Chapter 22 Summary of Arbitrage Pricing Theory A simple European derivative security makes a random payment at a time fixed in advance. The value at time t of such a security is the amount of wealth needed at time t in order to replicate the security by trading in the market. The hedging portfoliois a specification of how to do this trading. 22.1 Binomial model, Hedging Portfolio Let  be the set of all possible sequences of n coin-tosses. We have no probabilities at this point. Let r  0; ur+1;d=1=u be given. (See Fig. 2.1) Evolution of the value of a portfolio: X k+1 = k S k+1 +1+rX k ,  k S k : Given a simple European derivative security V ! 1 ;! 2  , we want to start with a nonrandom X 0 and use a portfolio processes  0 ;  1 H ;  1 T  so that X 2 ! 1 ;! 2 =V! 1 ;! 2  8! 1 ;! 2 : (four equations) There are four unknowns: X 0 ;  0 ;  1 H ;  1 T  . Solving the equations, we obtain: 223 224 X 1 ! 1 = 1 1+r 2 6 6 4 1+ r ,d u, d X 2 ! 1 ;H | z  V ! 1 ;H  + u , 1 + r u , d X 2 ! 1 ;T | z  V ! 1 ;T  3 7 7 5 ; X 0 = 1 1+r  1+r,d u,d X 1 H+ u, 1 + r u , d X 1 T   ;  1 ! 1 = X 2 ! 1 ;H, X 2 ! 1 ;T S 2 ! 1 ;H, S 2 ! 1 ;T ;  0 = X 1 H, X 1 T S 1 H, S 1 T : The probabilities of the stock price paths are irrelevant, because we have a hedge which works on every path. From a practical point of view, what matters is that the paths in the model include all the possibilities. We want to find a description of the paths in the model. They all have the property log S k+1 , log S k  2 =  log S k+1 S k  2 =log u 2 = log u 2 : Let  = log u0 .Then n,1 X k=0 log S k+1 , log S k  2 =  2 n: The paths of log S k accumulate quadratic variation at rate  2 per unit time. If we change u , then we change  , and the pricing and hedging formulas on the previous page will give different results. We reiterate that the probabilities are only introduced as an aid to understanding and computation. Recall: X k+1 = k S k+1 +1+rX k ,  k S k : Define  k =1+r k : Then X k+1  k+1 = k S k+1  k+1 + X k  k ,  k S k  k ; i.e., X k+1  k+1 , X k  k = k  S k+1  k+1 , S k  k  : In continuous time, we will have the analogous equation d  X t  t  =td  St t  : CHAPTER 22. Summary of Arbitrage Pricing Theory 225 If we introduce a probability measure f IP under which S k  k is a martingale, then X k  k will also be a martingale, regardless of the portfolio used. Indeed, f IE  X k+1  k+1     F k  = f IE  X k  k + k  S k+1  k+1 , S k  k      F k  = X k  k + k  f IE  S k+1  k+1     F k  , S k  k  : | z  =0 Suppose we want to have X 2 = V ,where V is some F 2 -measurable random variable. Then we must have 1 1+r X 1 = X 1  1 = f IE  X 2  2     F 1  = f IE  V  2     F 1  ; X 0 = X 0  0 = f IE  X 1  1  = f IE  V  2  : To find the risk-neutral probability measure f IP under which S k  k is a martingale, we denote ~p = f IP f! k = H g , ~q = f IP f! k = T g , and compute f IE  S k+1  k+1     F k  =~pu S k  k+1 +~qd S k  k+1 = 1 1+r ~pu +~qd S k  k : We need to choose ~p and ~q so that ~pu +~qd =1+r; ~p+~q=1: The solution of these equations is ~p = 1+r, d u,d ; ~q = u,1 + r u , d : 22.2 Setting up the continuous model Now the stock price S t; 0  t  T , is a continuous function of t . We would like to hedge along every possible path of S t , but that is impossible. Using the binomial model as a guide, we choose 0 and try to hedge along every path S t for which the quadratic variation of log S t accumulates at rate  2 per unit time. These are the paths with volatility  2 . To generate these paths, we use Brownian motion, rather than coin-tossing. To introduce Brownian motion, we need a probability measure. However, the only thing about this probability measure which ultimately matters is the set of paths to which it assigns probability zero. 226 Let B t; 0  t  T , be a Brownian motion defined on a probability space ; F ; P .Forany  2 IR , the paths of t + Bt accumulate quadratic variation at rate  2 per unit time. We want to define S t=S0 expft + Btg; so that the paths of log S t = log S 0 + t + Bt accumulate quadratic variation at rate  2 per unit time. Surprisingly, the choice of  in this definition is irrelevant. Roughly, the reason for this is the following: Choose ! 