- No category
BIEM/BIEF Math 2 Finance Notes 2025-2026
advertisement
BIEM/BIEF 2025-2026 MATH 2 FINANCE These notes contain cover everything for fnance except for the proofs, which I will upload separately. Many of the exercises solved are from past exams. The defnitions and theorems are highlighted on the side in blue: light blue indicates that they're more important. There are additional notes and exercises on my Dropbox and videos on my Youtube channel and Instagram. If you'd like to be updated when I post more notes and videos or have any questions regarding this course or micro, you can join my Whatsapp groups. BIEM: MATH MICRO BIEF: MATH MICRO ABOUT ME My name is Alice Sicconi and I've been tutoring Bocconi students in math, micro, statistics, and the GMAT since 2011. If you're interested in private lessons you can check out my web site or send me a message via whatsapp AT +39 345 1879591. FAQ ABOUT ME My name is Alice and I’m Italian-American. I enrolled in the BIEMF program at Bocconi in 2009 and began tutoring math, micro and statistics part-time in 2011. However, by the end of the academic year it had already become a full-time job! Since then, I've tutored approximately 500 students. HOW ARE YOUR PRIVATE LESSONS STRUCTURED? My lessons last 75 minutes, and I highly recommend doing a combination of theory and exercises. Bocconi exams can often be tricky, so I will test you continually to make sure that you don’t “fall into traps” and understand not just the theory, but also how to apply it properly through different examples. At the end of each lesson, I'll tell you which exercises you can solve on your own so that you can apply what you just learned and make sure that you're confident with the material. Then if you have any questions, we can cover them during the next lesson. I currently live outside Milan so the lessons will be via Zoom; I'll share my screen and write on my tablet, and send you the PDF of the notes at the end of each lesson. You'll also be able to record the lesson if you'd like so that you can watch it again later. If you'd like to get an idea of how I teach, you can check out my Youtube channel and Instagram. For math, there are videos on several different topics as well as many of the proofs that are in the syllabus. HOW MANY LESSONS WILL I NEED? My average student needs 15 lessons for each math partial in order to cover the entire syllabus; if you only need help with specific topics, you'll probably need less. Many of my students also choose to take extra lessons to solve additional exercises or have review sessions before the exam. HOW MUCH DO YOU CHARGE? The price per person (for a 75-minute lesson) is: €110 for 1 person €60 for 2 people Accumulation A P 100 Accumulated I principal Capital Accumulated Amount accumulated AI amount A Amount Interest usually interest isn't easy principal accumigate P Éo D FCT in practice accumulation ft 1 principal lifetime the value invested use this FaÉEEt of 1 euro that you after time has passed with interest value of 1 euro if you FCO 1 the invest for no time is still 1 be increasing must FG the higher the longer you invest the value of 1 euro 1 special cases Epee Interest Before inserting annual semi euro rate i and is 5 t t.IE that they 12 months 7 measure the time in semesters 26 for 1 i iE inEEEra match check 150 fed t 11 t rate if tal simple the same interest more convenient if tot compound is more convenient investment not a loan an assuming it's for Assuming is in a I EXAMPLE 1 the 0 amounts year to year COMPOUND You get interest on 100 Increases EEE.in TnitesfEte months accumulated compute 1 24 reinterest at a ftp i A cetts FEAR I É anideas Increases CONVEX amounts tumuqted rate FASTER directly years EEEYE.ir Finance Eis 120 100 the interest COMPOUND I 20 Definition Formula for the annual The rate interest 1 invested for 1 euro flott fall 1 00 21 Rate Interest interest on Annual the is year euro with interest after 1 year 11 FEEEnterest ÉITEE 511 0051271 ma Igemi to iz f 1121 fol iz feb 1 1 year annual rate year 1 semester EXAMPLE You invest The an 100 4 for euros rece.tt interestcoIstont f e'have 4 12 113 