Semi-Continuous Models

Chapter 12 Semi-Continuous Models 12.1 Discrete-time Brownian Motion Let fY j g n j =1 be a collection of independent,standard normal random variables defined on ; F ; P , where IPisthemarket measure. As before we denote the column vector Y 1 ;::: ;Y n  T by Y .We therefore have for any real colum vector u =u 1 ;::: ;u n  T , IEe u T Y = IE exp 8  : n X j =1 u j Y j 9 = ; = exp 8  : n X j =1 1 2 u 2 j 9 = ; : Define the discrete-time Brownian motion (See Fig. 12.1): B 0 = 0; B k = k X j =1 Y j ;k=1;::: ;n: If we know Y 1 ;Y 2 ;::: ;Y k , then we know B 1 ;B 2 ;::: ;B k . Conversely, if we know B 1 ;B 2 ;::: ;B k , then we know Y 1 = B 1 ;Y 2 = B 2 , B 1 ;::: ;Y k = B k , B k,1 . Define the filtration F 0 = f; g; F k =  Y 1 ;Y 2 ;::: ;Y k =B 1 ;B 2 ;::: ;B k ;k=1;::: ;n: Theorem 1.34 fB k g n k=0 is a martingale (under IP). Proof: IE B k+1 jF k  = IE Y k+1 + B k jF k  = IEY k+1 + B k = B k : 131 132 Y Y Y Y 1 2 3 4 k B k 0 12 34 Figure 12.1: Discrete-time Brownian motion. Theorem 1.35 fB k g n k=0 is a Markov process. Proof: Note that IE hB k+1 jF k =IEhY k+1 + B k jF k : Use the Independence Lemma. Define g b= IEhY k+1 + b= 1 p 2 Z 1 ,1 hy + be , 1 2 y 2 dy : Then IE hY k+1 + B k jF k =gB k ; which is a function of B k alone. 12.2 The Stock Price Process Given parameters:   2 IR ,themean rate of return.  0 ,thevolatility.  S 0  0 , the initial stock price. The stock price process is then given by S k = S 0 exp n B k +, 1 2  2 k o ;k=0;::: ;n: Note that S k+1 = S k exp n Y k+1 +, 1 2  2  o ; CHAPTER 12. Semi-Continuous Models 133 IE S k+1 jF k  = S k IE e Y k+1 jF k :e , 1 2  2 = S k e 1 2  2 e , 1 2  2 = e  S k : Thus  = log IE S k+1 jF k  S k = log IE  S k+1 S k     F k  ; and var  log S k+1 S k  =var  Y k+1 +, 1 2  2   =  2 : 12.3 Remainder of the Market The other processes in the market are defined as follows. Money market process: M k = e rk ;k=0;1;::: ;n: Portfolio process:   0 ;  1 ;::: ; n,1 ;  Each  k is F k -measurable. Wealth process:  X 0 given, nonrandom.  X k+1 =  k S k+1 + e r X k ,  k S k  =  k S k+1 , e r S k +e r X k  Each X k is F k -measurable. Discounted wealth process: X k+1 M k+1 = k  S k+1 M k+1 , S k M k  + X k M k : 12.4 Risk-Neutral Measure Definition 12.1 Let f IP be a probability measure on ; F  , equivalent to the market measure IP. If n S k M k o n k =0 is a martingale under f IP , we say that f IP is a risk-neutral measure. 134 Theorem 4.36 If f IP is a risk-neutral measure, then every discounted wealth process n X k M k o n k=0 is a martingale under f IP , regardless of the portfolio process used to generate it. Proof: f IE  X k+1 M k+1     F k  = f IE   k  S k+1 M k+1 , S k M k  + X k M k     F k  =  k  f IE  S k+1 M k+1     F k  , S k M k  + X k M k = X k M k : 12.5 Risk-Neutral Pricing Let V n be the payoff at time n ,andsayitis F n -measurable. Note that V n may be path-dependent. Hedging a short position:  Sell the simple European derivative security V n .  Receive X 0 at time 0.  Construct a portfolio process  0 ;::: ; n,1 which starts with X 0 and ends with X n = V n .  If there is a risk-neutral measure f IP ,then X 0 = f IE X n M n = f IE V n M n : Remark 12.1 Hedging in this “semi-continuous” model is usually not possible because there are not enough trading dates. This difficulty will disappear when we go to the fully continuous model. 12.6 Arbitrage Definition 12.2 An arbitrage is a portfolio which starts with X 0 =0 and ends with X n satisfying IP X n  0 = 1;IPX n 0  0: (IP here is the market measure). Theorem 6.37 (Fundamental Theorem of Asset Pricing: Easy part) Ifthere is a risk-neutralmea- sure, then there is no arbitrage. CHAPTER 12. Semi-Continuous Models 135 Proof: Let f IP be a risk-neutral measure, let X 0 =0 ,andlet X n be the final wealth corresponding to any portfolio process. Since n X k M k o n k=0 is a martingale under f IP , f IE X n M n = f IE X 0 M 0 =0: (6.1) Suppose IP X n  0 = 1 .Wehave IP X n  0 = 1 = IP X n  0 = 0 = f IP X n  0 = 0 = f IP X n  0 = 1: (6.2) (6.1) and (6.2) imply f IP X n =0=1 .Wehave f IP X n =0=1= f IPX n 0=0=IPX n 0=0: This is not an arbitrage. 