Derivatives Pricing and Stochastic Calculus

Transcription

1 Derivatives Pricing and Stochastic Calculus Romuald Elie LAMA, CNRS UMR 85 Université Paris-Est Marne-La-Vallée ensae.fr Idris Kharroubi CEREMADE, CNRS UMR 7534, Université Paris Dauphine ceremade.dauphine.fr October 11, 213

2 Contents 1 Probability reminder Generalities on probability spaces Measure theory and probability space Moments Gaussian random variables Scalar Gaussian variables Multivariate Gaussian variables Conditional expectation Sub-σ-algebra and conditioning Computations of conditional expectations Arbitrage theory Assumptions on the market The notion of arbitrage Portfolio comparison under (NFL) Call-Put parity relation Valuation of a forward contract Exercises Put and Call options Currency forward contract Binomial model with a single period Probabilistic model for the market Simple portfolio strategy Risk neutral probability Valuation and hedging of contingent claims Exercices Pricing of a call and a put option at the money Binomial tree with a single period Binomial model with multiple periods Some facts on discrete time processes and martingales

3 4.2 Market model Portfolio strategy Arbitrage and risk neutral probability Duplication of contingent claims Valuation et hedging of contingent claims Exercices Martingale transforms Trinomial model Stochastic Calculus with Brownian Motion General facts on random processes Random processes L p spaces Filtration Martingale Gaussian processes Brownian motion Total and quadratic variation Stochastic integration Ito s formula Ito s processes Stochastic differential equation Exercices Brownian bridge SDE with affine coefficients Ornstein-Ulhenbeck process Black & Scholes Model Assumptions on the market Probabilistic model for the market Risk neutral probability Self-financing portfolios Duplication of a financial derivative Black & Scholes formula Greeks Exercices Option valuation in the Black & Scholes model Put option with mean strike Asian Option

4 Chapter 1 Probability reminder 1.1 Generalities on probability spaces Measure theory and probability space Definition (σ-algebra). Let Ω be a set with P(Ω) the class of all subsets of Ω. A σ-algebra A on Ω is a sub-class of P(Ω) satisfying Ω, A, if A A then A c := {ω Ω : ω / A} A, if (A n ) n 1 is a sequence of elements of P(A) such that A n A and A n A n+1 for all n 1 then n 1 A. The couple (Ω, A) with A such a σ-algebra on Ω is called a measurable space. Proposition Let C be a class of elements of P(Ω). Then there exists a smallest σ-algebra σ(c) containing C. It is given by σ(c) = T. σ(c) is called the σ-algebra generated by C. {T σ algebra : C T } Example If we take Ω = R and C = {[x, ), x R} then σ(x) is called the Borel σ-algebra and is denoted by B(R). Il is also the σ-algebra generated by the open (resp. closed) subsets of R. 3

5 Definition (Probability measure). Let (Ω, A) be a measurable space. A probability measure P on (Ω, A) is a function P : A [, 1 such that for any sequence (A n ) n 1 of elements of A satisfying A i A j =, for any i j, we have ( P n 1 A n ) = lim N N n=1 P(A n ) The triplet (Ω, A, P) is called a probability space. Definition (independence). Let (Ω, A, P) be a probability space (i) Let A and B two elements of A. We say that A and B are independent if P(A, B) := P(A B) = P(A) P(B). (ii) Let B and C be two sub-σ-algebra of A. We say that B and C are independent if any B B and C C are independent. Definition (Measurability and random variables). Let (Ω, A, P) be a probability space. A function X : Ω R is A-measurable if for any x R we have X 1( [x, + ) ) := { ω Ω : X(ω) x } A. A random variable X on the probability space (Ω, A, P) is an A-measurable function. Proposition Let (Ω, A, P) be a probability space and X be a random variable on (Ω, A, P). Denote by σ(x) the σ-algebra σ ( {X 1 ([x, + ) : x R} ) (see Proposition 1.1.1). Then σ(x) is the smallest sub-σ-algebra of A such that X is σ(x)-measurable. Definition Let X and Y be two random variables defined on a probability space (Ω, A, P). We say that X and Y are independent if σ(x) and σ(y ) are independent. Proposition Let X and Y be two random variables on a probability space (Ω, A, P). Y is σ(x)-measurable if and only if it can be written f(x) with f : R R a B(R)-measurable function. Proof. Suppose that Y = f(x). Then, for any borel subset B of R we have f(x) 1 (B) = X 1 ( f 1 (B) ) σ(x). 4

6 Conversly, if Y is the indicator function of a σ(x)-measurable set A, A can then be written A = X 1 (B) for B B(R d ) and, we have Y = 1 A = 1 X 1 (B) = 1 B (X), and the identification Y = f(x) holds with f = 1 B. if Y is a finite sum of indicator functions 1 Ai with A i = σ(x), A i can then be written A i = X 1 (B i ) for B i B(R) andthe equality holds with f = i 1 B i convient. If Y is positive, it can be written as an increasing limit of random variables Y n which are finite sum of indicator functions. We then have Y n = f n (X) and we get Y = f(x) with f = limf n. If Y is not positive, we apply the previous result to its positive and negative parts Moments Definition Let X be a positive random variable on a probability space (Ω, A, P). The expectation E[X of X is defined by E[X = X(ω)dP(ω). Ω Let X be a random variable such that E[ X < + (we say that X is integrable). The expectation E[X of X is defined by E[X = E[X + E[X, where X + and X are the random variables defined by X + = max(x, ) and X = max( X, ) Definition For p 1, we define the space L p (Ω, A, P) (or L p (Ω) for short) by { L p (Ω) = X random variable on (Ω, A, P) such that E [ X p } <. Proposition For p 1, the space L p (Ω) endowed with the norm p defined by is a Banach (i.e. complete vector) space. X p = ( E[ X p ) p, X L p (Ω) Proposition (Jensen Inequality). Let X be a random variable and ϕ : R R be a convex function. Suppose that E[ ϕ(x) <, then we have ϕ ( E[X ) E [ ϕ(x). Definition Let (X n ) n be a sequence of random variables and X random variable on (Ω, A, P). We say that X n converges P-a.s. to X and we write X n X P-a.s. if P( {ω Ω : X n (ω) X(ω) as n }) = 1 5