a accumulated Find the I Yest 100 20 02 13 annual I 02 1 i e0 months compound years amount 100.67 commemorate 0.0202 OR test fifty 1 cumulation 20.02 I factors equal interest Capitalization Axioms I IIA 9579s P P2 20 invested ACP t ACP Pa t is result the and P on the separately samefthprincipals iEEEino am keeping P as constant If tyf.EE Eor no time so the accumulated use sometimes they C capital principal M If A ACP P f H E Flt is If amount the Accumulation of satisfies the where increasing f t there not necessarily is the no interest principal notation amount Accumulated Properties are together or II taclnita.it E strictly ACPa t is differentiable Factors capitalization then f t 20 axiom Eeytegrazle M depends C Ca 20 MCC Ca t c both on and MCC Ca t C C MCC MCC t i Jassios falsies Ca MCC2 t MCC the monotonicity property 20 rates sis CCI Falslos CCI Facs Os sing If 7 MCC 07 0 C MCC 0 Italsios Ito C t then 0 fats ds t cta MCC t us Jacsios Using alt M Ca t ta 20 Ifacs as ti if t 2 of definite i t.gl EEastet0120 f h i tmaxcot tarzan Is ACH Is 8.12 197s increasing on not s I strictly CORNER Not differentiab figantaopposite P f t present value P A I Interest Final nominal value Discount OR A ect A PIE ÉÉoII ftp.ectl 1 Discount coeters the present value of 1 euro that you will receive at time t interest 610,98 Tout is ect Simple Rational Discount Compound ect e eat eat eat flatly easier It's accumulation by flt to in decreasing think Discount hi at Ks Decreases more quickly of discounting reverse rather than by test ect as é EXAMPLE You will 100 euros in receive annual simple interest rate The a Find the present value the time measged usual they period and 18 months É 1.5 years interest rate the jacket Factor OR D too eat_P it 1101.1.5 fa 86.96 19 b Find what the Internal of Return Compound yield rate would accumulated amount factor gee Equate 1.5 ane Rate command interest same 86.96 are match differently 1 P match if 0 If is i 10 check if I P 18 months amount be equivaler orysamef.fi ation 100 86.96 itid's 1.15 Itic 1097653 ic 0.097653 s 1.097653 ltic ic 0.097653 iactor 1.5 c is which investment I annual simple 10 rate interest 9.9 rate compound interest Ñen of types them for convenient more so rates we can't 18 months compare Easter Compare factors accumulation the IEEE tenens EEEEJoaasei.se BETER OR From is ic the part b the simple 9.7653 compound yield OR Find is Eat yield for II His.is 10.1411 compare it to is 10 it's HIGHER for I Floating Rates he ftp.t.ttatts Interest fct.fr Iti it t Citi Act tz a 11 12 15 simple A Itis for 1 to the f 1 2 IFFEREN 1 Find 1 107 lti Is from kt Interest compound fct t iz years at at 0 1563 17 1.4 3 year If 1 2 10 iz 15 amount accumulated 1 0.1 1 i then A P 1 4 then P Invest for the 1st year at i 10 the amount and accumulated withdraw reinvest 2 for more at iz 15 years a p 11 0.111 b compound 1 0.11 fll 2 A Fact 2 554 1 0.15 p 1 1 1 152 172.5 is 3 it 4 is 3.5 15g It II t.int f 1 1 1 1 0.025 1 0.03 1 1 13 The I equivalent is 4 0.13 is 4 is 013 1 13 0.04 1 1.13 with compound interest 155061 194 1 194 ie 1.13 0.035.1 FV Empteth interests 1550 1 13 EiIti E E IIiI Answer annual A SIMPLE rate is inffans is fmuaimofiication.EE EiEijod the number NOT necessarily of periods per year the number of months Discount Interest simple Discount Interest Compound TEEI.LI i 10 Find Ex sitins i compound the period rate 1 3 month for 0.024114 Quarterly trimester Rate Nominal Annual An under approximation of convertible times rate mETÉis use I the Effie Find the nominal 4 0.024114 rate ONLY for compound TAN annual the year per annual resetting nominal quarters trimesters accumulation in it immediately compound 10 Find im Convert factors Ex m rate period in annual rate 0.096456 periods teroos per year 4 it 14 Rate compan discount to im jy ju underapproximation of 1 10 EXAMPLE You will The nominal 10 is year 100 receive Convert annual rate 5 compound rate semi annual Find the present value 100 4 D 6 4 Find the compound annual 115m't for 1 0.0512 1 a year which is 01025 10.2 the annua 878 rate iiiiiiIiii 2 periods isjm Find semesters 82.27 E a years t is in semesters t 1 i semesters