1 2  . Then, for  1 2 IR ,  1 t + Bt; ! 1 ; 0  t  T; is a continuous function of t . If we replace  1 by  2 ,then  2 t + Bt; ! 1  is a different function. However, there is an ! 2 2  such that  1 t + Bt; ! 1 = 2 t+Bt; ! 2 ; 0  t  T: In other words, regardless of whether we use  1 or  2 in the definition of S t ,wewillseethesame paths. The mathematically precise statement is the following: If a set of stock price paths has a positive probability when S t is defined by S t=S0 expf 1 t + Btg; then this set of paths has positive probability when S t is defined by S t=S0 expf 2 t + Btg: Since we are interested in hedging along every path, except possibly for a set of paths which has probability zero, the choice of  is irrelevant. The most convenient choice of  is  = r , 1 2  2 ; so S t=S0 expfrt + Bt , 1 2  2 tg; and e ,rt S t=S0 expfBt , 1 2  2 tg is a martingale under IP . With this choice of  , dS t=rS t dt + St dB t CHAPTER 22. Summary of Arbitrage Pricing Theory 227 and IP is the risk-neutral measure. If a different choice of  is made, we have S t=S0 expft + Btg; dS t=+ 1 2  2  | z   S t dt + St dB t: = rS t dt +  h ,r  dt + dB t i : | z  d e B t e B has the same paths as B . We can change to the risk-neutral measure f IP , under which e B is a Brownian motion, and then proceed as if  had been chosen to be equal to r , 1 2  2 . 22.3 Risk-neutral pricing and hedging Let f IP denote the risk-neutral measure. Then dS t= rS t dt + S t d e B t; where e B is a Brownian motion under f IP .Set  t=e rt : Then d  S t  t  =  S t  t d e B t; so S t  t is a martingale under f IP . Evolution of the value of a portfolio: dX t= tdS t+rXt,tS t dt; (3.1) which is equivalent to d  X t  t  =td  St t  (3.2) =t St t d e Bt: Regardless of the portfolio used, X t  t is a martingale under f IP . Now suppose V is a given F T  -measurable random variable, the payoff of a simple European derivative security. We want to find the portfolio process T ; 0  t  T , and initial portfolio value X 0 so that X T = V . Because X t  t must be a martingale, we must have X t  t = f IE  V  T      F t  ; 0  t  T: (3.3) This is the risk-neutral pricing formula. We have the following sequence: 228 1. V is given, 2. Define X t; 0  t  T , by (3.3) (not by (3.1) or (3.2), because we do not yet have t ). 3. Construct t so that (3.2) (or equivalently, (3.1)) is satisfied by the X t; 0  t  T , defined in step 2. To carry out step 3, we first use the tower property to show that X t  t defined by (3.3) is a martingale under f IP . We next use the corollary to the Martingale Representation Theorem (HomeworkProblem 4.5) to show that d  X t  t  =  t d e B t (3.4) for some proecss  . Comparing (3.4), which we know, and (3.2), which we want, we decide to define t= tt S t : (3.5) Then (3.4) implies (3.2), which implies (3.1), which implies that X t; 0  t  T ,isthevalueof the portfolio process t; 0  t  T . From (3.3), the definition of X , we see that the hedging portfolio must begin with value X 0 = f IE  V  T   ; and it will end with value X T = T f IE  V T     FT  = T V T = V: Remark 22.1 Although we have taken r and  to be constant, the risk-neutral pricing formula is still “valid” when r and  are processes adapted to the filtration generated by B . If they depend on either e B or on S , they are adapted to the filtration generated by B . The “validity”of the risk-neutral