7 Theorem (Dominated Convergence Theorem). Let Let (X n ) n be a sequence of random variables and X random variable on (Ω, A, P) such that X L p (Ω) and X n L p (Ω) for all n. Suppose that there exists a random variable Y L p (Ω) such that X Y and X n Y. If then X n X as n + P a.s. X n X as n + in L p (Ω). 1.2 Gaussian random variables We fix in the rest of this chapter a probability space (Ω, A, P) Scalar Gaussian variables Definition A real random variable X is a standard Gaussian random variable if it admits the density f X given by 1 ( ) f X (x) = exp x2, x R. 2π 2 We recall that a random variable X admits a density f if for any Borel bounded function ϕ we have [ E ϕ(x) = ϕ(x)f(x)dx. We also notice that for a standard Gaussian random variable X we have E[X = and E[X 2 = 1. Definition Let m and σ > be two constants. A random variable Y is a Gaussian random variable with mean m and variance σ 2 if the random variable Y m is a standard Gaussian random σ variable. We then write Y N (m, σ 2 ). If Y N (m, σ 2 ) then Y admits the density f Y given by Definition f Y (y) = Multivariate Gaussian variables R 1 ( exp y ) m 2, x R. 2πσ 2 2σ 2 Definition A random vector (X 1,..., X n ) is a Gaussien vector if any linear combination of the components X i is a Gaussian random variable i.e. the random variable a, X defined by n a, X := a i X i k=1 is a Gaussian random variable for all a = (a 1,..., a n ) R n. 6

8 We now give two useful properties related to Gaussian vectors. Proposition Let (X 1,..., X n ) be a Gaussian vector. The components X i and X j are independent if and only if cov(x i, X j ) = for any i, j {1,..., n}. Proposition Let X = (X 1,..., X n ) be a random vector such that the components X 1,..., X n are independent and follows Gaussian laws. Then X is a Gaussian vector Remark One should take care about the fact that it is possible to have Gaussian random variables X 1 and X 2 which are not independent and satisfy cov(x 1, X 2 ) =. 1.3 Conditional expectation Sub-σ-algebra and conditioning Theorem Let B be a sub-σ algebra of A. For any integrable random variable X, there exists an integrable B-measurable random variable Y such that E [ 1 B X = E [ 1 B Y for any B B. If Ỹ is another random variable with these properties we have Ỹ = Y, P-a.s. The almost surely uniquely defined random variable Y given by the previous theorem is called the conditional expectation of X given B and is denoted by E[X B. This conditional expectation satisfies the following properties. Proposition Let B be a sub-σ algebra of A. (i) If X is an integrable B-measurable random variable we have E[X B = X. (ii) If X is a random variable and Za bounded B-measurable random variable we have In particular, we have ZE [ X B = E [ ZX B. E [ ZE[X B = E [ ZX. (iii) Linearity: if X 1 and X 1 are two random variables and a 1 and a 2 two constants we have E [ X 1 a 1 + X 2 a 2 B = a 1 E [ X 1 B + a 2 E [ X 2 B. (iv) If C is a sub-σ-algebra of B and X a random variable we have E [ E[X B C = E [ X C. 7

9 (v) Conditional Jensen Inequality: if ϕ : R R is a convex function X a random variable such that ϕ(x) is integrable we have E [ ϕ(x) B ( ϕ E [ X B ). (vi) Conditional dominated convergence theorem: Let (X n ) n be a sequence of random variables and X and Y two random variables such that and then if Y L p (Ω) we have X n Y, for all n, X n X as n + P a.s. E [ X n B E [ X B as n + in L p (Ω) Computations of conditional expectations Proposition Let B be a sub-σ algebra of A. Let X be a random variable independent of B. Then we have E [ X B = E [ X. Proposition Let B be a sub-σ algebra of A. Let X and Y be two random variables such that X is independent of B and Y is B-measurable. Then if Φ is a measurable function such that Φ(X, Y ) is integrable we have where the function ϕ : R R is defined by by for all y R. E [ Φ(X, Y ) B = ϕ(y ) ϕ(y) = E[Φ(X, y) 8

10 Chapter 2 Arbitrage theory 2.1 Assumptions on the market In the sequel, we shall make the following assumptions. The assets are infinitely divisible: One can sell or buy an proportion of asset. Liquidity of the market: one can sell or buy any quantity of any asset at any time. One can short sell any asset. There is no transaction costs. One can borrow and lend money at the same interest rate r. 2.2 The notion of arbitrage To model the uncertainty of financial market, we use the probability theory. We consider a probability set Ω endowed with a probability measure P. We now give the definition of a portfolio. Definition A self-financing portfolio consists in a initial capital x and an investment strategy without bringing or consuming any wealth. For a portfolio X, we denote by X t its value at time t. We now consider the situation where one can earn money with no risk of loss. Definition An arbitrage between times and T is a self financing portfolio X with initial value X = and such that P(X T ) = 1 and P(X T > ) >. 9

11 We now introduce an assumption which ensures that such an opportunity does not happened. (NFL) (No Free Lunch) There is no arbitrage opportunity between times and T : for any portfolio X, we have [ X = and X T P(X T > ) =. This assumption simply means that there is no way to earn money without taking any risk. This assumption also correspond to the reality of the markets. Indeed, even if an arbitrage opportunity appears in a market, it is quickly taken by a trader. 2.3 Portfolio comparison under (NFL) Proposition Let X and Y be two self-financing portfolios. Under (NFL), we have X T = Y T P a.s. = X = Y. Proof. Suppose that X < Y and consider the following strategy. At time t =, we buy X, we sell Y and we put Y X > in the bank account at interest rate r. At the terminal time t = T the portfolio value is always positive. value at value at T Buy X X X T Sell Y Y Y T Put the difference in the bank account Y X > (Y X )/B(, T ) > Portfolio value > Therefore, (NFL) implies X Y. Replacing X by Y, we get X Y and X = Y. Remark To construct an arbitrage opportunity we buy the cheapest portfolio and sell the most expensive one. Since the portfolios have the same terminal value, this strategy provides a positive gain. Proposition Let X and Y be two self-financing portfolios. Under (NFL), we have X T = Y T P a.s. = X t = Y t P a.s. for all t T. This result is a direct consequence of the following proposition. Proposition Let X and Y be two self-financing portfolios. Under (NFL), we have X T Y T = X t Y t for all t T. 1

