CALCUL SOCHASTIQUE EN FINANCE

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1 CALCUL SOCHASTIQUE EN FINANCE Peter Tankov Nizar Touzi Ecole Polytechnique Paris Département de Mathématiques Appliquées Septembre 21

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3 Contents 1 Introduction European and American options No dominance principle and first properties Put-Call Parity Bounds on call prices and early exercise of American calls Risk effect on options prices Some popular examples of contingent claims A first approach to the Black-Scholes formula The single period binomial model The Cox-Ross-Rubinstein model Valuation and hedging in the Cox-Ross-Rubinstein model Continuous-time limit Some preliminaries on continuous-time processes Filtration and stopping times Filtration Stopping times Martingales and optional sampling Maximal inequalities for submartingales Complement: Doob s optional sampling for discrete martingales The Brownian Motion Definition of the Brownian motion The Brownian motion as a limit of a random walk Distribution of the Brownian motion Scaling, symmetry, and time reversal Brownian filtration and the Zero-One law Small/large time behavior of the Brownian sample paths Quadratic variation Complement

4 4 5 Stochastic integration with respect to the Brownian motion Stochastic integrals of simple processes Stochastic integrals of processes in H Construction The stochastic integral as a continuous process Martingale property and the Itô isometry Deterministic integrands Stochastic integration beyond H 2 and Itô processes Complement: density of simple processes in H Itô Differential Calculus Itô s formula for the Brownian motion Extension to Itô processes Lévy s characterization of Brownian motion A verification approach to the Black-Scholes model The Ornstein-Uhlenbeck process Distribution Differential representation Application to the Merton optimal portfolio allocation problem Problem formulation The dynamic programming equation Solving the Merton problem Martingale representation and change of measure Martingale representation The Cameron-Martin change of measure The Girsanov s theorem The Novikov s criterion Application: the martingale approach to the Black-Scholes model The continuous-time financial market Portfolio and wealth process Admissible portfolios and no-arbitrage Super-hedging and no-arbitrage bounds The no-arbitrage valuation formula Stochastic differential equations First examples Strong solution of a stochastic differential equation Existence and uniqueness The Markov property More results for scalar stochastic differential equations Linear stochastic differential equations An explicit representation The Brownian bridge Connection with linear partial differential equations Generator

5 Cauchy problem and the Feynman-Kac representation Representation of the Dirichlet problem The hedging portfolio in a Markov financial market Application to importance sampling Importance sampling for random variables Importance sampling for stochastic differential equations The Black-Scholes model and its extensions The Black-Scholes approach for the Black-Scholes formula The Black and Scholes model for European call options The Black-Scholes formula The Black s formula Option on a dividend paying stock The Garman-Kohlhagen model for exchange rate options The practice of the Black-Scholes model Hedging with constant volatility: robustness of the Black- Scholes model Complement: barrier options in the Black-Scholes model Barrier options prices Dynamic hedging of barrier options Static hedging of barrier options Local volatility models and Dupire s formula Implied volatility Local volatility models CEV model Dupire s formula Dupire s formula in practice Link between local and implied volatility Gaussian interest rates models Fixed income terminology Zero-coupon bonds Interest rates swaps Yields from zero-coupon bonds Forward Interest Rates Instantaneous interest rates The Vasicek model Zero-coupon bonds prices Calibration to the spot yield curve and the generalized Vasicek model Multiple Gaussian factors models Introduction to the Heath-Jarrow-Morton model Dynamics of the forward rates curve The Heath-Jarrow-Morton drift condition The Ho-Lee model

6 The Hull-White model The forward neutral measure Derivatives pricing under stochastic interest rates and volatility calibration European options on zero-coupon bonds The Black-Scholes formula under stochastic interest rates Introduction to financial risk management Classification of risk exposures Market risk Credit risk Liquidity risk Operational risk Model risk Risk exposures and risk limits: sensitivity approach to risk management Value at Risk and the global approach Convex and coherent risk measures Regulatory capital and the Basel framework A Préliminaires de la théorie des mesures 193 A.1 Espaces mesurables et mesures A.1.1 Algèbres, σ algèbres A.1.2 Mesures A.1.3 Propriétés élémentaires des mesures A.2 L intégrale de Lebesgue A.2.1 Fonction mesurable A.2.2 Intégration des fonctions positives A.2.3 Intégration des fonctions réelles A.2.4 De la convergence p.p. à la convergence L A.2.5 Intégrale de Lebesgue et intégrale de Riemann A.3 Transformées de mesures A.3.1 Mesure image A.3.2 Mesures définies par des densités A.4 Inégalités remarquables A.5 Espaces produits A.5.1 Construction et intégration A.5.2 Mesure image et changement de variable A.6 Annexe du chapitre A A.6.1 π système, d système et unicité des mesures A.6.2 Mesure extérieure et extension des mesures A.6.3 Démonstration du théorème des classes monotones

