Change of num´eraire

Chapter 33 Change of num ´ eraire Consider a Brownian motion driven market model with time horizon T  . For now, we will have one asset, which we call a “stock” even though in applications it will usually be an interest rate dependent claim. The price of the stock is modeled by dS t= rt Stdt +  tS t dW t; (0.1) where the interest rate process rt and the volatility process  t are adapted to some filtration fF t; 0  t  T  g . W is a Brownian motion relative to this filtration, but fF t; 0  t  T  g may be larger than the filtration generated by W . This is not a geometric Brownian motion model. We are particularly interested in the case that the interest rate is stochastic, given by a term structure model we have not yet specified. We shall work only under the risk-neutral measure, which is reflected by the fact that the mean rate of return for the stock is rt . We define the accumulation factor  t = exp  Z t 0 ru du  ; so that the discounted stock price S t  t is a martingale. Indeed, d  S t  t  = S t  t  t dW t: The zero-coupon bond prices are given by B t; T =IE " exp  , Z T t ru du      F t  = IE   t  T      F t  ; 325 326 so B t; T   t = IE  1  T      F t  is also a martingale (tower property). The T -forward price F t; T  of the stock is the price set at time t for delivery of one share of stock at time T with payment at time T . The value of the forward contract at time t is zero, so 0=IE  t T ST,Ft; T      F t  =  tIE  S T   T      Ft  , Ft; T IE   t  T      F t  =  t S t  t , F t; T B t; T  = S t , F t; T B t; T  Therefore, F t; T = St Bt; T  : Definition 33.1 (Num ´ eraire) Any asset in the model whose price is always strictly positive can be taken as the num´eraire. We then denominate all other assets in units of this num´eraire. Example 33.1 (Money market as num´eraire) The money market could be the num´eraire. At time t ,the stock is worth S t t units of money market and the T -maturity bond is worth Bt;T  t units of money market. Example 33.2 (Bond as num´eraire) The T -maturity bond could be the num´eraire. At time t  T , the stock is worth F t; T  units of T -maturity bond and the T -maturity bond is worth 1 unit. We will say that a probability measure IP N is risk-neutral for the num´eraire N if every asset price, divided by N , is a martingale under IP N . The original probability measure IP is risk-neutral for the num´eraire  (Example 33.1). Theorem 0.71 Let N be a num´eraire, i.e., the price process for some asset whose price is always strictly positive. Then IP N defined by IP N A= 1 N0 Z A N T    T   dIP; 8A 2FT  ; is risk-neutral for N . CHAPTER 33. Change of num´eraire 327 Note: IP and IP N are equivalent, i.e., have the same probability zero sets, and IP A=N0 Z A  T   N T   dIP N ; 8A 2FT  : Proof: Because N is the price process for some asset, N= is a martingale under IP . Therefore, IP N  = 1 N 0 Z  N T    T   dIP = 1 N 0 :IE  N T    T    = 1 N 0 N 0  0 =1; and we see that IP N is a probability measure. Let Y be an asset price. Under IP , Y= is a martingale. We must show that under IP N , Y=N is a martingale. For this, we need to recall how to combine conditional expectations with change of measure (Lemma 1.54). If 0  t  T  T  and X is F T  -measurable, then IE N  X     F t  = N 0 t N t IE  N T  N 0 T  X     F t  =  t N t IE  N T   T  X     F t  : Therefore, IE N  Y T  N T      F t  =  t N t IE  N T   T  Y T  N T      F t  =  t N t Y t  t = Y t N t ; which is the martingale property for Y=N under IP N . 