12 Proof. Fix t T and consider the following strategy: at time : we do nothing. at time t: on the set {ω Ω, X t (ω) > Y t (ω)}, we buy Y at price Y t, we sell X at price X t and we put the difference X t Y t > in the bank account, on the set {ω Ω, X t (ω) Y t (ω)}, we do nothing. Finaly, at time T, on {X t > Y t }, the portfolio contains Y T X T and the positive amount in the bank account, thus it has a positive value, on {X t Y t }, the portfolio value is equal to zero. at time t t en T On {X t > Y t } Buy Y at time t Y t Y T Sell X at time t X t X T Put the difference in the bank account X t Y t > (X t Y t )/B(t, T ) > Portfolio value > On {X t Y t } Portfolio value Therefore (NFL) implies P(X t > Y t ) =. 2.4 Call-Put parity relation A call option with strike K and maturity T on the underlying S is a contract that provides the payoff (S T K) + =: max{s T K, } at time T. We denote by C t the price of such a contract at time t. A put option with strike K and maturity T on the underlying S is a contract that provides the payoff (K S T ) + =: max{k S T, }. We denote by P t its price at time t. A zero-coupon bond with maturity T is a financial product with value 1 at time T. We denote by B(t, T ) its price at time t. We then have the following relation between C, P and B. Proposition Under (NFL), the call and put prices satisfy the parity relation for all t T. C t P t = S t KB(t, T ) 11

13 Proof. Consider the two following self-financing strategies: at at t at T Port. 1 Buy a put option at t P t (K S T ) + Buy the underlying asset at t S t S T Value P t + S t (K S T ) + + S T Port. 2 Buy a call option at t C t (S T K) + Buy K units of zero-coupon bond at t KB(t, T ) K Value C t + KB(t, T ) (S T K) + + K We then notice that (K S T ) + + S T = K 1 {ST K} + S T 1 {K ST } = (S T K) + + K. Therefore, these two portfolios have the same terminal value. Under (NFL), we get from Proposition that they have the same value at every time t T, which gives the call-put parity relation. Remark This parity relation is specific to the (NFL) assumption. In particular it does not depend no the model that we put on the market. 2.5 Valuation of a forward contract A forward contract is a contract signed at time t = between two parties to buy or sell an asset at a specified future time at a price F (, T ) agreed upon today. There is no change of money between the two parties at time t =. To determine the forward price F (, T ) we consider the two self-financing strategies: at time at time T Port. 1 Buy a unit of the underlying S at time S S T Sell F (, T ) units of zero-coupons bounds at time F (, T )B(, T ) F (, T ) Value S F (, T )B(, T ) S T F (, T ) Port. 2 Buy the forward contrat at time S T F (, T ) Under (NFL), we get from Proposition Remark More generally, we have F (, T ) = S B(, T ). for all t T. F (t, T ) = 12 S t B(t, T )

14 2.6 Exercises Put and Call options We suppose that there is not any arbitratage opportunity on the market i.e. (NFL) holds. We denote by B the price at time t = of the riskless asset with gain 1 at maturity t = T. We also denote by C and P the respective prices at time t = of a Call and a Put option on the underlying S with maturity T and strike K. 1. Prove by an arbitrage argument that 2. Deduce from the previous question that (S KB ) + C S. (KB S ) + P KB. 3. Prove that the Call option price is a nonincreasing function of the strike K and a nondecreasing function of the maturity. 4. What can you say about the Put option price? Currency forward contract We consider on the time interval [, T the market of the currency pair euro/us dollar. This market can be described as follows. 1. In the european economy, there is a riskless asset with continuous time interest rate r d. This european riskless asset pays 1 at maturity T, thus its price at time t is given by B d t = e r d(t t) e for all t [, T. 2. In the american economy, there also exists a riskless asset with continuous time interest rate r f. This american riskless asset pays 1 at maturity T, thus its price at time t is given by B f t = e r f (T t) $ for all t [, T. 3. To get 1 dollar at time t [, T, one has to pay S t. 4. Finaly, on this market there exist forward contracts for any date t [, T. A forward contract signed at time t [, T is defined by the following rules: no change of money at the entering date t, at time T, we get 1 $ and we pay F t e, where the amount F t is fixed at the initial time t. 1. Fix t [, T. Using S T, give the pay-off a time T in euro, of the following portfolios constructed at time t. 13

15 (a) Buy B f t $ at time t. this amount is invested in the american riskless asset. Borrow F t Bt d time t. e at (b) Sign a forward contract at time t. 2. Give the pay-off in euro at time T of the portfolio composed by buying a forward contract with forward price F and the short selling at time t a forward contract with forward price F t. By a no free lunch argument, deduce the value f t in euro at time t of the forward contract signed at time as a function of F t and F and then as a function of S t and S. 14

16 Chapter 3 Binomial model with a single period The binomial model is very useful for the computation of prices and most of its properties can be generalised to the continuous time case. 3.1 Probabilistic model for the market We consider a market with two assets and two dates: t = et t = 1. A riskless asset S with value 1 at t = and value R = (1 + r) at t = 1. The dynamics of the riskless asset S is therefore given by S = 1 S 1 = R = 1 + r. A risky asset S with initial value S at t =. At time t = 1, S can take two different values: an upper value S u 1 = u.s and a lower value S d 1 = d.s where u and d are two constants such that d < u. S 1 = us S S 1 = ds The probabilistic model consists in three objects: Ω, F and P. Ω is the set of all the states of the world. There are two possible states, depending on the value that can take the risky asset at time t = 1. We therefore take Ω = {ω u, ω d } P is the historical probability on Ω. It is defined by P(ω u ) = p and P(ω d ) = 1 p. The price S 1 of the risky asset at time 1 has a probability p to go up and a probability 1 p to go down. We take p, 1[ to allow both events ω u and ω d to occur. F = {F, F 1 } is a couple of σ-algebras representing to global information available a times t = et t = 1. 15