7 7 B Préliminaires de la théorie des probabilités 215 B.1 Variables aléatoires B.1.1 σ algèbre engendrée par une v.a B.1.2 Distribution d une v.a B.2 Espérance de variables aléatoires B.2.1 Variables aléatoires à densité B.2.2 Inégalités de Jensen B.2.3 Fonction caractéristique B.3 Espaces L p et convergences fonctionnelles des variables aléatoires B.3.1 Géométrie de l espace L B.3.2 Espaces L p et L p B.3.3 Espaces L et L B.3.4 Lien entre les convergences L p, en proba et p.s B.4 Convergence en loi B.4.1 Définitions B.4.2 Caractérisation de la convergence en loi par les fonctions de répartition B.4.3 Convergence des fonctions de répartition B.4.4 Convergence en loi et fonctions caractéristiques B.5 Indépendance B.5.1 σ algèbres indépendantes B.5.2 variables aléatoires indépendantes B.5.3 Asymptotique des suites d événements indépendants B.5.4 Asymptotique des moyennes de v.a. indépendantes

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9 Chapter 1 Introduction Financial mathematics is a young field of applications of mathematics which experienced a huge growth during the last thirty years. It is by now considered as one of the most challenging fields of applied mathematics by the diversity of the questions which are raised, and the high technical skills that it requires. These lecture notes provide an introduction to stochastic finance for the students of third year of Ecole Polytechnique. Our objective is to cover the basic Black-Scholes theory from the modern martingale approach. This requires the development of the necessary tools from stochastic calculus and their connection with partial differential equations. Modelling financial markets by continuous-time stochastic processes was initiated by Louis Bachelier (19) in his thesis dissertation under the supervision of Henri Poincaré. Bachelier s work was not recognized until the recent history. Sixty years later, Samuelson (Nobel Prize in economics 197) came back to this idea, suggesting a Brownian motion with constant drift as a model for stock prices. However, the real success of Brownian motion in the financial applications was realized by Fisher Black, Myron Scholes, et Robert Merton (Nobel Prize in economics 1997) who founded between 1969 and 1973 the modern theory of financial mathematics by introducing the portfolio theory and the no-arbitrage pricing arguments. Since then, this theory gained an important amount of rigour and precision, essentially thanks to the martingale theory developed in the eightees. Although continuous-time models are more demanding from the technical viewpoint, they are widely used in the financial industry because of the simplicity of the resulting formulae for pricing and hedging. This is related to the powerful tools of differential calculus which are available only in continuoustime. We shall first provide a self-contained introduction of the main concept from stochastic analysis: Brownian motion, stochastic integration with respect to the Brownian motion, Itô s formula, Girsanov change of measure Theorem, connection with the heat equation, and stochastic differential equations. We then consider the Black-Scholes continuous-time financial market where the noarbitrage concept is sufficient for the determination of market prices of derivative 9

10 1 CHAPTER 1. INTRODUCTION securities. Prices are expressed in terms of the unique risk-neutral measure, and can be expressed in closed form for a large set of relevant derivative securities. The final chapter provides the main concepts in interest rates models in the gaussian case. In order to motivate the remaining content of theses lecture notes, we would like to draw the reader about the following major difference between financial engineering and more familiar applied sciences. Mechanical engineering is based on the fundamental Newton s law. Electrical engineering is based on the Maxwell equations. Fluid mechanics are governed by the Navier-Stokes and the Bernoulli equations. Thermodynamics rest on the fundamental laws of conservation of energy, and entropy increase. These principles and laws are derived by empirical observation, and are sufficiently robust for the future development of the corresponding theory. In contrast, financial markets do not obey to any fundamental law except the simplest no-dominance principle which states that valuation obeys to a trivial monotonicity rule, see Section 1.2 below. Consequently, there is no universally accurate model in finance. Financial modelling is instead based upon comparison between assets. The Black-Scholes model derives the price of an option by comparison to the underlying asset price. But in practice, more information is available and one has to incorporate the relevant information in the model. For this purpose, the Black-Scholes model is combined with convenient calibration techniques to the available relevant information. Notice that information is different from one market to the other, and the relevance criterion depends on the objective for which the model is built (prediction, hedging, risk management...). Therefore, the nature of the model depends on the corresponding market and its final objective. So again, there is no universal model, and any proposed model is wrong. Financial engineering is about building convenient tools in order to make these wrong models less wrong. This is achieved by accounting for all relevant information, and using the only no-dominance law, or its stronger version of noarbitrage. An introdcution to this important aspect is contained in Chapter 1. Given this major limitation of financial modelling, a most important issue is to develop tools which measure the risk beared by any financial position and any model used in its management. The importance of this activity was highlighted by the past financial crisis, and even more emphasized during the recent subprime financial crisis. Chapter 12 provides the main tools and ideas in this area. In the remaining of this introduction, we introduce the reader to the main notions in derivative securities markets. We shall focus on some popular examples of derivative assets, and we provide some properties that their prices must satisfy independently of the distribution of the primitive assets prices. The only ingredient which will be used in order to derive these properties is the no dominance principle introduced in Section1.2 below.

