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Steven Shreve: Stochastic Calculus and FinancePRASADCHALASANICarnegie Mellon Universitychal@cs.cmu.eduSOMESHJHACarnegie Mellon Universitysjha@cs.cmu.eduTHIS IS A DRAFT: PLEASE DO NOT DISTRIBUTEcCopyright; Steven E. Shreve, 1996July 25, 1997 Contents1 Introduction to Probability Theory 111.1 TheBinomialAssetPricingModel 111.2 Finite Probability Spaces . 161.3 LebesgueMeasureandtheLebesgueIntegral 221.4 General Probability Spaces 301.5 Independence . 401.5.1 Independenceofsets . 401.5.2 Independence of-algebras . 411.5.3 Independence of random variables 421.5.4 Correlationandindependence 441.5.5 Independenceandconditionalexpectation. . 451.5.6 LawofLargeNumbers 461.5.7 CentralLimitTheorem 472 Conditional Expectation 492.1 ABinomialModelforStockPriceDynamics 492.2 Information 502.3 ConditionalExpectation . 522.3.1 Anexample 522.3.2 Definition of Conditional Expectation 532.3.3 FurtherdiscussionofPartialAveraging . 542.3.4 PropertiesofConditionalExpectation 552.3.5 ExamplesfromtheBinomialModel . 572.4 Martingales 581 23 Arbitrage Pricing 593.1 BinomialPricing . 593.2 Generalone-stepAPT . 603.3 Risk-Neutral Probability Measure 613.3.1 PortfolioProcess . 623.3.2 Self-financing Value of a Portfolio Process 623.4 Simple European Derivative Securities 633.5 TheBinomialModelisComplete . 644 The Markov Property 674.1 BinomialModelPricingandHedging 674.2 ComputationalIssues . 694.3 MarkovProcesses . 704.3.1 DifferentwaystowritetheMarkovproperty 704.4 ShowingthataprocessisMarkov 734.5 ApplicationtoExoticOptions 745 Stopping Times and American Options 775.1 AmericanPricing . 775.2 ValueofPortfolioHedginganAmericanOption . 795.3 Information up to a Stopping Time 816 Properties of American Derivative Securities 856.1 Theproperties . 856.2 ProofsoftheProperties 866.3 Compound European Derivative Securities 886.4 OptimalExerciseofAmericanDerivativeSecurity 897 Jensen’s Inequality 917.1 Jensen’s Inequality for Conditional Expectations . 917.2 OptimalExerciseofanAmericanCall 927.3 Stopped Martingales . 948 Random Walks 978.1 FirstPassageTime 97 38.2isalmostsurelyfinite 978.3 The moment generating function for 998.4 Expectation of 1008.5 TheStrongMarkovProperty . 1018.6 GeneralFirstPassageTimes . 1018.7 Example:PerpetualAmericanPut 1028.8 DifferenceEquation 1068.9 DistributionofFirstPassageTimes 1078.10TheReflectionPrinciple . 1099 Pricing in terms of Market Probabilities: The Radon-Nikodym Theorem. 1119.1 Radon-Nikodym Theorem 1119.2 Radon-Nikodym Martingales . . . 1129.3 TheStatePriceDensityProcess . 1139.4 Stochastic Volatility Binomial Model . 1169.5 Another Applicaton of the Radon-Nikodym Theorem . . 11810 Capital Asset Pricing 11910.1AnOptimizationProblem . 11911 General Random Variables 12311.1 Law of a Random Variable 12311.2 Density of a Random Variable . . 12311.3Expectation 12411.4 Two random variables . 12511.5MarginalDensity . 12611.6ConditionalExpectation . 12611.7ConditionalDensity 12711.8MultivariateNormalDistribution . 12911.9Bivariatenormaldistribution . 13011.10MGF of jointly normal random variables . 13012 Semi-Continuous Models 13112.1Discrete-timeBrownianMotion . 131 412.2TheStockPriceProcess 13212.3RemainderoftheMarket . 13312.4Risk-NeutralMeasure . 13312.5Risk-NeutralPricing . 13412.6Arbitrage . 13412.7StalkingtheRisk-NeutralMeasure 13512.8PricingaEuropeanCall 13813 Brownian Motion 13913.1 Symmetric Random Walk . 13913.2TheLawofLargeNumbers 13913.3CentralLimitTheorem 14013.4 Brownian Motion as a Limit of Random Walks . 14113.5BrownianMotion . 14213.6CovarianceofBrownianMotion . 14313.7Finite-DimensionalDistributionsofBrownianMotion 14413.8 Filtration generated by a Brownian Motion 14413.9MartingaleProperty 14513.10TheLimitofaBinomialModel 14513.11StartingatPointsOtherThan0 14713.12MarkovPropertyforBrownianMotion 14713.13Transition Density . 14913.14FirstPassageTime 14914 The Itˆo Integral 15314.1BrownianMotion . 15314.2FirstVariation . 15314.3QuadraticVariation 15514.4 Quadratic Variation as Absolute Volatility 15714.5 Construction of the ItˆoIntegral 15814.6 Itˆointegralofanelementaryintegrand 15814.7 Properties of the Itˆointegralofanelementaryprocess 15914.8 Itˆointegralofageneralintegrand . 162 514.9 Properties of the (general) Itˆointegral 16314.10Quadratic variation of an Itˆointegral . 