because it's just an approximation of the annual compound rate it immediately Path b per 2 iz iz a in 2 years twice convertible m 2 jfzm.j.im 10 euros as expected SETI affiliation factors equal semiannual Decomposability right left y xf right left xf ol xl x Decomposability consider ACH is an rangle langle right left yf 1 y accumulation decomposable factor if forall right cdot left 0 y xf y x Fct OR right cdot left sft s Multiplication s that i is law The variable only is COMPOUND interest decomposable st Interest simple show using is interest definition the whether decomposable simple right left 1tfi check right cdot left y xf whether right left y x1i 2 y x1i NO 0 2 y xi not simple true interest for every is geq 0 y xi NOT decomposableCompound Interest right left e s1tfi right cdot left y xf right cdot left y x1i interest is decomposable Relevant for present value final formulas later compound value langle geq 5t theorem suppose f is Equations Functional Cauchy least at continuous at one Xo ER then yer ix fix Theorem suppose f Then strictly one x yER 4 x emx not the of decomposability aer ax at least at is f I 9 positive point and definition continuous xoER dgeegtgsf.ly compandinter f mer Related to est the Relevant for proof interest decomposability point for compound Cash Flows Investments First outflows then inflows inflows then outflows fans First annuities ONLY inflows of outflows Annuities installments constant simple time periods to shortcuts calculate and later value final constant R finite number ts annual monthly etc the present value perpetuities ONLY inflows of outflows s infinitesImber Present 6 value Tto 70 i 10 compound years Find present the of value annuity the 103.31 t p Final value 50 70 6 V2 5061 0.112 OR 40 70 103.37 o V2 Vo 1 0.112 I 125 6 SHORTCUT ONLY for compon because it's decomposable 12 125 Accumulate Vo the 70 annuity in 1 12 vi 2 5011 0.17 OR 103.37 V0 z to cite f t change value Intermediate value of Find they CFFE.ineMFntoest just 7 115.83 years Ettite Éeest just change 1 2 1.2 f t SHORTCUT 103.31 1 0.1 115.83 Accumulate Vo only for compound OR V2 125 2 You E15dff SHORTCUT comes for compano Simple R Annuitiesconstant installments constant IF time have we decomposable ORDINARY POSTICIPATED z I Present ANTICIPATED R 0 2t Present value R 3t ddot right cdot frac left V 0 R n1i of INSTALLMENTS so I must match right left m 1i cdot R n ari ddot cdot R n 7 ai 1 period after value right cdot left V 0 n 1i RFinalddot right left V 0 n 1i Apply decomposability the end Final the value accumulate to right cdot frac left R n 1i at tn right cdot frac left R n1i i i right cdot left R n1i right cdot left R n 1i i ddot cdot neg R S ni cdot R S T ni TRICKS ORDINARY PV o FV ldots right left 0 n k 1 3t NOT the time can't be modified or im PVFVDUE R n i annual R value right cdot frac left V R 0 n1i i ldots right left n k 1 R 2tI 1t semi annual etc is interest which compound for shortcuts there are periods DUE right left n x1i right left x1i last constant annual compound v R installments time periods interest Eat iii i R citis I Citi 2500 2500 0.0715 1 1 Answer A 7 i P jiz 12 1 Vis R 1 R Citi fins 1 15 months 1 0.01 s 1 0.01 Itil 120 Answer C of will 3 constant EXAMPLEYou 100 The interest R 0 1 R 0 1 I I Simple constant SEMESTERS ordinary posticipated Annuity R 0 1 installments periods time 3 convert i 4 8 0 2i total number of installments annual installments semi right left m 2 n 1i nicefrac right left 0 2 1i to in of periodsper year semi annual rate present value the right cdot left V 0 R 3 2 1i 3 n interestnumberm 2 compound a Find months 10 3 2 constant 6 is R 0 1 rate installments annual in starting euros annual semi 2i matrix begin right cdot end left 9 0 4 8 3 2 71 b Find the value of V 0 V 4 2 1 0 4 2 1 time in right left V 9 0 4 3 5 2 71i the months 11 Intermediate value annuity yearsyears in Time in semestersOR right left V 6 9 0 4 8 3 5 2 71i M 8 n For one annuities the end is one period later 5 n right cdot left V 8 P F 1i a right cdot left R 8 1i i bigtriangleup Because it's one matrix begin cdot end 0 8 5 1 cdot circ 8 0 1 right cdot left 0 8 5 1 8 6 9 2 7 