11 1.1. European and American options European and American options The most popular examples of derivative securities are European and American call and put options. More examples of contingent claims are listed in Section 1.6 below. A European call option on the asset S i is a contract where the seller promises to deliver the risky asset S i at the maturity T for some given exercise price, or strike, K >. At time T, the buyer has the possibility (and not the obligation) to exercise the option, i.e. to buy the risky asset from the seller at strike K. Of course, the buyer would exercise the option only if the price which prevails at time T is larger than K. Therefore, the gain of the buyer out of this contract is B = (S i T K) + = max{s i T K, }, i.e. if the time T price of the asset S i is larger than the strike K, then the buyer receives the payoff ST i K which corresponds to the benefit from buying the asset from the seller of the contract rather than on the financial market. If the time T price of the asset S i is smaller than the strike K, the contract is worthless for the buyer. A European put option on the asset S i is a contract where the seller promises to purchase the risky asset S i at the maturity T for some given exercise price, or strike, K >. At time T, the buyer has the possibility, and not the obligation, to exercise the option, i.e. to sell the risky asset to the seller at strike K. Of course, the buyer would exercise the option only if the price which prevails at time T is smaller than K. Therefore, the gain of the buyer out of this contract is B = (K S i T ) + = max{k S i T, }, i.e. if the time T price of the asset S i is smaller than the strike K, then the buyer receives the payoff K ST i which corresponds to the benefit from selling the asset to the seller of the contract rather than on the financial market. If the time T price of the asset S i is larger than the strike K, the contract is worthless for the buyer, as he can sell the risky asset for a larger price on the financial market. An American call (resp. put) option with maturity T and strike K > differs from the corresponding European contract in that it offers the possibility to be exercised at any time before maturity (and not only at the maturity). The seller of a derivative security requires a compensation for the risk that he is bearing. In other words, the buyer must pay the price or the premium for the benefit of the contrcat. The main interest of this course is to determine this price. In the subsequent sections of this introduction, we introduce the no dominance principle which already allows to obtain some model-free properties of options which hold both in discrete and continuous-time models. In the subsequent sections, we shall consider call and put options with exercise price (or strike) K, maturity T, and written on a single risky asset with

12 12 CHAPTER 1. INTRODUCTION price S. At every time t T, the American and the European call option price are respectively denoted by C(t, S t, T, K) and c(t, S t, T, K). Similarly, the prices of the American and the Eurpoean put options are respectively denoted by P (t, S t, T, K) and p(t, S t, T, K). The intrinsic value of the call and the put options are respectively: C(t, S t, t, K) = c(t, S t, t, K) = (S t K) +. P (t, S t, t, K) = p(t, S t, t, K) = (K S t ) +, i.e. the value received upon immediate exercing the option. An option is said to be in-the-money (resp. out-of-the-money) if its intrinsic value is positive. If K = S t, the option is said to be at-the-money. Thus a call option is in-themoney if S t > K, while a put option is in-the-money if S t < K. Finally, a zero-coupon bond is the discount bond defined by the fixed income 1 at the maturity T. We shall denote by B t (T ) its price at time T. Given the prices of zero-coupon bonds with all maturity, the price at time t of any stream of deterministic payments F 1,..., F n at the maturities t < T 1 <... < T n is given by F 1 B t (T 1 ) F n B t (T n ). 1.2 No dominance principle and first properties We shall assume that there are no market imperfections as transaction costs, taxes, or portfolio constraints, and we will make use of the following concept. No dominance principle Let X be the gain from a portfolio strategy with initial cost x. If X in every state of the world, Then x. 1 Notice that, choosing to exercise the American option at the maturity T provides the same payoff as the European counterpart. Then the portfolio consisting of a long position in the American option and a short position in the European counterpart has at least a zero payoff at the maturity T. It then follows from the dominance principle that American calls and puts are at least as valuable as their European counterparts: C(t, S t, T, K) c(t, S t, T, K) and P (t, S t, T, K) p(t, S t, T, K) 2 By a similar easy argument, we now show that American and European call (resp. put) options prices are decreasing (resp. increasing) in the exercise price,

