Martingale Approach to Pricing and Hedging

Transcription

1 Introduction and echniques Lecture 9 in Financial Mathematics UiO-SK451 Autumn 15 eacher:s. Ortiz-Latorre Martingale Approach to Pricing and Hedging 1 Risk Neutral Pricing Assume that we are in the basic Black-Scholes model. We have one riskless asset and one risky asset with prices given, respectively, by B = fb t = e rt g t[; ] and S t = fs t g t[; ] satisfying ds t = S t + SdW t ; S > : We can setup strategies = f t ; 1 t g with value V t () = t B t + 1 t S t and that satisfy the self- nancing condition dv t () = t db t + 1 t ds t = re rt t dt + 1 t ds t ; or equivalently 1.1 Change of Numeraire V t () = V () + sdb s + 1 sds s : As the price processes are strictly positive, in particular B t > ; one can always normalize the market by considering ~B t = Bt 1 B t = 1 and S ~ t = Bt 1 S t = e rt S t : hus normalization corresponds to regarding the price B t of the safe investment (riskless asset) as the unit of price (the numeraire) and computing the other prices in terms of this unit. Alternatively, one can look at the normalized market as a discounted market where all assets are quoted (priced) in terms of its the present value. Moreover, we can consider the discounted portfolio ~V t () = B 1 t V t () = e rt t B t + 1 t S t = t + 1 t ~ S t ; and, applying integration by parts, one has that d V ~ t () = Bt 1 dv t () re rt V t ()dt + dbt 1 (dvt ()) {z } = = e rt dv t () r ~ V t ()dt: (1) If we assume that dv t () = t db t + 1 t ds t ; i.e., is self- nancing, then d V ~ t () = e rt fr t e rt dt + 1 t ds t g r n t + 1 ~ o t S t dt = 1 t e rt ds t r 1 t e rt S t dt = 1 t e rt ds t + d(e rt )S t = 1 t d S ~ t ; which yields that V ~ t () is self- nancing because d B ~ t = d(1) = : Using formula (1) and assuming that d V ~ t () = 1 t d S ~ t (i.e. is self- nancing in the normalized market) one gets that is self- nancing in the unnormalized market. Hence, self- nancing portfolios are invariant by a change of numeraire or discounting, in other words, a portfolio is self- nancing if and only if is self- nancing in the normalized market. From a nancial point of view this makes sense because the self- nancing property, that is, the fact that changes in the value of the portfolio are due only to changes in the asset prices, does not depend on which unit we measure the prices. Note that, in the discounted market, a self- nancing portfolio is written in integral form as ~V t () = V () + 1 sd ~ S s : 1 Last updated: November 3, 15

2 1. Equivalent Martingale Measures and Arbitrage he process M t = exp = exp r r dw s W t 1 r ds! r t! ; is a martingale with respect to P. By Girsanov s theorem, we can de ne a probability measure Q by setting dq dp = M and the process 1 ~W t = r t + W t; is a Brownian motion under Q: In addition, as M > ; we have that Q P: In the previous lecture, we showed that the dynamics of S under Q is that of a geometric Brownian motion with drift r and volatility ; i.e., ds t = rs t dt + S t d W ~ t : If we now compute d ~ S t ; we get that or in explicit form d S ~ t = d(e rt S t ) = re rt S t dt + e rt ds t = r S ~ t dt + e rt rs t dt + S t d W ~ t = r ~ S t dt + r ~ S t dt + ~ S t d ~ W t = ~ S t d ~ W t ; () ~S t = S ~ exp W ~ t t ; which is a martingale under Q: his motivates the following general de nition. De nition 1 An equivalent martingale measure (EMM) is a probability measure Q equivalent to P (P Q) such that the discounted price of any asset in the market is a martingale under Q: Remark We just have shown that in the basic Black-Scholes model there is at least one equivalent probability measure Q, given by! dq dp = exp r W 1 r : Note also that the discounted price of the riskless asset is constant and, hence, a martingale under any probability measure. Moreover, we have that ~V t () = ~ V () + 1 sd ~ S s = ~ V () + 1 s ~ S s d ~ W t ; and V ~ t () is a stochastic integral with respect to a Brownian motion under Q. Hence, under the integrability condition " Z # E Q 1 t S ~ t dt < 1; we have that ~ V t () is a martingale under Q: his property motivates the following de nition of admissibility. Last updated: November 3, 15