b right cdot left V 9 4 8 R 3 P 2 F1i i right left V S P 2 1i i spddot cdot R a sto we since right cdot left 0 8 R 5 1i matrix begin cdot end 0 8 5 1 ddot cdot 8 0 5 7 a 1 R 8 0 5 7 a 1 ddot 0 5 7 a 1 can we accumulate have PV decomposability 2 1j a 0l time at annual compound rate cdot m ij cdot 2ij cdot 2 1i 6 2i Find i right left m 1i right left m 6 0 3 2 1i 150 2 1j Higher than 150b 18 0 1 8 n 1 right cdot left V 0 8 R 2 nz1iright cdot left 8 6 0 5 1 18 semi m 2 SEMESTERS annual installments 1 6 0 4 6 2 1 d rangle langle right left M V 8 0 2 1i right left M V 9 6 0 4 8 3 5 2 1i right left 4 8 6 0 2 1 4 8 6 3 5 d 6 0 rangle langle right cdot left 6 4 0 R 2 1 R 6 0 2 7 Perpetuities Annuities perpetual to n never end that If are installments constant simple they with time intere constant compound periods shorcuts for are the PV there Perpetuities annuities are ORDINARY POSTICIPATED P R 1 I tz t O Present V value Is R.li 0 i or im No final value Find the expressio jiffy value I Citi 11th make installments R erpetrityvo 1mg R.it I Fil in n Citi Bials IFFpresent value with a atty.IE femer uneeiE.i egrs DUE semi annual instalments me Find is b constant c i installments a constant R rate appeppe Ims R p.at compound k oh Present n R it Vo DUE 2 at Hi 2 1 E II tyggnopyears.unee.EE installments Find i4 Hilty 1 will receive EXAMPLEYou constant in starting The a n 1 nominal annual rate Calculate 3 n suppose with quarter is quarterly installments V 0 1 present value 12 assume it's jy jm value of value of the each installment convert jm cdot m n 1rij im to cdot 4ij cdot 4 2 1i 3 yi R 0 R R z 3 QUARTERS right cdot left V 0 4 R 3 1i 0 3 7 a R 0 3 5 1 0 3 7 a b suppose rightarrow infty n Calculate the eachinstallment rightarrow right cdot infty lim_ frac left V 0 4 R n1i cdot 0 4 R 3 1i c suppose right cdot left 0 3 n 1 infty 0 3 7 a R 0 1 Find the number of installments 0 3 right frac left ln 0 4 8 3 5 2 7 n1 lln 4 8 0 3 5 2 n 7 1n ln 4 0 8 3 5 2 7 n1 approx 2 1 installments Financial Evaluations Fostercattta Interia.EEiictrateofreturntieijq i opportunity Yield to Effective Rate compound overall NEW value present INVESTMENT 100 50 Net YTM maturity Annual Global TAEG rate of this new 70 1 10 0 Investment t investment opportunity cost 50 70 Since 9 100 is interest 0 so to 3.31 2 92115,1 1 opportunity cost there GOOD investment usual price 710 pop Determine whether 100 G 13 1 13 1 treat it as an opportunity cost 1781312 BAD investment 1213 NPV is evaluated cost NPV the with Discounted Cash the individual's capital of criteria assignee criteria NPV D Different Ci BAD a INDIFFERENT for a loan I Flow opportunity a 100 Discounted 61 7 cash a 50 100 Ifoffflow DCF cleave ftp 19 opportunity cost Investment Flow as 70 as variable NOT Im GCx Entree.ae Yo a co opportunity cost yggy.gr Prigge s 100 610 1 100 usually take you 28 any 11 8 1 012 no receive investment 100 Is 20 70 interest so interest with 8,9 712 to 50 you would 150 v30 100 10 Tet fire out for decreasing unique funich is standard a and strictly the x esse intercept convex is strictly remain one then outflow INVESTMENT inflows I a 02 1 100 17 660 100 50 Tea 70 a intercept y ET b d 4 5100 19 1 0 11 for s answer ANY operation THE fee Effassic investment true with investment also for ONE true classic a Outflow for a loan B Éiet Investment 1 0 2 640.037 5 3 660.061 660.097 3 3 6 70 9 x useless a a tazt Aoc 7 If Gota B Answer 610 0 act as for 40 an so then outflows and investment Inflows 0 8 100000 30 1 082 100000 R 9 4 8 30000 T.EE Ey58 i 1082 4 1 08 30000 1.08 100000 1.082 Pfi e 100000 PV of 3988 the 8 45000 installments 4 For R 1 08 lusing 8 not i R 700 Find 100 the R R 8 12 of value 1 which 10 inflows Eat t for R 112 112 T.ee 19 0.1 R years 1 O 1 o saE a or R simple a annuity Loans I j opportunity usual cost compound Internal Implicit of capital loan implies it's rate of Financing an Rate cost of Financing Effective cost overall compound rate of this new loan it's the equivalent of it but for a loan for 100 50 70 i 10 I 6610 1 100 88 3.31 78 BAD loan j 70 NPV NPV NPV NPV D criteria a a a a cost Effective vs GOOD j i it INDIFFERENT j f BAD j i criteria