13 1.2. No dominance and first properties 13 i.e. for K 1 K 2 : C(t, S t, T, K 1 ) C(t, S t, T, K 2 ) and c(t, S t, T, K 1 ) c(t, S t, T, K 2 ) P (t, S t, T, K 1 ) P (t, S t, T, K 2 ) and p(t, S t, T, K 1 ) p(t, S t, T, K 2 ) Let us justify this for the case of American call options. If the holder of the low exrecise price call adopts the optimal exercise strategy of the high exercise price call, the payoff of the low exercise price call will be higher in all states of the world. Hence, the value of the low exercise price call must be no less than the price of the high exercise price call. 3 American/European Call and put prices are convex in K. Let us justify this property for the case of American call options. For an arbitrary time instant u [t, T ] and λ [, 1], it follows from the convexity of the intrinsic value that λ (S u K 1 ) + + (1 λ) (S u K 2 ) + (S u λk 1 + (1 λ)k 2 ) +. We then consider a portfolio X consisting of a long position of λ calls with strike K 1, a long position of (1 λ) calls with strike K 2, and a short position of a call with strike λk 1 + (1 λ)k 2. If the two first options are exercised on the optimal exercise date of the third option, the resulting payoff is non-negative by the above convexity inequality. Hence, the value at time t of the portfolio is non-negative. 4 We next show the following result for the sensitivity of European call options with respect to the exercise price: B t (T ) c (t, S t, T, K 2 ) c (t, S t, T, K 1 ) K 2 K 1 The right hand-side inequality follows from the decrease of the European call option c in K. To see that the left hand-side inequality holds, consider the portfolio X consisting of a short position of the European call with exercise price K 1, a long position of the European call with exercise price K 2, and a long position of K 2 K 1 zero-coupon bonds. The value of this portfolio at the maturity T is X T = (S T K 1 ) + + (S T K 2 ) + + (K 2 K 1 ). By the dominance principle, this implies that c (S t, τ, K 1 ) + c (S t, τ, K 2 ) + B t (τ)(k 2 K 1 ), which is the required inequality. 5 American call and put prices are increasing in maturity, i.e. for T 1 T 2 : C(t, S t, T 1, K) C(t, S t, T 2, K)and P (t, S t, T 1, K 1 ) P (t, S t, T 2, K 2 ) This is a direct consequence of the fact tat all stopping strategies of the shorter maturity option are allowed for the longer maturity one. Notice that this argument is specific to the American case.

14 14 CHAPTER 1. INTRODUCTION 1.3 Put-Call Parity When the underlying security pays no income before the maturity of the options, the prices of calls and puts are related by p(t, S t, T, K) = c(t, S t, T, K) S t + KB t (T ). Indeed, Let X be the portfolio consisting of a long position of a European put option and one unit of the underlying security, and a short position of a European call option and K zero-coupon bonds. The value of this portfolio at the maturity T is X T = (K S T ) + + S T (S T K) + K =. By the dominance principle, the value of this portfolio at time t is non-negative, which provides the required inequality. Notice that this argument is specific to European options. We shall se in fact that the corresponding result does not hold for American options. Finally, if the underlying asset pays out some dividends then, the above argument breaks down because one should account for the dividends received by holding the underlying asset S. If we assume that the dividends are known in advance, i.e. non-random, then it is an easy exercise to adapt the put-call parity to this context. However, id the dividends are subject to uncertainty as in real life, there is no direct way to adapt the put-call parity. 1.4 Bounds on call prices and early exercise of American calls 1 From the monotonicity of American calls in terms of the exercise price, we see that c(s t, τ, K) C(S t, τ, K) S t When the underlying security pays no dividends before maturity, we have the following lower bound on call options prices: C(t, S t, T, K) c(t, S t, T, K) (S t KB t (T )) +. Indeed, consider the portfolio X consisting of a long position of a European call, a long position of K T maturity zero-coupon bonds, and a short position of one share of the underlying security. The required result follows from the observation that the final value at the maturity of the portfolio is non-negative, and the application of the dominance principle. 2 Assume that interest rates are positive. Then, an American call on a security that pays no dividend before the maturity of the call will never be exercised early. Indeed, let u be an arbitrary instant in [t, T ), - the American call pays S u K if exercised at time u,