17 At time t =, there is no information, so F is the trivial σ-algebra: F = {, Ω}. At time t = 1, one knows if the the asset S went up or down: F 1 = P(Ω) = {, Ω, {ω u }, {ω d }}. This σ-algebra represents all the subsets of Ω that can be said to have occurred or not at time t = 1. Remark We have F F 1. Indeed, we get more and more information with time. Remark F 1 is the σ-algebra generated by S 1 : F 1 = σ(s 1 ). Indeed, by definition, the σ-algebra generated by S 1 is the class of inverse images by S 1 of Borel subsets of R, i.e. {S1 1 (B), B B(R)}. It is also the smallest σ-algebra that makes S 1 measurable. For any Borel subset B of R, if us and ds belong to B, we have S 1 1 (B) = Ω, if us belongs to B, we have S 1 1 (B) = {ω u }, if ds belongs to B, we have S 1 1 (B) = {ω d }, and if any of these two values is in B, we have S 1 1 (B) =. Therefore F 1 is the σ-algebra generated by S 1. Definition A contingent claim (or financial derivative) is an F 1 -measurable random variable. The value of a contingent claim depends on the state of the world at time t = 1. From Proposition and Remark 3.1.6, any contigent claim C can be written C = φ(s 1 ) where φ : R R is a deterministic measurable function. For instance, if C is a Call option with strike K, we have C = φ(s 1 ) with φ : x R (x K) + We consider the problem of the valuation of a contingent claim at time t =. To this end we construct a hedging portfolio for our contigent claim. Under Assumption (NFL) we obtain that all the replicating portfolios have the same initial capital which is the economic definition of the price of the contingent claim. 16

18 3.2 Simple portfolio strategy Definition A simple portfolio strategy (x, ) consists in an initial amount x and a quantity (number of units) of risky asset S. We denote by X x, t the value of the associated self-financing portfolio at time t =, 1. For a simple portfolio strategy (x, ), we have at time t = X x, = S + (x S ) 1 = x. Since X x, is a self-financing portfolio, its value at time t = 1 is given by X x, 1 = S 1 + (x S )R = (S 1 S R) + x R. Such a strategy (x, ) is said to be simple since it consists in investing only in the assets S and S 1. Definition A contingent claim C is said to be replicated if there exists a simple strategy (x, ) such that X x, 1 = C P-a.s. Theorem The binomial model with a single period is complete: any contingent claim C can be replicated by a simple portfolio strategy (x, ). Proof. Consider a contingent claim C. We look for a couple couple (x, ) satisfying { C(ωu ) = S1 u + (x S )R = xr + (u R) S, C(ω d ) = S1 d + (x S )R = xr + (d R) S. Since u > d, this system of two equations with two unknowns is invertible. It therefore admits a unique solution (x, ) which is given by = C(ω u) C(ω d ) and x = 1 ( R d (u d)s R u d C(ω u) + u R ) u d C(ω d). Under (NFL) assumption, all the replication portfolios of the a contingent claim C have the same initial value given by C = 1 ( R d R u d C 1(ω u ) + u R ) u d C 1(ω d ). The economic definition of the price of a contingent claim at time t = is therefore C. 17

19 3.3 Risk neutral probability Definition A simple arbitrage opportunity is a simple portfolio strategy (x, ) with initial amount equal to zero, x =, and a nonnegative terminal wealth which is positive with positive probability: [ X, 1 et P X, 1 > > We then introduce the no free lunch assumption in this context. (NFL ) For any simple strategy (, ), we have [ X, 1 P a.s. = [ X, 1 = P a.s. Proposition Under (NFL ) we have d < R < u. Proof. Suppose that d R. An arbitrage strategy is then given by (, ) with = 1 (i.e. buying a unit of a risky asset). Indeed, at time t = 1, we have { X, 1 (ω u ) = S (u R) >, X, 1 (ω d ) = S (d R). Suppose that u R. An arbitrage strategy is then given by (, ) with = 1 (i.e. selling a unit of a risky asset). Indeed, at time t = 1, we have { X, 1 (ω u ) = S (R u), X, 1 (ω d ) = S (R d) >. For a simple portfolio strategy (x, ), let us introduce the discounted value X x, of the portfolio X x, defined by X x, t := Xx, t R t x, x, for t =, 1. We then have X = x and X 1 = S 1 + (x S ). The self-financing condition can be rewritten with the discounted values under the following form: X x, 1 X x, = ( S 1 S ). Definition A risk neutral probability is a probability measure Q defined on Ω and equivalent to P, under which any discounted value X x, of self-financing portfolio is a martingale, i.e. X x, = E Q x, [ X 1 or equivalently x = 1 R EQ [X x, 1. 18

20 Remark Two probability measures P and Q are said to be equivalent if they have the same negligible sets, i.e. P(A) > Q(A) >, for any event A. In the binomial, this simply means Q(ω d ) > and Q(ω u ) >. A first result is that we cannot have multiple risk neutral probabilities. Proposition Since the market is complete, there is at most a unique risk neutral probability. Proof. Indeed, suppose that we have two risk neutral probability measures Q 1 and Q 2. Fix B F 1 = P(Ω), and consider the contingent claim C = 1 B. Since the market is complete there exists a simple portfolio strategy (x, ) that replicates C. We then get Q 1 (B) = E Q 1 [1 B = E Q 1 [X x 1 = Rx Q 2 (B) = E Q 2 [1 B = E Q 2 [X x 1 = Rx. Therefore, Q 1 (B) = Q 2 (B), and since B is arbitrarily chosen in F 1 we have Q 1 = Q 2. Proposition Suppose that d < R < u, then there exists a unique risk neutral probability Q. Proof. Let us take a simple portfolio strategy (x, ). We have the following equations: { X x, 1 (ω u ) = S 1 (ω u ) + (x S )R = xr + (u R) S X x, 1 (ω d ) = S 1 (ω d ) + (x S )R = xr + (d R) S which give x = 1 R ( u R u d Xx, 1 (ω d ) + R d ) u d Xx, 1 (ω u ). (3.3.1) Introduce the probability measure Q defined on Ω by Q(ω u ) := R d u d := q and Q(ω d) := u R u d = 1 q. Since d < R < u, we get q, 1[ and Q is equivalent to P. From the definition of Q, equation rewrites x = 1 ( ) Q(ω d ) X x, 1 (ω d ) + Q(ω u ) X x, 1 (ω u ) = 1 R R EQ [X x, 1. The uniqueness of Q follows from Proposition Proposition Suppose that there exists a (unique) risk neutral probability Q, then assumption (NFL ) holds true. 19