switches compared investments Discounted 100 a 50 a 6 Flow 70 az 100 so investments If cost co X _G 111 a 100 opportunity cost I a Inflow Different from Fx opportunity Loan Outflows 12 950 1 G Glx Cash tally 100 ftp usually you 660 loan 111s standards for a unique get free loans 2s 100 79 will increasing if so inconvenient is your you GLX can 20 be become more standard and If 100 LOWER satisfied SO inflow is loan strictly concave only one strictly this new x so NO info it can flows cash about the be an a investment NEITHERa is NDV usually xi IF then R 1t constant 2t installments constant time o for be the represents YEARS nt periodsc G Lx ldots right left G 0 ax1 R R b 1 or evaluated with i NOT X cost opportunity the loan DCF investment an which a loan can or neither INVESTMENT one LOAN with one outflow strictly inflow NEITHER Answer strictly decreasing convexwith increasing strictly concave strictly Depends C a Et NPV 5 1350 For an For a 150 1 Sm 5 investment loan MAY 2 the LESS what than is the of loan It could even that operation be is which case DCF may there borrow so ye back rate in The C butflows Inflow of pay Answer multiple m No x may x s not another not we be be an can't not an you 40 type of financial investment or loan say and monotonic for maybe hn humble x s NOT IMPORTANT Finding 100 50 70 d Discounted 100 Ian PTice P 100 0 Ffs Ffg 72 Effy 100 2 of value Efg 100 0 after present 100 0 Flow Cash Ffp 501 7 50 70 inflows the call 2 1912 55 17 be Fac S 2A 52 410147 51385 5155 5 505 ES 1 47 Es 20 5 5 5 605 20 12.321 1 txt using Fixed Income Bonds 2555cm BQEtttiatfacebivafue tt is Rate the of annual Yield to maturity YTM is IRR Return which COMPOUND rate Internal the ieice ruofthet.pt sfegg t fight_ incest T incl E into Into I Po c Ca t Cnt R t t In tn t a on not coupons R Reimbursement value Po fat tri 2 1 constant fix tn annual constant EI 18 constant capons R Po YEARS For 10 semi I I Ch necessarily cases Special 0 Bonds Coupon ANNUAL annual coupons coupons R Po SEMESTERS is.net em iitaiiiual 1 18012 1 1 12 1 21 coupons Duration is Duration maturities the weighted the inflows a time weights as It's Between in discounted the using of the values of average years NOT an amount of money th one for with ONLY outflow investments It risk sensitivity of the bond measures the due i to in changes and EXAMPLE weighted Average Grade 30 Credits 10 2 fit t EIEI.is I 100 8 12 cho.tl ten Special TPO O case sum of credits grades 1 107 D sum of credits grades of crests gingdiscount them 18 ti grades 20 Credits 5 Grade 30 10 20 5 10 5 GPA P of eernm E fits_to E Fits 2 1.8197 years 88.1 ZCB's IN T DCI CHIET T DCI MATURA Bond SOEXAMPHE pA A Bond B PB Bond C PS 2 90 2 0 the Assuming is same least the sensitive t.EE EEEEst Da Bond B ECB duration 100 ᵈ years ZCB Maturity 1 9,28 41 172811 b a LOWEST sensitive risky 0 80 it's the has 780 a c which of the bonds to changes in i BondC 2 least the i 12 181228 of Duration Answer C a 1 282 so it's usually albl 170 100 41 12 0 years Shit PT if I 1 19814442 1 108 EEE d If 121 then than 3 Answer B times the face the value price would since be more 3 years volatility of interest rate Pli Bonds a or change in Rare in exercises in rates inverse negative and withstate have prices of the in variation volatility Price the New Price Approximation of the New or in approximate interest and price changes Fi Δi ΔP of HINTS estimate prices of Approximation Interest PI it Yty compound interest IRR Approximation an As relationship BBETT.I.is Tuas Absolute variation in price in euros NOT Be careful with the sign Δi a for specific terms of Duration mooned Duration Interest Relative relative change in price BE ΔB to Definition Pli aappliggiorg.to the due price pcffperif.gs Prices of is prices in change 59 Volatility Effe P Rate rate ΔP it Δi I a they ftp.wani p 12 IIthe 1113 approximation P EE 9 EE.in Easi 120172 b 1 01115 1112117514 t t t 11 0 1751 D 120 If i YTM then d 4 01 5,4 Pli 4 120 P d i 11.75 I Δ 12 since don't we use pyfq.ee Pci Ai an Dp an ACTUAL it ΔI 12 Pli P 11.75 P 12 PC 1.75 PV of the inflows using change i Approximate Formula 1000 1000 1000 years a P 5 b 