16515 Itˆo’s Formula 16715.1 Itˆo’sformulaforoneBrownianmotion 16715.2 Derivation of Itˆo’sformula 16815.3GeometricBrownianmotion . 16915.4QuadraticvariationofgeometricBrownianmotion . 17015.5 Volatilityof Geometric Brownian motion 17015.6FirstderivationoftheBlack-Scholesformula 17015.7MeanandvarianceoftheCox-Ingersoll-Rossprocess 17215.8 Multidimensional Brownian Motion . 17315.9Cross-variationsofBrownianmotions 17415.10Multi-dimensional Itˆoformula 17516 Markov processes and the Kolmogorov equations 17716.1StochasticDifferentialEquations . 17716.2MarkovProperty . 17816.3 Transition density . 17916.4 The Kolmogorov Backward Equation 18016.5ConnectionbetweenstochasticcalculusandKBE 18116.6Black-Scholes . 18316.7 Black-Scholes with price-dependent volatility 18617 Girsanov’s theorem and the risk-neutral measure 18917.1 Conditional expectations underfIP 19117.2Risk-neutralmeasure . 19318 Martingale Representation Theorem 19718.1MartingaleRepresentationTheorem . 19718.2Ahedgingapplication . 19718.3d-dimensionalGirsanovTheorem 19918.4d-dimensionalMartingaleRepresentationTheorem . 20018.5 Multi-dimensional market model . 200 619 A two-dimensional market model 20319.1 Hedging when,1 1 20419.2 Hedging when =1 . 20520 Pricing Exotic Options 20920.1ReflectionprincipleforBrownianmotion 20920.2UpandoutEuropeancall. 21220.3Apracticalissue 21821 Asian Options 21921.1Feynman-KacTheorem 22021.2Constructingthehedge 22021.3PartialaveragepayoffAsianoption 22122 Summary of Arbitrage Pricing Theory 22322.1Binomialmodel,HedgingPortfolio . 22322.2 Setting up the continuous model . 22522.3Risk-neutralpricingandhedging . 22722.4Implementationofrisk-neutralpricingandhedging . 22923 Recognizing a Brownian Motion 23323.1 Identifying volatility and correlation . 23523.2Reversingtheprocess . 23624 An outside barrier option 23924.1Computingtheoptionvalue 24224.2ThePDEfortheoutsidebarrieroption 24324.3Thehedge . 24525 American Options 24725.1PreviewofperpetualAmericanput 24725.2FirstpassagetimesforBrownianmotion:firstmethod 24725.3Driftadjustment 24925.4Drift-adjustedLaplacetransform . 25025.5Firstpassagetimes:Secondmethod . 251 725.6PerpetualAmericanput 25225.7ValueoftheperpetualAmericanput . 25625.8Hedgingtheput 25725.9PerpetualAmericancontingentclaim . 25925.10PerpetualAmericancall 25925.11Putwithexpiration 26025.12Americancontingentclaimwithexpiration . 26126 Options on dividend-paying stocks 26326.1Americanoptionwithconvexpayofffunction 26326.2Dividendpayingstock 26426.3 Hedging at timet1 26627 Bonds, forward contracts and futures 26727.1Forwardcontracts . 26927.2Hedgingaforwardcontract 26927.3Futurecontracts 27027.4Cashflowfromafuturecontract . 27227.5Forward-futurespread . 27227.6Backwardationandcontango . 27328 Term-structure models 27528.1 Computing arbitrage-free bond prices: first method . . . 27628.2Someinterest-ratedependentassets . 27628.3Terminology 27728.4Forwardrateagreement 27728.5 Recovering the interestrtfromtheforwardrate 27828.6 Computing arbitrage-free bond prices: Heath-Jarrow-Morton method 27928.7Checkingforabsenceofarbitrage 28028.8ImplementationoftheHeath-Jarrow-Mortonmodel . 28129 Gaussian processes 28529.1Anexample:BrownianMotion 28630 Hull and White model 293 830.1Fiddlingwiththeformulas 29530.2 Dynamics of the bond price 29630.3CalibrationoftheHull&Whitemodel 29730.4 Option on a bond . 29931 Cox-Ingersoll-Ross model 30331.1 Equilibriumdistribution ofrt 30631.2 Kolmogorov forward equation . . 30631.3 Cox-Ingersoll-Ross equilibrium density . 30931.4BondpricesintheCIRmodel 31031.5 Option on a bond . 31331.6DeterministictimechangeofCIRmodel . 31331.7Calibration 31531.8 Tracking down'00inthetimechangeoftheCIRmodel . 31632 A two-factor model (Duffie & Kan) 31932.1 Non-negativity ofY 32032.2 Zero-coupon bond prices . 32132.3Calibration 32333 Change of num´eraire 32533.1 Bond price as num´eraire . 32733.2 Stock price as num´eraire . 32833.3Mertonoptionpricingformula 32934 Brace-Gatarek-Musiela model 33534.1 Review of HJM under risk-neutralIP . 33534.2 Brace-Gatarek-Musiela model . . 33634.3LIBOR 33734.4ForwardLIBOR 33834.5 The dynamics ofLt; . 33834.6ImplementationofBGM . 34034.7Bondprices 34234.8 Forward LIBOR under more forward measure 343 934.9Pricinganinterestratecaplet . 34334.10Pricinganinterestratecap 34534.11CalibrationofBGM 34534.12Longrates . 34634.13Pricingaswap . 346 [...]