21 Proof. Fix R such that X, 1. Since Q is a risk neutral probability, we have E Q [X, 1 = R. =. So, X, 1 is a nonnegative random variable with expectation under Q equal to zero. Therefore Q(X, 1 > ) =. Since P and Q are equivalent we get P(X, 1 > ) =. We have finaly proved (NFL ) = d < R < u = There exists a unique risk neutral probabilty = (NFL ), therefore, all this implications become equivalences (NFL ) d < R < u There exists a unique risk neutral probabilty 3.4 Valuation and hedging of contingent claims We can now give the definition of the price of a contingent claim. Theorem Let C be a contingent claim. Under (NFL ), all the strategies (x, ) that replicate C have the same initial capital P (C) given by P (C) = r EQ [C, where Q is the unique probability measure. P (C) is called the price of the contingent claim C. Proof. Let (x, ) and (x, ) be two simple strategies replicating C. Then we have X x, 1 = X x, 1 = C. Under (NFL ), there exists a unique risk neutral probability measure Q. Since X x, and X x, are martingales under Q we get Therefore, we get x = x. x = x = r EQ [X x, 1 = r EQ x, [X1 = r EQ [C, r EQ [C. The price P (C) can be expressed with the parameter q as follows: P (C) = 1 ( q C(ωu ) + (1 q) C(ω d ) ) = 1 R 1 + r EQ [C 2

22 Remark The risk neutral probability does not depend on the historical probability P. Therefore, The price of a contingent claim does not depend on the historical probability P. Remark To compute the price of a cotigent claim C, we only need to know the parameters r, u et d. The coefficients u et d correspond to what we call the volatility of the asset in the next chapters. Remark In the hedging portfolio of a contigent claim C, the number of shares of S held is given by: = C(ω ( ) u) C(ω d ) = φ(su 1 ) φ(s1) d, (u d)s S1 u S1 d which is the relative variation of the contingent claim w.r.t. the risky asset S. 3.5 Exercices Pricing of a call and a put option at the money Let us take S = 1, r =.5, d =.9 et u = What is the price and the hedging strategy of a Call option at the money i.e. K = S = 1? We compute the risk neutral probability: q = 1 + r d u d We deduce the price and the hedging strategy: P (C) = =.75. = and = Cu 1 C1 d = 1 (u d)s 2 =, 5 Hedging strategy: Buy 1/2 unit of risky asset S and put (7.14 5) in the riskless asset S. 2. What about the Put option at the money? Since we know the risk neutral probability, we can compute the price and the hedging strategy of the put option: P (P ) = = and = P 1 u P1 d = 1 (u d)s 2 =, 5 Hedging strategy: sell 1/2 unit of risky asset S and put ( ) in the riskless asset S. 3. Is the Call-Put parity relation satisfied? P (C) P (P ) = = = = 1 1 R = S KB(, T ).

23 3.5.2 Binomial tree with a single period We consider a financial market with two dates and composed by a riskless asset S and a risky asset S. The dynamics of these assets are given by Riskless asset: S = 1 S 1 = 15 Risky asset: S = 1 S 1 = 12 with probability.75 S 1 = 9 with probability Describe the probability space (Ω, F, P). 2. Give the definition of the risk neutral probability. Calculate it. 3. Calculate the prices of a Call and a Put Option with strike Recall the Call-Put parity and show that it is satisfied. 22

24 Chapter 4 Binomial model with multiple periods With this model, also called Cox Ross Rubinstein model, we get similar results to those obtained in the previous chapter for the one period binomial model. 4.1 Some facts on discrete time processes and martingales We fix in this section a probability space (Ω, A, P). Definition A (discrete time) process is a finite sequence of random variables (Y k ) k n defined on (Ω, A, P). Definition A (discrete time) filtration is a nondecreasing sequence (F k ) k n of σ-algebras included in A, i.e. for all k {,..., n 1}. F k F k+1 A, Definition A process (Y k ) k n, is adapted to the filtration F (or F-adapted) if Y k is F k -measurable for all k n. Proposition Let (Y k ) k n be an F-adapted process. Then, the random variable Y i is F k -measurable for all i k. Proof. In terms of information, if the result of the random variable Y i is included in the information given by F i, it is included in the information given by F k F i. In the formalism of measure theory, one notice that the reciprocal image Y 1 i (B) of any Borel set B by Y i is in F i. Since F i F k for k i, we get Y 1 i (B) F k. Definition The filtration generated by a process (Y k ) k n is the smallest filtration F Y such that (Y k ) k n is F Y -adapt. It is given by for all k {,..., n}. F Y k := σ(y,..., Y k ), 23

25 Remark A a consequence of measurability properties, a random variable is F Y k -measurable if and only if it can be written as a (Borel) function of (Y,..., Y k ). Definition A discrete time process (M k ) k n is an F-martingale under P if it satisfies: (i) M is F-adapted, (ii) E[ M k < for all k {,..., n}, (iii) E[M k+1 F k = M k for all k {,..., n 1}. Remark The process M is said to be a supermartingale (resp. submartingale) if it satisfies conditions (i) and (ii) of the previous definition and E[M k+1 F i (resp. ) M k for all k {,..., n 1}. The martingale (resp. supermartingale, submartingale) property means that the best estimate (in the least-square sense) of M k+1 given all the past is (resp. is lower than, is greater than) M k. Remark If M is an F-martingale under P we have E[M k F i = M i, for all i k. In particular E[M k = M for all k. 4.2 Market model We keep the same model as in the previous chapter but with n periods (n 2). We consider a time interval [, T divided in n periods = t < t 1 <... < t n = T. We suppose that the market is composed by two assets. The first one is a riskless asset S. We denote by S t k it value at time t k. We suppose that the dynamics of the process (S t k ) k n is given by: S = 1 S t 1 = (1 + r) S t 2 = (1 + r) 2... S T = (1 + r) n. The second one is a risky asset S. We denote by S tk it value at time t k. We fix two constants u > d >, and we suppose that the dynamics of the process (S tk ) k n is given by the following tree: 24