33.1 Bond price as num ´ eraire Fix T 2 0;T   and let B t; T  be the num´eraire. The risk-neutral measure for this num´eraire is IP T A= 1 B0;T Z A BT; T  T  dIP = 1 B 0;T Z A 1 T dIP 8A 2FT: 328 Because this bond is not defined after time T , we change the measure only “up to time T ”, i.e., using 1 B 0;T  B T;T  T  and only for A 2FT . IP T is called the T -forward measure. Denominated in units of T -maturity bond, the value of the stock is F t; T = St Bt; T  ; 0  t  T: This is a martingale under IP T , and so has a differential of the form dF t; T = F t; T F t; T  dW T t; 0  t  T; (1.1) i.e., a differential without a dt term. The process fW T ;0tTg is a Brownian motion under IP T . We may assume without loss of generality that  F t; T   0 . We write F t rather than F t; T  from now on. 33.2 Stock price as num ´ eraire Let S t be the num´eraire. In terms of this num´eraire, the stock price is identically 1. The risk- neutral measure under this num´eraire is IP S A= 1 S0 Z A S T    T   dIP; 8A 2FT  : Denominated in shares of stock, the value of the T -maturity bond is B t; T  S t = 1 F t : This is a martingale under IP S , and so has a differential of the form d  1 F t  =  t; T   1 F t  dW S t; (2.1) where fW S t; 0  t  T  g is a Brownian motion under IP S . We may assume without loss of generality that  t; T   0 . Theorem 2.72 The volatility  t; T  in (2.1) is equal to the volatility  F t; T  in (1.1). In other words, (2.1) can be rewritten as d  1 F t  =  F t; T   1 F t  dW S t; (2.1’) CHAPTER 33. Change of num´eraire 329 Proof: Let g x=1=x ,so g 0 x=,1=x 2 ;g 00 x=2=x 3 .Then d  1 F t  = dg F t = g 0 F t dF t+ 1 2 g 00 Ft dF t dF t = , 1 F 2 t  F t; T F t; T  dW T t+ 1 F 3 t  2 F t; T F 2 t; T  dt = 1 F t h , F t; T  dW T t+ 2 F t; T  dt i =  F t; T   1 F t  ,dW T t+ F t; T  dt: Under IP T ; ,W T is a Brownian motion. Under this measure, 1 F t has volatility  F t; T  and mean rate of return  2 F t; T  . The change of measure from IP T to IP S makes 1 F t a martingale, i.e., it changes the mean return to zero, but the change of measure does not affect the volatility. Therefore,  t; T  in (2.1) must be  F t; T  and W S must be W S t=,W T t+ Z t 0  F u; T  du: 33.3 Merton option pricing formula The price at time zero of a European call is V 0 = IE  1  T  S T  , K  +  = IE  S T   T  1 fS T K g  , KIE  1 T  1 fST K g  = S 0 Z fS T K g S T  S 0 T  dIP , KB0;T Z fSTK g 1 B 0;TT dIP = S 0IP S fS T  Kg,KB0;TIP T fST Kg =S0IP S fF T  Kg,KB0;TIP T fFT Kg =S0IP S  1 F T   1 K  , KB0;TIP T fFT Kg: 330 This is a completely general formula which permits computation as soon as we specify  F t; T  .If we assume that  F t; T  is a constant  F , we have the following: 1 F T  = B 0;T S0 exp n  F W S T  , 1 2  2 F T o ; IP S  1 F T   1 K  = IP S   F W S T  , 1 2  2 F Tlog S 0 KB0;T  = IP S  W S T p T  1  F p T log S 0 KB0;T + 1 2  F p T  = N 1 ; where  1 = 1  F p T  log S 0 KB0;T + 1 2  2 F T  : Similarly, F T = S0 B 0;T exp n  F W T T  , 1 2  2 F T o ; IP T fF T  Kg=IP T   F W T T, 1 2  2 F Tlog KB0;T S0  = IP T  W T T  p T  1  F p T  log KB0;T S0 + 1 2  2 F T  = IP T  ,W T T  p T  1  F p T  log S 0 KB0;T , 1 2  2 F T  = N  2 ; where  2 = 1  F p T  log S 0 KB0;T , 1 2  2 F T  : If r is constant, then B 0;T=e ,rT ,  1 = 1  F p T  log S 0 K +r+ 1 2  2 F T  ;  2 = 1  F p T  log S 0 K +r, 1 2  2 F T  ; and we have the usual Black-Scholes formula. When