3 De nition 3 A self- nancing trading strategy is called admissible if ~ V t () is a martingale under Q. he next proposition shows that the class of admissible strategies is a good class in terms of arbitrage. Proposition 4 he Black-Scholes model is free of arbitrage in the sense that there exists no admissible arbitrage portfolios. Proof. Assume that is an arbitrage portfolio, i.e. V () ; V () ; P -a.s. and P (V () > ) > : As Q P; we also have that V () ; Q-a.s. and Q(V () > ) > : hen, h i Z # E ~V Q () = E Q "V () + 1 sd S ~ s " Z # = V () + E Q 1 t d S ~ t = V () ; because the integral is a martingale with zero expectation, h under i Q. his is a contradiction, because V () ; Q-a.s. and Q(V () > ) > yields that E ~V Q () > : he following, imprecisely stated, theorem is one of the cornerstones of mathematical nance. heorem 5 (First Fundamental heorem of Asset Pricing) A market model is free of arbitrage if and only if there exists at least one equivalent martingale measure. he di cult part is to show that if the model is free of arbitrage then there exists an equivalent martingale measure. 1.3 Equivalent Martingale Measures and Completeness Our goal is to give an arbitrage free price to any sensible contingent claim H (a positive F - measurable random variable satisfying some integrability constraints) that pays some amount at time : We just have shown that in the Black-Scholes model there are no arbitrage opportunities because there exists an EMM Q: Moreover, we have seen that if is admissible then ~ V () is a martingale under Q: hen, if we combine these facts with the martingale representation theorem we have all the ingredients for pricing and hedging. heorem 6 (Risk Neutral Pricing) Let H L (; F ; Q) be a contingent claim. arbitrage free price of H is given by hen the t (H) = E Q [e r( t) HjF t ]; (3) and the price at time is given by Moreover, the hedging strategy is given by (H) = E Q [e r H]: t = t (H) 1 t ~ S t ; 1 t = h t ~ S t ; where h is the unique process in L a; such that Z e r H = E Q [e r H] + h s d W ~ s : Proof. We have that t (H); the arbitrage free price at time t of any replicable contingent claim H; is given by V t () the value of its admissible hedging portfolio at time t. Hence, if is a hedging strategy for H we have that Z Z H = V () + t db t + 1 t ds t ; 3 Last updated: November 3, 15