26 u n 1 S... u n S ds u 2 S du n 1 S... us... S uds d 2 S d n 1 us... d n 1 S d n S Since the tree is recombining, the risky asset can take i + 1 values at each time t i. To complete the description of the dynamics of S, we have to define the set Ω and the law of the evolution of the asset on each branch of the tree. Set of the possible states of the world: Ω is actually the set of possible trajectories for the process S. ω is therefore the set of n-tuples (ω 1,..., ω n ) such that each ω i can take only two possible values ω d i or ω u i. Ω := {(ω 1,..., ω n ) : ω i = ω d i ou ω i = ω u i, for all i = 1,..., n} We then are given an historical probability measure P for the occurrence of of each event. We assume that and we make the fundamental assumption: P(ω i = ω u i ) = p et P(ω i = ω d i ) = 1 p The risky the asset returns Y i := S t i S ti 1, i = 1,..., n, are independent random variables. From the previous option we deduce that: P(ω 1,..., ω n ) = p #{j,ω j=ω u j } (1 p) #{j,ω j=ω d j }. 25

27 We can the rewrite the dynamics of the risky asset S as follow: i S ti = S. Y k, i = 1,..., n, k=1 where Y 1,..., Y n are independent random variable defined on Ω such that: P(Y i = u) = P(ω i = ωi u ) = p and P(Y i = d) = P(ω i = ωi d ) = 1 p. The information available at each time t i, i =,..., n, is given by the filtration (F ti ) i n defined by and F t = {, Ω}, F ti := σ(y 1, Y 2,..., Y i ) = σ(s t, S t1, S t2,..., S ti ), i = 1,..., n. Thus, an F ti -measurable random variable is given by the information accumulated until time t i. It therefore can be written as a function of (S t1,..., S ti ) or equivalently as a function of (Y 1,..., Y i ). Definition A contingent claim (or financial derivative) C T variable. Such a contigent claim can therefore be written an F T -measurable random C T = φ(s t1,..., S tn ), with φ a Borel function. We now look for the price of such a financial derivative. As previously, we use the (NFL) property to show that all the hedging portfolios of this contingent claim have the same initial value which is the economic definition of the claim s price. 4.3 Portfolio strategy Definition A (simple) portfolio strategy (x, ) consists in an initial capital x and an F- adapted process (,..., n 1 ) where i represents the number of shares of risky asset S held by the investor at time t i, for i =,..., n 1. In the previous definition, we ask the process to be F-adapted since at each time t i, the investor does not have more information than F ti. Therefore, the investments done at time t i should be measurable w.r.t. the information F ti available at time t i. Denote by X x, t i the value of the portfolio with strategy (x, ) at time t i for i =,..., n. At time t i, the portfolio X x, contains i shares of S. Therefore, it contains (X x, i i S ti )/St i units of the riskless asset S. From the definition of S, this can be written as t i = i S ti + (Xx, t i i S ti ) (1 + r) i. (1 + r) i X x, 26

28 We suppose that the portfolio is self-financing i.e. no bringing or consuming any wealth. We therefore get at time t i+1. t i+1 = i S ti+1 + (Xx, t i i S ti ) (1 + r) i+1 (1 + r) i X x, Let us introduce the discounted processes: X x, t i := Xx, t i The previous equation can be rewritten as follow and (1 + r) S i ti := S t i. (1 + r) i X x, t i = i S ti + ( X x, t i i Sti ) 1, and Xx, t i+1 = i Sti+1 + ( X x, t i i Sti ). We then get the self-financing relation: X x, t i+1 X x, t i = i ( S ti+1 S ti ). Since this equation holds for any i =,..., n 1, we get by induction for all i =,..., n 1. X x, t i+1 = x + i k ( S tk+1 S tk ), k= Remark From the previous equation we deduce that the portfolio value process X x, is F-adapted. 4.4 Arbitrage and risk neutral probability Definition An arbitrage opportunity is a portfolio strategy (x, ) with initial amount equal to zero, x =, and a nonnegative terminal wealth which is positive with positive probability: [ X, t n et P X, t n > >. Recall that t n = T. In the binomial model with multiple periods, the no arbitrage opportunity assumption takes the following form. (NFL ) We have the following implication X, T = X, T = P a.s. for any F-adapted process = (,..., n 1 ). 27

29 Proposition Under Assumption (NFL), we have d < 1 + r < u. Proof. Suppose for instance that 1 + r d and consider the portfolio strategy (, ), where = 1 and i = for i = 1,..., n (we buy the risky asset at time t, we sell it at time t 1 and we put our gain in the riskless asset). The process is F-adapted since it is deterministic and the portfolio value at time T = t n is given by: X, T = + i k ( S tk+1 S tk ) = S t1 S t k= From the definition of S t1, X, T (1 + r) n S ( u 1 + r 1 can take only two possible values: ) ( d >, and (1 + r) n S 1 + r 1 ) with respective probabilities p > and 1 p >. This strategy is therefore an arbitrage opportunity. In the case u 1 + r we construct an arbitrage opportunity by considering the strategy = 1 and i = for i = 1,..., n 1. Remark As in the one period binomial model, if 1 + r u or 1 + r d one of the two assets S and S always provides a better gain than the other. Therefore, this generates an arbitrage opportunity. Remark Under Assumption (NFL), there is no arbitrage opportunity on every subtree. Indeed, if there is an arbitrage opportunity on a subtree, the strategy consisting in doing nothing out of the subtree, following the arbitrage strategy on the subtree, is a a global arbitrage opportunity, since the probability of reaching any sub tree is positive. Following the results obtained in the previous chapter, we introduce the probability measure Q on Ω equal to the risk neutral probability for the binomial model with single period on each one period subtree and keeping the random variables Y i independent. This probability measure is therefore given by Q(ω 1,..., ω n ) = q #{j,ω j=ω u j }.(1 q) #{j,ω j=ω d j } with q := Remark From the definition of the measure Q we have: (1 + r) d u d. Q(S ti = us ti 1 ) = Q(Y i = u) = q et Q(S ti = ds ti 1 ) = Q(Y i = d) = 1 q Definition A risk neutral probability is a probability measure equivalent to the historical probability P under which any simple portfolio discounted value process is a martingale. 28