12.7 Stalking the Risk-Neutral Measure Recall that  Y 1 ;Y 2 ;::: ;Y n are independent, standard normal random variables on some probabilityspace ; F ; P .  S k = S 0 exp n B k +, 1 2  2 k o .  S k+1 = S 0 exp n  B k + Y k+1 + , 1 2  2 k +1 o = S k exp n Y k+1 +, 1 2  2  o : Therefore, S k+1 M k+1 = S k M k : exp n Y k+1 +,r , 1 2  2  o ; IE  S k+1 M k+1     F k  = S k M k :IE exp fY k+1 gjF k :expf , r , 1 2  2 g = S k M k : expf 1 2  2 g: expf , r , 1 2  2 g = e ,r : S k M k : If  = r , the market measure is risk neutral. If  6= r , we must seek further. 136 S k+1 M k+1 = S k M k : exp n Y k+1 +,r, 1 2  2  o = S k M k :exp n Y k+1 + ,r   , 1 2  2 o = S k M k : exp n  ~ Y k+1 , 1 2  2 o ; where ~ Y k+1 = Y k+1 + ,r  : The quantity ,r  is denoted  and is called the market price of risk. We want a probability measure f IP under which ~ Y 1 ;::: ; ~ Y n are independent, standard normal ran- dom variables. Then we would have f IE  S k+1 M k+1     F k  = S k M k : f IE h expf ~ Y k+1 gjF k i : expf, 1 2  2 g = S k M k : expf 1 2  2 g: expf, 1 2  2 g = S k M k : Cameron-Martin-Girsanov’s Idea: Define the random variable Z = exp 2 4 n X j =1 ,Y j , 1 2  2  3 5 : Properties of Z :  Z  0 .  IEZ = IE exp 8  : n X j =1 ,Y j  9 = ; : exp  , n 2  2  = exp  n 2  2  : exp  , n 2  2  =1: Define f IP A= Z A ZdIP 8A2F: Then f IP A  0 for all A 2F and f IP  = IEZ =1: In other words, f IP is a probability measure. CHAPTER 12. Semi-Continuous Models 137 We show that f IP is a risk-neutral measure. For this, it suffices to show that ~ Y 1 = Y 1 + ; ::: ; ~ Y n = Y n +  are independent, standard normal under f IP . Verification:  Y 1 ;Y 2 ;::: ;Y n : Independent, standard normal under IP, and IE exp 2 4 n X j =1 u j Y j 3 5 = exp 2 4 n X j =1 1 2 u 2 j 3 5 :  ~ Y = Y 1 + ; ::: ; ~ Y n = Y n + :  Z0 almost surely.  Z = exp h P n j =1 ,Y j , 1 2  2  i ; f IP A= Z A ZdIP 8A2F; f IEX = IEXZ for every random variable X .  Compute the moment generating function of  ~ Y 1 ;::: ; ~ Y n  under f IP : f IE exp 2 4 n X j =1 u j ~ Y j 3 5 = IE exp 2 4 n X j =1 u j Y j + + n X j=1 ,Y j , 1 2  2  3 5 = IE exp 2 4 n X j =1 u j ,  Y j 3 5 : exp 2 4 n X j =1 u j  , 1 2  2  3 5 = exp 2 4 n X j =1 1 2 u j ,   2 3 5 : exp 2 4 n X j =1 u j  , 1 2  2  3 5 = exp 2 4 n X j =1   1 2 u 2 j , u j  + 1 2  2 +u j  , 1 2  2   3 5 = exp 2 4 n X j =1 1 2 u 2 j 3 5 : 138 12.8 Pricing a European Call Stock price at time n is S n = S 0 exp n B n +, 1 2  2 n o = S 0 exp 8  :  n X j =1 Y j +, 1 2  2 n 9 = ; = S 0 exp 8  :  n X j =1 Y j + ,r   ,  , rn +, 1 2  2 n 9 = ; = S 0 exp 8  :  n X j =1 ~ Y j +r, 1 2  2 n 9 = ; : Payoff at time n is S n , K  + . Price at time zero is f IE S n , K  + M n = f IE 2 4 e ,rn 0 @ S 0 exp 8  :  n X j =1 ~ Y j +r, 1 2  2 n 9 = ; ,K 1 A + 3 5 = Z 1 ,1 e ,rn  S 0 exp n b +r, 1 2  2 n o ,K  + : 1 p 2n e , b 2 2n 2 db since P n j =1 ~ Y j is normal with mean 0, variance n , under f IP . This is the Black-Scholes price. It does not depend on  . [...]... n 2 2 j =1 = exp n 2 : exp , n 2 = 1: 2 2 Define f IP A = f Then I A  0 for all A 2 F and P Z A Z dIP 8A 2 F : f IP   = IEZ = 1: f In other words, I is a probability measure P CHAPTER 12 Semi-Continuous Models 137 f We show that I is a risk-neutral measure For this, it suffices to show that P ~ ~ Y 1 = Y1 + ; : : : ; Yn = Yn + f are independent, standard normal under I P Verification: Y1 ; Y2; . Chapter 12 Semi-Continuous Models 12.1 Discrete-time Brownian Motion Let fY j g n j =1 be a collection. Note that S k+1 = S k exp n Y k+1 +, 1 2  2  o ; CHAPTER 12. Semi-Continuous Models 133 IE S k+1 jF k  = S k IE e Y k+1 jF k :e , 1 2  2 =