1,08 1,088 1000 1,98 Foot 1 0.08 1 1988 188 2 198 257710 1.95 years c 2577.101 3 Di ΔP P D 20 254 10 Δ 0004301 The approximated 19 8 Δ new rate is it D 0.08 0.00430 Annual Δ coupons C 60 R 1060 1000 60 don't we 0.001 if know Po R a Po 98g OR 60 Po 11883 9802 11003 3109800 1 1.063 0.06 1 15 sees that 9800 0.06 PI R Efi i b i c ΔP psi FAP Pli If 19993 6 PCI i 6 PII Po B 3 98oz 2 1000 8863.3 add p 6 Pci to Δi Pci 056 then 1000 9ft 2 It TE intteauation notos If DP Di P 6 7 86 both 0.001 sides Financial Immunization a.tv t.to Tan Po vain time Po C only amounts if i is want which Estee the rate This new at z the IRR Ytm not always we fontinow union irsi.org it at In eeii.ge nntEiiEE It EEYEEE We an effect to find will perfect the YEEE.tn If immunization EDC z date is DURATION prevail time z Review Linear AlgebraRankRank is maximum the and rows in columns a of independent matrixindependent rows columns D COBank a Rank number row column 2 dep.independent 3 6 2 ind24 dep3 ind.Rank 1 2 3 2 2 independent a 3 2 6 4 5 1 b 6i 4 5 d n frac columns rows subseteq 5 3 2 1 564 7 9 indindinodep dep uparrow 3 3 leftharpoonup uparrow ind 3 uparrow ind subseteq b a b c a Kronecker R n NO there are W is NOT R k Capelli's theorem Ranky no Ranky I YES Ranks solutions replicable n NO unique solutionsYES infinitely many infinitely many portfolio vectors n Ranky k a o R is INJECTIVE SURJECTIVE R is solutioRanky a solution there is a unique portfolio vector Every system has AtMOST one solutio Every system has at LEAST one a 8 9 Of Ranka 2 Lowest Ranka I FI least 1 1 one 0 Rankazz At 0 0 between rows columns number least one 0 0 2 2 1 0 Ranka 2 O RanKA Ranka I 3 Rankaz 1 0 RANKA 22 4 I 2 Ranka 4 0 4 2 and 10 2 3 t.EE The 8 1st row 1st column a nearer iit Ei E RanEA 2 are DEPENDENT 22 is since Rank Y 3 R k R n Domain codomain input TOYTUH R 3 2 cdot y x Operator Financial MarketsPortfolio vector X Quantities weights of each assetPayoff Y 2 3 2 1 45 Total payoff in state I contingent claim X 2 x1 42x1 X times 3 2 3 5 2 x1 times 2 1 state contingent times 3 1 payoff vector portfolio ofthe state vector for of unit 41Asset y 1 1 Payoff in state I recession 2 payoff in 2 stasis state 3 payoff in state 3 growth payoff 1 payoff vector for one unit right left 2 L y1 contingent the represents assets nabla frac y Replicable times frac iw R n if is_ contingent claim replicable a exists sum_ cdot frac W in X Y R n Jyx1rIj vdots n x A vector state so contingent the Possibilities X 0 J 0 5 x which linear for Buy sell can payoff vectorsquantities be obtained payoff the y J sqrt y contingent claim vector has system as solutions individual quantities long on ys short on yo a x matrix begin end 6 Y 0 4 3 2 x1 4 6 I 6 0 4 3 2 x1 Kronecker RANKY 2 4 1 6 Capelli's which 3 b k right left X R 2 R w end matrix begin 4 1 to Dim w determine whether it's replicabl RANKY 6 4 times 3 2 2 Rank geq 1 6 RANKY 2 ank YW is least as gh as anky R times 3 is 3 NOT surjective the financial market is NOT complete SO som w's are not replicable neq 4 0 2 1 Rank Y W 2 4 6 1 LI 6 2 k 10 13 14 leq Z R ank underline leq w 3 y 2 geq R 3 n a k 6 6 1 4 0 1 1 3 2 1 0 4 6 0 3 5 2 3 1 4 right left 6 4 0 8 3 5 2 7 1 3 Rankneq 4 6 0 1 6 1 underline w Y 2 Rank backslash I Rankylw Ranky Check Rank Y if since Ranks Rankyla If Ranky 50 w There the Rank YlW is Find 2 n are solution is UNIQUE there NO are solutions NOT replicable solution s the rowls Remove_any_dependent 2 solutions matrix begin end 6 0 4 2 x1 431 1 varphi 2 Rank Since Rank 3 the be DEPENDENT Solve the_system 6 2 x1 times 4 0 2 x1 other row column must Removethedependentrows 3 1 subtract the dependentcolumns spddot right left 6 0 4 3 2 x1 6 0 4 8 2 x1 underline simeq right cdot left p v x LOP holds since R is INJECTIVE In cdot 6 3 2 x1 x1 Market cdot 3 5 2 1 value of the portfolio 3 1 this Ranky YIN s 2Rank 2 for case or W is replicabled financial so the any w 3 is not replicable is NOT complete market W Complete financial A market claims we Rk W span L Rk represents the The Market is complete Financial are I all ones that market is acttailytossibits vectors replicable in RK are All complete Dim W Y possible