Martingale Representation Theorem

Chapter 18 Martingale Representation Theorem 18.1 Martingale Representation Theorem See Oksendal, 4th ed., Theorem 4.11, p.50. Theorem 1.56 Let B t; 0  t  T; be a Brownian motion on ; F ; P .Let F t; 0  t  T ,be the filtration generated by this Brownian motion. Let X t; 0  t  T , be a martingale (under IP ) relative to this filtration. Then there is an adapted process  t; 0  t  T , such that X t= X0 + Z t 0  u dB u; 0  t  T: In particular, the paths of X are continuous. Remark 18.1 We already know that if X t is a process satisfying dX t= tdB t; then X t is a martingale. Now we see that if X t is a martingale adapted to the filtration generated by the Brownian motion B t , i.e, the Brownian motion is the only source of randomness in X ,then dX t= t dB t for some  t . 18.2 A hedging application Homework Problem 4.5. In the context of Girsanov’s Theorem, suppse that F t; 0  t  T; is the filtration generated by the Brownian motion B (under IP ). Suppose that Y is a f IP -martingale. Then there is an adapted process  t; 0  t  T , such that Y t=Y0 + Z t 0  u d e B u; 0  t  T: 197 198 dS t= tStdt + tS t dB t;  t = exp  Z t 0 ru du  ; t= t,rt t ; e Bt= Z t 0 udu + B t; Z t = exp  , Z t 0 u dB u , 1 2 Z t 0  2 u du  ; f IP A= Z A ZTdIP; 8A 2F: Then d  S t  t  = S t  t  t d e B t: Let t; 0  t  T; be a portfolio process. The corresponding wealth process X t satisfies d  X t  t  =tt St t d e Bt; i.e., X t  t = X 0 + Z t 0 uu S u  u d e B u; 0  t  T: Let V be an F T  -measurable random variable, representing the payoff of a contingent claim at time T . We want to choose X 0 and t; 0  t  T ,sothat X T = V: Define the f IP -martingale Y t= f IE  V T     Ft  ; 0t T: According to Homework Problem 4.5, there is an adapted process  t; 0  t  T , such that Y t=Y0 + Z t 0  u d e B u; 0  t  T: Set X 0 = Y 0 = f IE h V  T  i and choose u so that uu S u  u =  u: CHAPTER 18. Martingale Representation Theorem 199 With this choice of u; 0  u  T ,wehave X t  t = Y t= f IE  V T     Ft  ; 0t T: In particular, X T   T  = f IE  V  T      F T   = V  T  ; so X T = V: The Martingale Representation Theorem guarantees the existence of a hedging portfolio, although it does not tell us how to compute it. It also justifies the risk-neutral pricing formula X t=t f IE  V T     Ft  = t Zt IE  ZT T V     Ft  = 1 t IE  TV     Ft  ; 0 t T; where  t= Zt t = exp  , Z t 0 u dB u , Z t 0 ru+ 1 2  2 u du  18.3 d -dimensional Girsanov Theorem Theorem 3.57 ( d -dimensional Girsanov)  B t=B 1 t;::: ;B d t; 0  t  T ,a d - dimensional Brownian motion on ; F ; P ; Ft;0tT; the accompanying filtration, perhaps larger than the one generated by B ;  t= 1 t;::: ; d t; 0  t  T , d -dimensionaladapted process. For 0  t  T; define e B j t= Z t 0  j udu + B j t; j =1;::: ;d; Zt = exp  , Z t 0 u:dBu, 1 2 Z t 0 jjujj 2 du  ; f IP A= Z A ZTdIP: 200 Then, under f IP , the process e B t= e B 1 t;::: ; e B d t; 0  t  T; is a d -dimensionalBrownian motion. 18.4 d -dimensional Martingale Representation Theorem Theorem 4.58  B t=B 1 t;::: ;B d t; 0  t  T; a d -dimensional Brownian motion on ; F ; P ; Ft;0tT; the filtration generated by the Brownian motion B . If X t; 0  t  T , is a martingale (under IP ) relative to F t; 0  t  T , then there is a d -dimensionaladpated process  t= 1 t;::: ; d t , such that X t=X0 + Z t 0  u:dBu; 0  t  T: Corollary 4.59 If we have a d -dimensionaladapted process t= 1 t;::: ; d t; then we can define e B; Z and f IP as in Girsanov’s Theorem. If Y t; 0  t  T , is a martingale under f IP relative to F t; 0  t  T , then there is a d -dimensional adpated process  t= 1 t;::: ; d t such that Y t=Y0 + Z t 0  u:d e Bu; 0 t  T: 18.5 Multi-dimensional market model Let B t=B 1 t;::: ;B d t; 0  t  T ,bea d -dimensional Brownian motion on some ; F ; P ,andlet F t; 0  t  T ,bethefiltration generated by B . Then we can define the following: Stocks dS i t= i tS i tdt + S i t d X j =1  ij t dB j t; i =1;::: ;m Accumulation factor  t = exp  Z t 0 ru du  : Here,  i t; ij t and rt are adpated processes. CHAPTER 18. Martingale Representation Theorem 201 Discounted stock prices d  S i t  t  = i t,rt | z  Risk Premium S i t  t dt + S i t  t d X j =1  ij t dB j t ? = S i t  t d X j =1  ij t j t+ dB