r is not constant, we still have the explicit formula V 0 = S 0N  1  , KB0;TN 2 : CHAPTER 33. Change of num´eraire 331 As this formula suggests, if  F is constant, then for 0  t  T , the valueof a European call expiring at time T is V t=StN 1 t , KBt; T N  2 t; where  1 t= 1  F p T ,t  log F t K + 1 2  2 F T , t  ;  2 t= 1  F p T ,t  log F t K , 1 2  2 F T , t  : This formula also suggests a hedge: at each time t , hold N  1 t shares of stock and short KN  2 t bonds. We want to verify that this hedge is self-financing. Suppose we begin with $ V 0 and at each time t hold N  1 t shares of stock. We short bonds as necessary to finance this. Will the position in the bond always be ,KN  2 t ? If so, the value of the portfolio will always be S tN  1 t , KBt; T N  2 t = V t; and we will have a hedge. Mathematically, this question takes the following form. Let t=N 1 t: At time t , hold t shares of stock. If X t is the value of the portfolio at time t ,then X t , tS t will be invested in the bond, so the number of bonds owned is X t,t B t;T  S t and the portfolio value evolves according to dX t= t dS t+ Xt, t B t; T  S t dB t; T : (3.1) The value of the option evolves according to dV t=N 1 t dS t+St dN  1 t + dS t dN  1 t , KN  2 t dB t; T  , KdBt; T  dN  2 t , KBt; T  dN  2 t: (3.2) If X 0 = V 0 , will X t=Vt for 0  t  T ? Formulas (3.1) and (3.2) are difficult to compare, so we simplify them by a change of num´eraire. This change is justified by the following theorem. Theorem 3.73 Changes of num´eraire affect portfolio values in the way you would expect. Proof: Suppose we have a model with k assets with prices S 1 ;S 2 ;::: ;S k . At each time t , hold  i t shares of asset i , i =1;2;::: ;k , 1 , and invest the remaining wealth in asset k . Begin with a nonrandom initial wealth X 0 ,andlet X t be the value of the portfolio at time t . The number of shares of asset k held at time t is  k t=  Xt, P k,1 i=1  i tS i t  S k t ; 332 and X evolves according to the equation dX = k,1 X i=1  i dS i +  X , k,1 X i=1  i S i ! dS k S k = k X i=1  i dS i : Note that X k t= k X i=1  i tS i t; and we only get to specify  1 ;::: ; k,1 , not  k , in advance. Let N be a num´eraire, and define b X t= Xt Nt ; c S i t= S i t Nt ; i=1;2;::: ;k: Then d b X = 1 N dX + Xd  1 N  +dX d  1 N  = 1 N k X i=1  i dS i +  k X i=1  i S i ! d  1 N  + k X i=1  i dS i d  1 N  = k X i=1  i  1 N dS i + S i d  1 N  + dS i d  1 N  = k X i=1  i d c S i : Now  k =  X , P k,1 i=1  i S i  S k =  X=N , P k,1 i=1  i S i =N  S k =N = b X , P k,1 i=1  i c S i c S k : Therefore, d b X = k X i=1  i d c S i +  b X , k,1 X i=1  i c S i ! d c S k c S k CHAPTER 33. Change of num´eraire 333 This is the formula for the evolutionof a portfoliowhich holds  i shares of asset i , i =1;2;::: ;k, 1 , and all assets and the portfolio are denominated in units of N . We return to the European call hedging problem (comparison of (3.1) and (3.2)), but we now use the zero-coupon bond as num´eraire. We still hold t=N 1 t shares of stock at each time t . In terms of the new num´eraire, the asset values are Stock: S t B t; T  = F t; Bond: B t; T  B t; T  =1: The portfolio value evolves according to d b X t=tdF t+ b Xt,t d1 1 =tdF t: (3.1’) In the new num´eraire, the option value formula V t=N 1 tS t , KBt; T N  2 t becomes b V t= Vt Bt; T  = N  1 tF t , KN  2 t; and d b V = N  1 t dF t+Ft dN  1 t + dN  1 t dF t , KdN 2 t: (3.2’) To show that the hedge works, we must show that F t dN  1 t + dN  1 t dF t , KdN 2 t = 0: This is a homework problem. 