L Market of variables of the value VCI ET pox price vector portfolio vector cost the X 0 0 y subspace of W a of quantities a y of benefit of LOP may not hold market frictionless the is v1 R portfolio which is a linear function there is value in portfolio Suppose market Rk sidecar Market financial complete Financial Market students Incomplete a k Ranky is vectors w BASICALLY Rank Y K X R Rr teetents nonsig w the Dimw replicabl are Rank Y Dim IMR L Égt ar claims operator payoff Dim SpanL The so replicable Theorem consider the REX E So all contingent if portfolio the portfolio R Janningsetofw usually they call it a alb Dimw 4 1 3 9 541 242 R so if check is 3 Ranky 2 the R3 R Ranky 2 2 the market is replicable 3 Ranky not every since c Ranky Dim span L w system B Answer is is incomplete possible a 1 Xs I 1 Apply Algorithm Kronecker's 1 3 6 1 1 1 4 8 5 W 34 42 a Ranky w 2 Ranky w n 3 Ranky w with replicable many infinitely Since 2 is portfolios of Law the same prices Ranky n icing ormula if w MAY they guarantees in Pricing Rule f W a market W spank suppose consider the pricing then α part The a replicable dm same unique a rpgs Pw the portfolios for same price the many WEW VIII linear is a combination the assets replicating portfolio XER W that Pw the R Pw COP is unique is Erial ti ioenerane Pricing Formula OnPoem of linear price be the have FEET EE will satisfies f W rule with tag infinitely are os unique cant we of w price of the prices a replicating of On contingent payoff R is injective LOP The the Éfo two any Ranky n if there single is of state Even a er portfolios two BUT Price one a replicable combination claims wt claim w of the wm with weights I Pantsx suppose Eaten can R 3 ftp Ex v14 b holds YEESE IF 2 ts s 1l 6 4 10 1 2 3 1 3 1 2 R we can't EVERY safy if all replicable 811 the we the LOP DOES NOT hold Lop will be are Pw 5 and price of w FIÉ I same p ie hold for 3 Enticing 5 3 F the pie I since MAY replicating portfolios that tie 5 with_Frices Find states E E p VC 1 I LOP R 2 assets a iswhether isthe conclude we T for_in portfolios replicating both are 8 frac w Y x R 3 2 matrix begin times end 0 3 5 2 x1 times 3 2 3 x times 3 1 Ranky 2 Not LOD MAY not hold Market is complete injective surjectiveEvery is replicable w many a with portfolios right left 6 3 5 2 x1 infinitely 2 x1 3 x1 x1 underline in R 2 x1 x1 cdot frac 4 P 2 x 2 x1 right left 4 2 x1 x1 Not constantLOP doesn't hold y 0 1 1 mdlgwhtsquare 0 1 0 times 3 3 leq 1 RANKY leq RankV21 rightarrow R 3 RANKYgeq2 o 1 1 1 RankY 3 matrix begin end 0 1 2Since R 3 Ranky is the market the 3 Ranky LOP MAY is Rank Y surjective not not complete.Since check for if replicating matrix begin end w 3 y1 o R is not not hold replicable have portfolios can a be replicated so injective different different prices w with x1 frac 0 x o Text p 2 1 0 p 1 O N 0 P 1 LOP 2 0 1 underline right cdot left X V 3 P 2 p x1 There are BkindAnswer of the I arbitrages so does NOT hold d y I equiv 3 2 y 1 2 y 1 2 1 2 1 rightarrow R 3 I 4 times 3 1 Rank Y 3 Ranky 2 market Incomplete since x 3 2 y 1 underline and 2 1 0 0 x replicate matrix begin end 0 1 w 0 3 y 4 ldots underline rangle langle right cdot left V p x1 cdot O 3 P 2 p1 0 2 p1 spddot right cdot frac left V 0 3 2 p x 1 P 3 underline right neq left V x so the LOP doesn't hold theorem consider LOP the then omega right cdot left pi f frac where pi and omega right frac left f the which MAY not market financial The a market Vomega W in is the pricing a linear is satisfies be complete which is unique kernel function.If matrix begin ldots right end left pi 2 1 of o complete underline omega forall in R k omega right cdot frac left pi f prices is market I the represents arrow the vector contingent etc frac w 0 2 e 1 of pure the claims vdots find To the omega right left X V F pricing kernel omega cdot frac pi R x cdot pi P y x cdot frac pi R y A unique II existsLOP holdsSince w must be replicable omega cdot X Y and underline omega rangle langle forall right cdot left pi in R p kf 3rd pricing FormulaEXAMPLE and holds LOP suppose the with pi 3 2 complete omega 4 Find the price of ldots cdot frac p w 4 3 5 P 2i S the