Chapter 18 MartingaleRepresentationTheorem 18.1 MartingaleRepresentationTheorem 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= X0 + 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= tdB 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=Y0 + Z t 0 u d e B u; 0 t T: 197 198 dS t= tStdt + tS t dB t; t = exp Z t 0 ru du ; t= t,rt t ; e Bt= Z t 0 udu + B t; Z t = exp , Z t 0 u dB u , 1 2 Z t 0 2 u du ; f IP A= Z A ZTdIP; 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 =tt St t d e Bt; i.e., X t t = X 0 + Z t 0 uu 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 Ft ; 0t T: According to Homework Problem 4.5, there is an adapted process t; 0 t T , such that Y t=Y0 + 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 uu S u u = u: CHAPTER 18. MartingaleRepresentationTheorem 199 With this choice of u; 0 u T ,wehave X t t = Y t= f IE V T Ft ; 0t T: In particular, X T T = f IE V T F T = V T ; so X T = V: The MartingaleRepresentationTheorem 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 Ft = t Zt IE ZT T V Ft = 1 t IE TV Ft ; 0 t T; where t= Zt t = exp , Z t 0 u dB u , Z t 0 ru+ 1 2 2 u du 18.3 d -dimensional Girsanov TheoremTheorem 3.57 ( d -dimensional Girsanov) B t=B 1 t;::: ;B d t; 0 t T ,a d - dimensional Brownian motion on ; F ; P ; Ft;0tT; 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 udu + B j t; j =1;::: ;d; Zt = exp , Z t 0 u:dBu, 1 2 Z t 0 jjujj 2 du ; f IP A= Z A ZTdIP: 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 MartingaleRepresentationTheoremTheorem 4.58 B t=B 1 t;::: ;B d t; 0 t T; a d -dimensional Brownian motion on ; F ; P ; Ft;0tT; 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=X0 + Z t 0 u:dBu; 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=Y0 + Z t 0 u:d e Bu; 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 tS i tdt + S i t d X j =1 ij t dB j t; i =1;::: ;m Accumulation factor t = exp Z t 0 ru du : Here, i t; ij t and rt are adpated processes. CHAPTER 18. MartingaleRepresentationTheorem 201 Discounted stock prices d S i t t = i t,rt | 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,rt; 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 MartingaleRepresentationTheorem 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 ; B t = B1 t ; : : : ; Bd t ; 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 t Si t dt + Si t Accumulation factor t = exp Zt 0 r u du : Here, i t ; ij t and r t 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 Si tt = i t , r t Si tt dt + Si tt 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 = 1 t ; : : : ; d t such that Z Y t = Y 0 + t 0 e u : dB u ; 0 t ...2 u du 2 18.3 d-dimensional Girsanov TheoremTheorem 3.57 (d-dimensional Girsanov) B t 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