pricing formula means: 1. If you start with X 0 = f IE  V  T   ; then there is a hedging portfolio t; 0  t  T , such that X T = V ; 2. At each time t ,thevalue X t of the hedging portfolio in 1 satisfies X t  t = f IE  V  T      F t  : Remark 22.2 In general, when there are multiple assets and/or multiple Brownian motions, the risk-neutral pricing formula is valid provided there is a unique risk-neutral measure. A probability measure is said to be risk-neutral provided CHAPTER 22. Summary of Arbitrage Pricing Theory 229  it has the same probability-zero sets as the original measure;  it makes all the discounted asset prices be martingales. To see if the risk-neutral measure is unique, compute the differential of all discounted asset prices and check if there is more than one way to define e B so that all these differentials have only d e B terms. 22.4 Implementation of risk-neutral pricing and hedging To get a computable result from the general risk-neutral pricing formula X t  t = f IE  V  T      F t  ; one uses the Markov property. We need to identifysome state variables,the stock price and possibly other variables, so that X t= t f IE  V T     Ft  is a function of these variables. Example 22.1 Assume r and  are constant, and V = hS T  . We can take the stock price to be the state variable. Define vt; x= e IE t;x h e ,rT ,t hS T  i : Then X t=e rt e IE  e ,rT hST      F t  = vt; S t; and X t t = e ,rt vt; S t is a martingale under e IP . Example 22.2 Assume r and  are constant. V = h  Z T 0 S u du ! : Take S t and Y t= R t 0 Sudu to be the state variables. Define vt; x; y= e IE t;x;y h e ,rT ,t hY T  i ; where Y T =y+ Z T t Sudu: 230 Then X t=e rt e IE  e ,rT hST      F t  = vt; S t;Y t and X t  t = e ,rt vt; S t;Y t is a martingale under e IP . Example 22.3 (Homework problem 4.2) dS t=rt; Y t S tdt + t; Y tS t d e B t; dY t=t; Y t dt +  t; Y t d e B t; V = hS T : Take S t and Y t to be the state variables. Define vt; x; y= e IE t;x;y 2 6 6 6 6 6 6 4 exp  , Z T t ru; Y u du  | z  t T  hS T  3 7 7 7 7 7 7 5 : Then X t=t e IE  hST  T      F t  = e IE " exp  , Z T t ru; Y u du  hS T      F t  = vt; S t;Y t; and X t  t = exp  , Z t 0 ru; Y u du  vt; S t;Y t is a martingale under e IP . In every case, we get an expression involving v to be a martingale. We take the differential and set the dt term to zero. This gives us a partial differential equation for v , and this equation must hold wherever the state processes can be. The d e B term in the differential of the equation is the differential of a martingale, and since the martingale is X t  t = X 0 + Z t 0 u S u  u d e B u we can solve for t . This is the argument which uses (3.4) to obtain (3.5). CHAPTER 22. Summary of Arbitrage Pricing Theory 231 Example 22.4 (Continuation of Example 22.3) X t  t = exp  , Z t 0 ru; Y u du  | z  1=t vt; S t;Y t is a martingale under e IP .Wehave d  X t  t  = 1  t  ,rt; Y tvt; S t;Y t dt + v t dt + v x dS + v y dY + 1 2 v xx dS dS + v xy dS d Y + 1 2 v yy dY dY  = 1 t  ,rv + v t + rSv x + v y + 1 2  2 S 2 v xx + Sv xy + 1 2  2 v yy  dt +Sv x + v y  d e B  The partial differential equation satisfied by v is ,rv + v t + rxv x + v y + 1 2  2 x 2 v xx + xv xy + 1 2  2 v yy =0 where it should be noted that v = vt; x; y , and all other variables are functions of t; y .Wehave d  X t  t  = 1  t Sv x + v y  d e Bt; where  = t; Y t ,  =  t; Y t , v = vt; S t;Y t ,and S = S t . We want to choose t so that (see (3.2)) d  X t  t  =tt; Y t S t  t d e B t: Therefore, we should take t to be t=v x t; S t;Y t +  t; Y t t; Y t S t v y t; S t;Y t: 232 [...]... the state processes can be The dB term in the differential of the equation is the differential of a martingale, and since the martingale is X t = X 0 + Z t
u S u dBu e t u 0 we can solve for