30 We next aim at proving that the measure Q is a risk neutral probability. Proposition S is an F-martingale under Q. Proof. S is integrable and F-adapted. We then have E Q[ Ftk 1 Stk+1 = (q u 1 + r S tk + (1 q) d S ) tk ( 1 (1 + r) d u (1 + r) = u r u d u d for any k =,..., n 1. = S tk ) d S tk Proposition The discounted value X x, of a portfolio with strategy (x, ) is an F- martingale under Q. Proof. Xx, est intgrable F-adapted and E Q[ x, Xx, k+1 X [ k Ftk = E Q k ( S tk+1 S tk ) Ftk = k E Q[ ( S tk+1 S tk ) Ftk =. Remark The revious proposition tells us that any portfolio discounted value process is a martingale as soon as the discounted assets are martingales. This is mainly due to the fact that the number of shares of S held by the investor between t k and t k+1 is constant (from the self-financing condition) and F tk -measurable. This property can actually be extended to some transformation of martingales w.r.t. predictable processes called stochastic integration and which remains a martingale as we will see in the next chapter. Theorem Suppose that d < 1 + r < u. Then there exists a risk neutral probability (Q). Proof. We juste have seen that X x, is an F-martingale under Q for any strategy (x, ). Moreover Q is equivalent to P. Indeed since d < R < u, q and (1 q) are positive. Therefore Q(ω 1,..., ω n ) is positive for any (ω 1,..., ω n ) Ω. We now focus on the link between the existence of a risk neutral probability Q and Assumption (NFL). Proposition Suppose that there exists a risk neutral probability. Then Assumption (NFL ) is satisfiyed. Proof. Let Q be a risk neutral probability measure and (, ) a strategy such that, Since Q is a risk neutral probability, we have E Q[ X, T X, T = (1 + r) n X, = 29

31 Hence, we obtain Thus X, T Q(X, T = ) = 1. = Q-a.s. and also P-a.s. since Q and P are equivalent We therefore get the same result as in the binomial model with a single period: (NFL ) d < R < u there exists a unique risk neutral probability. At each time t i the value of a portfolio with strategy (x, )and with terminal value X x, T given by is X x, t i = 1 (1 + r) EQ[ X x, n i T Fi. Once we have a hedging portfolio for a contingent claim, we get from (NFL ) that its value at any time t i time is given by the conditional expectation given F ti of the discounted contingent claim under the risk neutral probability. We now concentrate on the existence of such a hedging portfolio. 4.5 Duplication of contingent claims Theorem any contingent claim C is duplicable by a portfolio strategy (x, ) i.e. there exists a strategy (x, ) such that X x, T = C. The market is said to be complete. Analysis of the problem. We look for a portfolio strategy (x, ) that replicates the contigent claim C at time T. Since C is F tn -measurable, it can be rewritten φ(s t1,..., S tn ) for some borel fuction φ. We therefore look for (x, ) such that X x, t n = φ(s t1,..., S tn ). Since any discounted portfolio value process is a martingale under the risk neutral probability Q the hedging portfolio process X x, satisfies X x, t k = 1 (1 + r) n k EQ[ φ(s t1,..., S tn ) Ftk, k =,..., n. In particular, its initial value x is necessarily given by x := 1 (1 + r) n EQ[ φ(s t1,..., S tn ). 3

32 We then notice that 1 (1+r) n k E Q [φ(s t1,..., S tn ) F tk, as an F tk -measurable random variable, can be rewritten as V k (S t1,..., S tk ) with V k a deterministic (i.e. nonrandom) function. We then have V k (S t1,..., S tk ) := 1 (1 + r) n k EQ[ φ(s t1,..., S tn ) Ftk, k =,..., n. In the previous chapter, we have seen that in the hedging portfolio, the number of shares of the risky asset held by the investor is equal to the relative variation of the contingent claim value w.r.t. the risky asset. We then propose the hedging process defined by k := V k+1(s t1,..., S tk, u.s tk ) V k+1 (S t1,..., S tk, d.s tk ) u.s tk d.s tk, for all k {1,..., n 1}. We notice that this process is F-adapted. Therefore (x, ) defines une a simple portfolio strategy. Solution of the problem. We now prove that the strategy (x, ) defined previously is a hedging strategy for the contingent claim C. To this end, we prove by induction that for all k {,..., n}. X x, t k = V k (S t1,..., S tk ) By construction of the strategy (x, ), we have x := Thus the result holds true for k =. 1 (1 + r) n EQ[ φ(s t1,..., S tn ) = V. Suppose that the result holds true for k and let us prove it for k + 1. We first notice that X x, t k = V k (S t1,..., S tk ) 1 = [ (1 + r) n k EQ φ(s t1,..., S tn ) Ftk [ 1 = [E (1 + r) EQ Q 1 = (1 + r) EQ[ V k+1 (S t1,..., S tk+1 ) F tk = = = 1 (1 + r) n (k+1) φ(s t 1,..., S tn ) Ftk+1 F tk 1 (1 + r) EQ[ V k+1 (S t1,..., S tk, u S tk ) 1 {Xk+1 =u} +V k+1 (S t1,..., S tk, d S tk ) 1 {Xk+1 =d} F tk 1 { Q(Xk+1 = u).v k+1 (S t1,..., S tk, u S tk ) (1 + r) +Q(X k+1 = d).v k+1 (S t1,..., S tk, d S tk ) } 1 { q.vk+1 (S t1,..., S tk, u S tk ) + (1 q).v k+1 (S t1,..., S tk, d S tk ) }. (1 + r) 31