15 1.5. Risk effect on options 15 - but S u K < S KB u (T u) because interest rates are positive. - Since C(S u, u, K) S u KB u (T u), by the lower bound, the American option is always worth more than its exercise value, so early exercise is never optimal. 3 Assume that the security price takes values as close as possible to zero. Then, early exercise of American put options may be optimal before maturity. Suppose the security price at some time u falls so deeply that S u < K KB u (T ). - Observe that the maximum value that the American put can deliver when if exercised at maturity is K. - The immediate exercise value at time u is K S u > K [K KB u (T u)] = KB u (T u) the discounted value of the maximum amount that the put could pay if held to maturity, Hence, in this case waiting until maturity to exercise is never optimal. 1.5 Risk effect on options prices 1 The value of a portfolio of European/american call/put options, with common strike and maturity, always exceeds the value of the corresponding basket option. Indeed, let S 1,..., S n be the prices of n security, and consider the portfolio composition λ 1,..., λ n. By sublinearity of the maximum, n λ i max { Su i K, } { n } max λ i Su i K, i=1 i.e. if the portfolio of options is exercised on the optimal exercise date of the option on the portfolio, the payoff on the former is never less than that on the latter. By the dominance principle, this shows that the portfolio of options is more maluable than the corresponding basket option. 2 For a security with spot price S t and price at maturity S T, we denote its return by R t (T ) := S T S t. Definition Let Rt(T i ), i = 1, 2 be the return of two securities. We say that security 2 is more risky than security 1 if i=1 R 2 t (T ) = R 1 t (T ) + ε where E [ ε R 1 t (T ) ] =. As a consequence, if security 2 is more risky than security 1, the above definition implies that Var [ R 2 t (T ) ] = Var [ R 1 t (T ) ] + Var[ε] + 2Cov[R 1 t (T ), ε] = Var [ R 1 t (T ) ] + Var[ε] Var [ R 1 t (T ) ]

16 16 CHAPTER 1. INTRODUCTION 3 We now assume that the pricing functional is continuous in some sense to be precised below, and we show that the value of an European/American call/put is increasing in its riskiness. To see this, let R := R t (T ) be the return of the security, and consider the set of riskier securities with returns R i := R i t(t ) defined by R i = R + ε i where ε i are iid and E [ε i R] =. Let C i (t, S t, T, K) be the price of the American call option with payoff ( S t R i K ) +, and C n (t, S t, T, K) be the price of the basket option defined by the payoff ( 1 n n i=1 S tr i K ) + ( = ST + 1 n n i=1 S tε i K ) +. We have previously seen that the portfolio of options with common maturity and strike is worth more than the corresponding basket option: C 1 (t, S t, T, K) = 1 n n C i (t, S t, T, K) C n (t, S t, T, K). i=1 Observe that the final payoff of the basket option C n (T, S T, T, K) (S T K) + a.s. as n by the law of large numbers. Then assuming that the pricing functional is continuous, it follows that C n (t, S t, T, K) C(t, S t, T, K), and therefore: that C 1 (t, S t, T, K) C(t, S t, T, K). Notice that the result holds due to the convexity of European/American call/put options payoffs. 1.6 Some popular examples of contingent claims Example 1.1. (Basket call and put options) Given a subset I of indices in {1,..., n} and a family of positive coefficients (a i ) i I, the payoff of a Basket call (resp. put) option is defined by ( + ( B = a i ST i K) resp. K ) + a i ST i. i I i I Example 1.2. (Option on a non-tradable underlying variable) Let U t (ω) be the time t realization of some observable state variable. Then the payoff of a call (resp. put) option on U is defined by B = (U T K) + resp. (K U T ) +. For instance, a Temperature call option corresponds to the case where U t is the temperature at time t observed at some location (defined in the contract).