t This is the argument which uses (3.4) to obtain (3.5) CHAPTER 22 Summary of Arbitrage Pricing Theory 231 Example 22.4 (Continuation of Example 22.3) Xt = exp , Z t ru; Y u du vt; St;... CHAPTER 22 Summary of Arbitrage Pricing Theory 229 it has the same probability-zero sets as the original measure; it makes all the discounted asset prices be martingales To see if the risk-neutral measure is unique, compute the differential of all discounted asset prices e e and check if there is more than one way to define B so that all these differentials have only dB terms 22.4 Implementation of risk-neutral...CHAPTER 22 Summary of Arbitrage Pricing Theory and IP is the risk-neutral measure If a different choice of 227 is made, we have S t = S 0 expf t + Btg; dS t =  + z1 2  S t dt + S t dBt: | 2  = rS t dt + e B has the same paths as B i h ,r... “validity” of the risk-neutral pricing formula means:   f V ; X 0 = IE T  then there is a hedging portfolio
t; 0  t  T , such that X T  = V ; 1 If you start with 2 At each time t, the value X t of the hedging portfolio in 1 satisfies X t = IE  V F t : f t T  Remark 22.2 In general, when there are multiple assets and/or multiple Brownian motions, the risk-neutral pricing formula... then proceed as if had been chosen to be equal to r , 2 2 e B is a 22.3 Risk-neutral pricing and hedging f Let I denote the risk-neutral measure Then P e dS t = rS t dt + S t dB t; e f where B is a Brownian motion under I Set P t = ert : Then d f P so S t is a martingale under I t Evolution of the value of a portfolio: S t t e = St dB t; t dX t =
tdS t + rX t ,
tS... t + rX t ,
tS t dt; which is equivalent to d X t S t t =
td t t e =
t St dB t: (3.1) (3.2) f Regardless of the portfolio used, Xtt is a martingale under I P Now suppose V is a given F T -measurable random variable, the payoff of a simple European derivative security We want to find the portfolio process
T ; 0  t  T , and initial portfolio value X 0 so that... only dB terms 22.4 Implementation of risk-neutral pricing and hedging To get a computable result from the general risk-neutral pricing formula X t = IE  V F t ; f t T  one uses the Markov property We need to identify some state variables, the stock price and possibly other variables, so that   f V X t = tIE T  F t is a function of these variables Example 22.1 Assume r and variable... portfolio process
t; 0  t  T (3.5)  t  T , is the value of From (3.3), the definition of X , we see that the hedging portfolio must begin with value   f V ; X 0 = IE T  and it will end with value   V f V X T  = T IE T  F T  = T  T  = V: Remark 22.1 Although we have taken r and to be constant, the risk-neutral pricing formula is still “valid” when r and are processes adapted... process
T ; 0  t  T , and initial portfolio value X 0 so that X T  = V Because Xtt must be a martingale, we must have X t = IE  V F t ; 0  t  T: f t T  This is the risk-neutral pricing formula We have the following sequence: (3.3) 228 1 V is given, 2 Define X t; 0  t  T , by (3.3) (not by (3.1) or (3.2), because we do not yet have
t) 3 Construct
t so that (3.2) (or equivalently,... vyy  dt The partial differential equation satisfied by v is ,rv + vt + rxvx + vy + 1 2x2 vxx + xvxy + 1 2 vyy = 0 2 2 where it should be noted that v = vt; x; y, and all other variables are functions of t; y We have Xt 1 e d t = t Svx + vy dBt; = t; Y t, = t; Y t, v = vt; St; Y t, and S = St We want to choose
t so that where (see (3.2)) d Xt Therefore, we should take . of  , dS t=rS t dt + St dB t CHAPTER 22. Summary of Arbitrage Pricing Theory 227 and IP is the risk-neutral measure. If a different choice of. 22 Summary of Arbitrage Pricing Theory A simple European derivative security makes a random payment at a time fixed in advance. The value at time t of such