market is 8 dy R2 3 60 ranky 2 surjective The market so every I P is w not but injective complete is replicable W II Y X P R 560 3 1401T 2H 211 II Y 3 15 assets 2 60112 60 states 2 3 40 60 099T 3512515 32 15 75 50.75 0.55 unique II so the since I's 0 the are no LOD holds FTF arbitrages holds of Type so I there II matrix begin right geq end left 0 3 2 1 POSITIVEAt matrix begin right end left 0 3 2 1 POSITIVEAll matrix begin right end left gg 0 3 2 1 Any of combination or least one others are so Arbitrage or not all STRICTLY STRONGLY The the of POSITIVETypes Arbitrage kind I right cdot geq left X Y R 0 is ER portfolio called if contingent state of the portfolio payoffmarket value right cdot circ left X V P 0 x sales revenue from costs market value is the so The the ER portfolio II underline Omega rangle langle right cdot left X Y R kind underline right cdot left leq V 0 p x market which called costs sales arbitrage an has it from arbitrages satisfies has underline Omega right cdot left X Y R x no I of the underline rangle langle right cdot geq left 0 p v x the the of holdsNo kind forall in R x i no right cdot geq left X Y R 0 x implies market of conditions arbitrages The there is a benefit purchasesNo Arbitrage forall in R x i I is purchases from if from revenue The of arbitrage an kind if LOP LOP arbitrages of the underline rangle langle right cdot left D V 0 x I kind if Properties with 1 L R no and f W there rule exists II W W ER Riesz Markov f Rn f is R Markov LOD f Rk a market value of the than linear vector IE RE unique pricing kernel X price state of Fc ax the contigent ALL WERE for and increasing fositive vector Pricing a no is R theorem linear unique the function rather w Theorem consider It has aesz a as a the find portfolio claim the pricing can of arbitrages Markets Financial I kind Complete a satisfies 2 The we of off is Rule and Linear financial complete orpitrages of Increasing market the VXER kind I with It Sicist.se Ewstsiner and increasing W W ERK ERK FW I D Fundamental Theorem consider It has complete a no positive of Finance financial orpitrages of the market I kind with EQ or kin Ie see.tosmetefew5PisTiear and strictly increasing ERK FCU II W W ERK Q strongly positive a No b market c arbitrages is B II assume if FTE there but twerk are arbitrages Every is holds LOP claim w contingent replicable s market he of the of arbitrages a If specific mentioned assume so EVERY to W the and complete x kind I sofigafsig arent there the complete a Answer 0 of pricing that is complete holds I the so kind II kind kernel is it's applicable the market LOP holds is Ranky k I show that the system is possible HEY 3 1 Kronecite Ranky IRONKY RAKY Ranky 3 1 caper 3 SIGN MATRIX 2 I 8 34 3 Ranky R3 R R is Rankyla ict f to Ronny 3 market in case is complete 6 3 RONKYIW 3 Ranky I The 3 system is Kronecker Capelli 3 this n 3 a unique market is complete possible replicable Ranks W so the is w ANY for solution since sevens pinterest.ee aenfewiiiiave because it's surjective LIP a holds niafefi.ie If I II 2 E.IE iEi x1 p If R en f is injective 2 5 but IEr.ee sIEtpiictes 3 6 NOT although surjective not every only possible 12 113 if the LOP holds EE EEtit 2 3 5 IT EI.IE EEEa iEEE ÉÉÉÉEsE EE EEss t.tt giggy D w Since the I ftp s i.n e s.e É Trice of of weiff E 4f 6 market is complete and of finance theorem fundamental satisfied the and there are no arbitrages is 0 tort 45 8 13L I R R 1 1 11 R 3 RANKY Ranky Rank Y Z Rank Y 3 3 3 103 106 26 91 106 20 151 1825010 3 rdna 2 Lanky R is NOT surjective R is also not so COP can't we holds the 3 market since injective whether say is incomplete Ranky the 3 the same I pa I pfÉp replicate plx p p X 41 f so 42 1,8 Pg Lop does not hold There are Type I 2ps Effete text arbitrages p eat v Pi Bff Egg since R there is t f f t 2Pa f P3 2ps and an arbitrage of P3 2Pa vexico the I and II kin
0
advertisement
Download
advertisement
Add this document to collection(s)
You can add this document to your study collection(s)
Sign in Available only to authorized usersAdd this document to saved
You can add this document to your saved list
Sign in Available only to authorized users