j t | z  d e B j t (5.1) For 5.1 to be satisfied, we need to choose  1 t;::: ; d t ,sothat d X j =1  ij t j t= i t,rt; i=1;::: ;m: (MPR) Market price of risk. The market price of risk is an adapted process t= 1 t;::: ; d t satisfying the system of equations (MPR) above. There are three cases to consider: Case I: (Unique Solution). For Lebesgue-almost every t and IP -almost every ! ,(MPR)hasa unique solution t .Using t in the d -dimensional Girsanov Theorem, we define a unique risk-neutral probability measure f IP . Under f IP , every discounted stock price is a martingale. Consequently, the discounted wealth process corresponding to any portfolio process is a f IP - martingale, and this implies that the market admits no arbitrage. Finally, the Martingale Representation Theorem can be used to show that every contingent claim can be hedged; the market is said to be complete. Case II: (No solution.) If (MPR) has no solution, then there is no risk-neutral probability measure and the market admits arbitrage. Case III: (Multiple solutions). If (MPR) has multiplesolutions, then there are multiplerisk-neutral probability measures. The market admits no arbitrage, but there are contingent claims which cannot be hedged; the market is said to be incomplete. Theorem 5.60 (Fundamental Theorem of Asset Pricing) Part I. (Harrison and Pliska, Martin- gales and Stochasticintegralsin the theoryof continuoustrading, Stochastic Proc. and Applications 11 (1981), pp 215-260.): If a market has a risk-neutral probability measure, then it admits no arbitrage. PartII. (Harrisonand Pliska, A stochasticcalculus model of continuoustrading: complete markets, Stochastic Proc. and Applications 15 (1983), pp 313-316): The risk-neutral measure is unique if and only if every contingent claim can be hedged. 202 [...]... Using t in the d-dimensional Girsanov Theorem, we define a unique f f risk-neutral probability measure I Under I , every discounted stock price is a martingale P P f Consequently, the discounted wealth process corresponding to any portfolio process is a I P martingale, and this implies that the market admits no arbitrage Finally, the Martingale Representation Theorem can be used to show that every... t; : : : ; Bd t; 0  t  T; is a d-dimensional Brownian motion 18.4 d-dimensional Martingale Representation Theorem Theorem 4.58 on  ; F ; P; Bt = B1 t; : : : ; Bdt; 0  t  T; a d-dimensional Brownian motion F t; 0  t  T; the filtration generated by the Brownian motion B If X t; 0  t  T , is a martingale (under IP ) relative to F t; 0  t  T , then there is a d-dimensional adpated... Stocks dSi t = i tSit dt + Si t Accumulation factor t = exp Zt 0 ru du : Here, i t; ij t and rt are adpated processes d X j =1 ij t dBj t; i = 1; : : : ; m CHAPTER 18 Martingale Representation Theorem 201 Discounted stock prices d X d Sitt = i t , rt Sitt dt + Sitt ij t dBj t | z j =1 Risk Premium d ? Si t X t t + dB t = t ij | j z j j =1 e dBj... multiple risk-neutral probability measures The market admits no arbitrage, but there are contingent claims which cannot be hedged; the market is said to be incomplete Theorem 5.60 (Fundamental Theorem of Asset Pricing) Part I (Harrison and Pliska, Martingales and Stochastic integrals in the theory of continuous trading, Stochastic Proc and Applications 11 (1981), pp 215-260.): If a market has a risk-neutral... t = X 0 + Zt 0 u: dB u; 0  t  T: Corollary 4.59 If we have a d-dimensional adapted process t = 1 t; : : : ; d t; then we can e f f define B; Z and I as in Girsanov’s Theorem If Y t; 0  t  T , is a martingale under I relative P P to F t; 0  t  T , then there is a d-dimensional adpated process t =  1t; : : : ; dt such that Z Y t = Y 0 + t 0 e u: dB u; 0  t ...2 u du 2 18.3 d-dimensional Girsanov Theorem Theorem 3.57 (d-dimensional Girsanov) Bt dimensional Brownian motion on  ; F ; P; = B1 t; : : : ; Bd t; 0  t  T , a d- F t; 0  t  T; the accompanying filtration, perhaps larger than . Chapter 18 Martingale Representation Theorem 18.1 Martingale Representation Theorem See Oksendal, 4th ed., Theorem 4.11, p.50. Theorem 1.56 Let B. T; is a d -dimensionalBrownian motion. 18.4 d -dimensional Martingale Representation Theorem Theorem 4.58  B t=B 1 t;::: ;B d t; 0  t  T; a d