17 1.6. Examples of contingent claims 17 Example 1.3. (Asian option)an Asian call option on the asset S i with maturity T > and strike K > is defined by the payoff at maturity: ( S i T K) +, where S i T is the average price process on the time period [, T ]. With this definition, there is still choice for the type of Asian option in hand. One can define S i T to be the arithmetic mean over of given finite set of dates (outlined in the contract), or the continuous arithmetic mean... Example 1.4. (Barrier call options) Let B, K > be two given parameters, and T > given maturity. There are four types of barrier call options on the asset S i with stike K, barrier B and maturity T : When B > S : an Up and Out Call option is defined by the payoff at the maturity T : UOC T = (S T K) + 1 {max[,t ] S t B}. The payoff is that of a European call option if the price process of the underlying asset never reaches the barrier B before maturity. Otherwise it is zero (the contract knocks out). an Up and In Call option is defined by the payoff at the maturity T : UIC T = (S T K) + 1 {max[,t ] S t>b}. The payoff is that of a European call option if the price process of the underlying asset crosses the barrier B before maturity. Otherwise it is zero (the contract knocks out). Clearly, UOC T + UIC T = C T is the payoff of the corresponding European call option. When B < S : an Down and In Call option is defined by the payoff at the maturity T : DIC T = (S T K) + 1 {min[,t ] S t B}. The payoff is that of a European call option if the price process of the underlying asset never reaches the barrier B before maturity. Otherwise it is zero (the contract knocks out).

18 18 CHAPTER 1. INTRODUCTION an Down and Out Call option is defined by the payoff at the maturity T : DOC T = (S T K) + 1 {min[,t ] S t<b}. The payoff is that of a European call option if the price process of the underlying asset crosses the barrier B before maturity. Otherwise it is zero (the contract knocks out). Clearly, DOC T + DIC T = C T is the payoff of the corresponding European call option. Example 1.5. (Barrier put options) Replace calls by puts in the previous example

19 Chapter 2 A first approach to the Black-Scholes formula 2.1 The single period binomial model We first study the simplest one-period financial market T = 1. Let Ω = {ω u, ω d }, F the σ-algebra consisting of all subsets of Ω, and P a probability measure on (Ω, F) such that < P(ω u ) < 1. The financial market contains a non-risky asset with price process S = 1, S 1(ω u ) = S 1(ω d ) = e r, and one risky asset (d = 1) with price process S = s, S 1 (ω u ) = su, S 1 (ω d ) = sd, where s, r, u and d are given strictly positive parameters with u > d. Such a financial market can be represented by the binomial tree : time time 1 Risky asset S = s Su Sd Non-risky asset S = 1 R = e r In the terminology of the Introduction Section 1, the above model is the simplest wrong model which illustrates the main features of the valuation theory in financial mathematics. The discouted prices are defined by the value of the prices relative to the nonrisky asset price, and are given by S := S, S := 1, and S1 := S 1 R, S 1 := 1. 19

20 2 CHAPTER 2. FIRST APPROACH TO BLACK-SCHOLES A self-financing trading strategy is a pair (x, θ) R 2. The corresponding wealth process at time 1 is given by : or, in terms of discounted value X x,θ 1 := (x θs )R + θs 1, X x,θ 1 := x + θ( S 1 S ). (i) The No-Arbitrage condition : An arbitrage opportunity is a portfolio strategy θ R such that X,θ 1 (ω i), i {u, d}, and P[X,θ 1 > ] >. It can be shown that excluding all arbitrage opportunities is equivalent to the condition d < R < u. (2.1) Exercise 2.1. In the context of the present one-period binomial model, prove that the no-arbitrage condition is equivalent to (2.1). then, introducing the equivalent probability measure Q defined by Q[S 1 = us ] = 1 Q[S 1 = ds ] = q := R d u d, (2.2) we see that the discounted price process satisfies S is a martingale under Q, i.e. E Q [ S 1 ] = S. (2.3) The probability measure Q is called risk-neutral measure, or equivalent martingale measure. (ii) Hedging contingent claims : A contingent claim is defined by its payoff B u := B(ω u ) and B d := B(ω d ) at time 1. In the context of the binomial model, it turns out that there exists a pair (x, θ ) R A such that X x,θ T = B. Indeed, the equality X x,θ 1 = B is a system of two (linear) equations with two unknowns which can be solved straightforwardly : x (B) = q B u R + (1 q)b d R = EQ [ B] and θ (B) = B u B d su sd. The portfolio (x (B), θ (B)) satisfies X x,θ T perfect replication strategy for B. = B, and is therefore called a (iii) No arbitrage valuation : Suppose that the contingent claim B is available for trading at time with market price p(b), and let us show that, under

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