33 The self-financing condition writes X x, t k+1 = X x, t k + k ( S tk+1 S tk ). Writing the undiscounted value we get X x, t k+1 = q.v k+1 (S t1,..., S tk, u S tk ) + (1 q).v k+1 (S t1,..., S tk, d S tk ) + V k+1 (S t1,..., S tk, u S tk ) V k+1 (S t1,..., S tk, d S tk ) ( ) Stk+1 (1 + r)s tk. u S tk d S tk Replacing S tk+1 by Y k+1 S tk and q by 1+r d, we deduce u d X x, t k+1 = V k+1 (S t1,..., S tk, u S tk ) Y k+1 d u d Since Y k+1 takes only the values d and u, we get alors + V k+1 (S t1,..., S tk, d S tk ) u Y k+1 u d. X x, t k+1 = V k+1 (S t1,..., S tk, Y k+1 S tk ) = V k+1 (S t1,..., S tk, S tk+1 ), which concludes the induction. A consequence of the completeness of the market is the unicity of the risk neutral probability measure. Proposition Since the market is complete there is at most one risk neutral probability measure. Proof. Let Q 1 and Q 2 be two risk neutral probability measures. For any B F T = P(Ω), 1 B is a contingent claim since it is F T -measurable. It can therefore be duplicated by a simple strategy (x, ) and we have Therefore Q 1 = Q 2. Q 1 (B) = E Q 1 [1 B = (1 + r) n x = E Q 2 [1 B = Q 2 (B). 4.6 Valuation et hedging of contingent claims Theorem Let C be a contingent claim. Under (NFL ), all the strategies (x, ) that replicate C have the same initial capital P (C) given by P (C) = 1 (1 + r) n EQ [C, where Q is the unique probability measure. P (C) is called the price of the contingent claim C. 32

34 Proof. Let (x, ) and (x, ) be two simple strategies replicating C. Then we have X x, t n = X x, t n = C. Under (NFL ), there exists a unique risk neutral probability measure Q. Since X x, and X x, are martingales under Q we get x = x = 1 (1 + r) n EQ [X x, t n = 1 (1 + r) n EQ x, [Xt n = 1 (1 + r) n EQ [C, 1 (1 + r) n EQ [C. Therefore, we get x = x. Finally, since the market is complete such a replicating strategy exists. Remark As in the binomial model with a single period the price of the contingent claim depends only on its payoff,u, r et d. In particular the price does not depend on the historical probability P!!! To remember: - Le price of a contingent claim can be written as the expectation of the discounted gain under the risk neutral probability Q. - Under the risk neutral probability, the discounted values of simple self-financing portfolios are martingales. In the case where n goes to infinity, and for a good choice of the parameters u d and r, we can prove that this model converges to a continuous time model called the Black & Scholes model. To study such a model, we need to define similar objects in continuous time. This will be done in the next chapter through the stochastic calculus theory. 4.7 Exercices Martingale transforms Let (X k ) k n be an F-martingale and (H k ) k n 1 be a bounded F-adapted process. We define the process (M k ) k n by M = x and M k := x + k H i 1 (X i X i 1 ), 1 k n. i=1 1. Prove that the process M is an F-martingale. 2. In the binomial model with n periods if the discounted risky asset is a martingale under the risk neutral probability, what can you say about the self financing strategies? 33

35 4.7.2 Trinomial model We consider an extended version of the Cox Ross Rubinstein model allowing the asset price to take three different values at each time step. We suppose that there are n periods and that the market is composed by two assets. The first one is a riskless asset S. We denote by S t k it value at time t k. We suppose that the dynamics of the process (S t k ) k n is given by: S = 1 S t 1 = (1 + r) S t 2 = (1 + r) 2... S T = (1 + r) n. The second one is a risky asset S. We denote by S tk it value at time t k. We fix three constants u > m > d > 1, and we suppose that the dynamics of the process (S tk ) k n is given by S tk = Y k S tk 1, 1 k n, where (Y k ) 1 k n is an IID sequence of random variable with law P(Y 1 = u) = p u, P(Y 1 = d) = p d and P(Y 1 = m) = p m := 1 p u p d, with p u, p m, p d (, 1). 1. (a) Suppose that 1 + r d. Construct an arbitrage opportunity. (b) Suppose that 1 + r u. Construct an arbitrage opportunity. (c) Give a necessary condition to ensure the viability of the market. 2. We look for a risk neutral probability measure. We firstly suppose that n = 1. (a) Denote by Q such a probability measure with Q(Y 1 = u) = q u, Q(Y 1 = d) = q d and Q(Y 1 = m) = q m := 1 q u q d, Use the martingale relation to get an equation satisfied by (q u, q m, q d ). E Q[ S1 = S (b) Give the expressions of q m and q d as functions of q u, r, u, d and m. (c) By distinguishing the cases m 1 + r and m < 1 + r prove that there exists an infinity of risk neutral probability under the condition u > 1 + r > d. Extend this result to a multiple period model. 3. (a) Prove that the existence of a risk neutral probability measure ensures the viability of the market. (b) Deduce a necessary and sufficient condition for the viability of the market. 34

36 Chapter 5 Stochastic Calculus with Brownian Motion We fix a complete probability space (Ω, A, P). 5.1 General facts on random processes Random processes Definition A (random) process X on the probability space (Ω, A, P) is a family of random variables. (X t ) t [,T. Such a process X can also be seen as a function of two variables X : { [, T Ω R (t, ω) X t (ω). The functions t X t (ω), for ω varying in Ω, are called the trajectories of the process X Remark Since we aim at representing the dynamic evolution of a sysem by a random process X, the variable t [, T correspond to the time. However, if one want to represent more general systems, t can be chosen as an element of R, R 2... The space where X is valued can also be more complex than R. Definition A random process X is a continuous process its trajectories t X t (ω) are continuous for almost every ω Ω, i.e. P( {ω Ω : t Xt (ω) is a continuous function }) = L p spaces Notation: For p R +, we denote by: } L p (Ω) := {X random variable on (Ω, A, P) such that X p := E[ X p 1 p < 35