4 and the discounted portfolio will be a replicating portfolio for the claim ~ H = e r H; i.e., Z ~H = V ~ () = V () + 1 t d S ~ t : It follows from the martingale properties of ~ V () under Q that E Q [e r HjF t ] = E Q [ ~ V ()jf t ] = ~ V t () = e rt V t (); which yields V t () = E Q [e r( t) HjF t ]: he equality t (H) = V t () yields the pricing formula (3) : he second step is to prove that a su cient condition for a claim H to be replicable is that H L (; F ; Q): Consider the discounted claim ~ H = e r H; which also belongs to L (; F ; Q): Consider the square integrable martingale M t = E Q [e r HjF t ] under Q: As ~ W is a F-Brownian motion under Q we can apply the martingale representation theorem to write M t = E Q [e r H] + for h L a; : De ne the trading strategy given by h s d ~ W s ; he discounted value of this portfolio is t = M t 1 t ~ S t ; 1 t = h t ~ S t : ~V t () = t + 1 t ~ S t = M t ; which is a martingale under Q; so it is admissible. Its nal value will be V () = e r ~ V () = e r M = E Q [HjF ] = H; therefore V t () replicates H: Finally, is self- nancing because and, on the other hand, by equation () we get d ~ V t () = dm t = h t d ~ W t ; 1 t d ~ S t = h t ~ S t d ~ S = h t d ~ W t : Remark 7 he previous theorem provides a very general pricing and hedging formulae and it is very useful to prove theoretical results in the eld of mathematical nance. However, in practical terms, it may be di cult to use because it involves the computation of a conditional expectation. he computation of the hedging strategy is even more di cult as there are no general formulas for computing the kernels in a martingale representation. If the random variable H is smooth in the sense of Malliavin, then one can use the Clark-Ocone formula for those kernels but even in that case it appears a conditional expectation to compute. Remark 8 A su cient condition for H L (; F ; Q) is that H L +" (; F ; P ) for some " > : Remark 9 In the basic Black-Scholes model the ltration F W = F ~W and, hence, we can apply directly the martingale representation theorem with ~ W. In more general cases, when the drift and volatility of S are random, we only have that F ~W F W and in order to apply the martingale representation theorem with ~ W we would need H to be ~ F -measurable. Nevertheless, one can prove that such martingale representation still holds in those cases but it needs additional proof. 4 Last updated: November 3, 15

5 Example 1 Assume that H = h(s ) then t (H) = E Q h e r( t) h(s )jf t i : S t solves the following s.d.e. ds t = rs t + S t d ~ W t ; under Q: Hence, S t is a Markov process, and we have that h i h E Q e r( t) h(s )jf t = E Q e r( i t) h(s t;x ) j x=st : Moreover, by the Feynman-Kac representation, v(t; x) := E Q e r( Scholes PDE with terminal condition h(x): t) f(s t;x ) solves the Black- herefore, the Black-Scholes market is complete for all contingent claims that are square integrable under an equivalent martingale measure. Up till now we have proved that in Black-Scholes model there exists an EMM measure Q given by! dq dp = exp r W 1 r : But, does there exist another ^Q P such that under ^Q the discounted price process ~ S t = e rt S t is a martingale? he answer is no. Lemma 11 In the Black-Scholes model Q is the unique EMM. Proof. I m going to sketch the proof. Assume that ^Q is another h probability i measure such that ^Q P: hen, we can consider the density process D t = E d ^Q dp jf t : Assume that d ^Q dp L (P ). herefore, by the martingale representation theorem we have that " d D t = E ^Q # + s dw s = 1 + s dw s : dp By a similar reasonings as in the Girsanov s theorem, one can prove that a process X t is a martingale under ^Q if and only if D t X t is a martingale under P: Assume that ~ S t is martingale under ^Q then D t ~ St must be a martingale under P: Let us compute the dynamics of D t ~ St under P: First, we have that d ~ S t = ( dd t = t dw t : hen, by the integration by parts formula, we get d(d t ~ St ) = D t d ~ S t + ~ S t dd t + dd t d ~ S t r) ~ S t dt + ~ S t dw t = D t ( r) S ~ t dt + D t S ~ t dw t + S ~ t t dw t + t S ~ t dt = nd t ( r) S ~ t + t S ~ o t dt + nd t S ~ t + S ~ o t t dw t : Hence, for D t ~ St to be a martingale we must have that D t ( r) ~ S t + t ~ S t = ; P ; a.e. which is equivalent to have hus, D t has the representation ( r) t = D t : D t = 1 ( r) D s dw s : 5 Last updated: November 3, 15

6 But the unique solution of this s.d.e. is D t = exp ( r) 1 W t r t! : As D = dq dp we get that, actually, ^Q = Q: he only point left is the assumption that d ^Q dp L (P ); but this is just a technical point that can be addressed. here exists a deep result that links the uniqueness of a martingale measure and the completeness of a market model. heorem 1 (Second Fundamental heorem of Asset Pricing) If a market model admits an EMM Q; then the market is complete if and only if Q is the unique EMM. 6 Last updated: November 3, 15