... of S2 alone 20 Definition 1.5 Let be a nonemtpy finite set and let F be the -algebra of all subsets of Let X be a random variable on ; F The -algebra X generated by X is defined to be the collection of all sets of the form f! 2 ; X ! 2 Ag, where A is a subset of IR Let G be a sub- -algebra of F We say that X is G -measurable if every set in X is also in G Note: We normally write simply... c a; b = ,1; a b; 1 : This shows that every closed interval is Borel In addition, the closed half-lines 1 a; 1 = and n=1 ,1; a = a; a + n 1 n=1 a , n; a are Borel Half-open and half-closed intervals are also Borel, since they can be written as intersections of open half-lines and closed half-lines For example, a; b = ,1; b a; 1 : Every set which contains only one real number is Borel Indeed,... more The -algebra F1 is said to contain the “information of the first toss”, which is usually called the “information up to time 1” Similarly, F2 contains the “information of CHAPTER 1 Introduction to Probability Theory 19 the first two tosses,” which is the “information up to time 2.” The -algebra F3 = F contains “full information” about the outcome of all three tosses The so-called “trivial” -algebra... nothing about ! Definition 1.3 Let be a nonempty finite set A filtration is a sequence of -algebras F0 ; F1; F2; : : :; Fn such that each -algebra in the sequence contains all the sets contained by the previous -algebra Definition 1.4 Let be a nonempty finite set and let random variable is a function mapping into IR F be the -algebra of all subsets of A Example 1.3 Let be given by (2.1) and consider the binomial... pricing and its relation to risk-neutral pricing is clearly illuminated Secondly, the model is used in practice because with a sufficient number of steps, it provides a good, computationally tractable approximation to continuous-time models Thirdly, within the binomial model we can develop the theory of conditional expectations and martingales which lies at the heart of continuous-time models With this third... CHAPTER 1 Introduction to Probability Theory 23 Definition 1.9 The Borel -algebra, denoted B IR , is the smallest -algebra containing all open intervals in IR The sets in B IR are called Borel sets Every set which can be written down and just about every set imaginable is in B IR The following discussion of this fact uses the -algebra properties developed in Problem 1.3 By definition, every open interval... a; b is in B IR , where a and b are real numbers Since B IR is a -algebra, every union of open intervals is also in B IR For example, for every real number a, the open half-line 1 a; 1 = n=1 is a Borel set, as is a; a + n 1 ,1; a = a , n; a : n=1 For real numbers a and b, the union ,1; a b; 1 is Borel Since B IR is a -algebra, every complement of a Borel set is Borel, so B IR contains... as preimages of sets in IR is: ;; ; AHH; AHT ATH ; ATT ; and sets which can be built by taking unions of these This collection of sets is a -algebra, called the -algebra generated by the random variable S2, and is denoted by S2 The information content of this -algebra is exactly the information learned by observing S2 More specifically, suppose the coin is tossed three times and you do not know the... 1 T S2 TH + 1 + r X1 T , 1 T S1 T ; V2 TT = 1 T S2 TT + 1 + r X1 T , 1 T S1 T : Subtracting one of these from the other and solving for mula” 1 T , we obtain the “delta-hedging for- V 1 T = S2 TH , V2 TT ; TH , S TT 2 2 (1.12) and substituting this into either equation, we can solve for 1 ~ X1 T = 1 + r pV2 TH + qV2 TT : ~ (1.13) 16 Equation (1.13), gives... determine IP for all the other sets in F Definition 1.2 Let be a nonempty set A -algebra is a collection following three properties: (i) ; 2 G, G of subsets of with the 18 (ii) If A 2 G , then its complement Ac 2 G, (iii) If A1; A2; A3; : : : is a sequence of sets in G , then 1 Ak is also in G k=1 Here are some important -algebras of subsets of the set F0 = F1 = F2 = ;; in Example 1.2: ; ;; . ............................ 32934 Brace-Gatarek-Musiela model 33534.1 Review of HJM under risk-neutralIP......................... 33534.2 Brace-Gatarek-Musiela model . . ........................... arbitrage-free bond prices: Heath-Jarrow-Morton method ........ 27928.7Checkingforabsenceofarbitrage .......................... 28028.8ImplementationoftheHeath-Jarrow-Mortonmodel.................