334 [...]... volatility F t; T  and mean  rate of return 2 t; T  The change of measure from IPT to IPS makes F 1t a martingale, i.e., it F  changes the mean return to zero, but the change of measure does not affect the volatility Therefore, t; T  in (2.1) must be F t; T  and WS must be WS t = ,WT t + Zt 0 F u; T  du: 33.3 Merton option pricing formula The price at time zero of a European call is  1  +... (3.1) and (3.2) are difficult to compare, so we simplify them by a change of num´ raire e This change is justified by the following theorem Theorem 3.73 Changes of num´ raire affect portfolio values in the way you would expect e Proof: Suppose we have a model with k assets with prices S1 ; S2; : : : ; Sk At each time t, hold
i t shares of asset i, i = 1; 2; : : : ; k , 1, and invest the remaining wealth... Sk N CHAPTER 33 Change of num´ raire e 333 This is the formula for the evolution of a portfolio which holds
i shares of asset i, i = 1; 2; : : : ; k , 1, and all assets and the portfolio are denominated in units of N We return to the European call hedging problem (comparison of (3.1) and (3.2)), but we now use the zero-coupon bond as num´raire We still hold
t = N  1t shares of stock at each... CHAPTER 33 Change of num´ raire e 331 As this formula suggests, if F is constant, then for 0  t  T , the value of a European call expiring at time T is V t = S tN  1t , KB t; T N  2t; where   F t 1 2 T , t ; log K + 2 F 1t = F T ,t 1 F t , 1 2 T , t : log K 2 F 2t = p T ,t p1 F This formula also suggests a hedge: at each time t, hold KN  2t bonds N  1t shares of stock... hold N  1t shares of stock We short bonds as necessary to finance this Will the position in the bond always be ,KN  2t? If so, the value of the portfolio will always be S tN  1t , KB t; T N  2t = V t; and we will have a hedge Mathematically, this question takes the following form Let
t = N  1t: At time t, hold
t shares of stock If X t is the value of the portfolio at... time t, hold
i t shares of asset i, i = 1; 2; : : : ; k , 1, and invest the remaining wealth in asset k Begin with a nonrandom initial wealth X 0, and let X t be the value of the portfolio at time t The number of shares of asset k held at time t is  , X t , Pk=11
i tSi t i
k t = ; Sk t 332 and X evolves according to the equation dX = = k,1 X i=1 k X i=1 
i dSi + X , k,1 X i=1 !... If X t is the value of the portfolio at time t, then X t , t
tS t will be invested in the bond, so the number of bonds owned is X B,
t S t and the t;T  portfolio value evolves according to t ,
 dX t =
t dS t + X Bt; T  t S t dBt; T : (3.1) The value of the option evolves according to dV t = N  1t dS t + S t dN  1t + dS t dN  1t , KN  2t dBt;... We return to the European call hedging problem (comparison of (3.1) and (3.2)), but we now use the zero-coupon bond as num´raire We still hold
t = N  1t shares of stock at each time t e In terms of the new num´ raire, the asset values are e Stock: Bond: S t = F t; Bt; T  Bt; T  = 1: Bt; T  The portfolio value evolves according to b b dX t =
t dF t + X t ,
t d1 =
t . compare, so we simplify them by a change of num´eraire. This change is justified by the following theorem. Theorem 3.73 Changes of num´eraire affect portfolio. rate of return  2 F t; T  . The change of measure from IP T to IP S makes 1 F t a martingale, i.e., it changes the mean return to zero, but the change