( 0 ,1 )
FE-Wilmott-IntroQF Ch7
  N ( 0 ,1 )
t

slide 22
Mean-Reverting Processes
dS     S   dt    dX
dS     S   dt  
Zvi Wiener
FE-Wilmott-IntroQF Ch7
S  dX
slide 23
Mean-Reverting Processes
dS     S   dt  
F 
 4  
dF  
 8F
Zvi Wiener
2

S  dX
S
F 

  dt 

dX

2 
2
FE-Wilmott-IntroQF Ch7
slide 24
Simulating Yields
GBM processes are widely used for stock prices
and currencies (not interest rates). A typical model
of interest rates dynamics:

 rt  k ( b  rt )  t   rt  z t
Speed of mean reversion Long term mean
Zvi Wiener
FE-Wilmott-IntroQF Ch7
slide 25
Simulating Yields

 rt  k ( b  rt )  t   rt  z t
 = 0 - Vasicek model, changes are normally distr.
 = 1 - lognormal model, RiskMetrics.
 = 0.5 - Cox, Ingersoll, Ross model (CIR).
Zvi Wiener
FE-Wilmott-IntroQF Ch7
slide 26
Mean Reverting Process

dS = (-S)dt + S dX
S

time
Zvi Wiener
FE-Wilmott-IntroQF Ch7
slide 27
Other models
 rt   ( t )  t    z t
Ho-Lee term-structure model
HJM (Heath, Jarrow, Morton) is based on forward
rates - no-arbitrage type.
Hull-White model:
 rt   ( t )  ar t  t    z t
Zvi Wiener
FE-Wilmott-IntroQF Ch7
slide 28
Home Assignment
Read chapter 7 in Wilmott.
Follow Excel files coming with the book.
Zvi Wiener
FE-Wilmott-IntroQF Ch7
slide 29