Chapter 33 Changeof 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= rt Stdt + tS t dW t; (0.1) where the interest rate process rt 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 rt . We define the accumulation factor t = exp Z t 0 ru 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 ru 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 ST,Ft; T F t = tIE S T T Ft , Ft; 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 = St Bt; 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 Bt;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 N0 Z A N T T dIP; 8A 2FT ; is risk-neutral for N . CHAPTER 33. Changeofnum´eraire 327 Note: IP and IP N are equivalent, i.e., have the same probability zero sets, and IP A=N0 Z A T N T dIP N ; 8A 2FT : 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 changeof 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 B0;T Z A BT; T T dIP = 1 B 0;T Z A 1 T dIP 8A 2FT: 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 2FT . IP T is called the T -forward measure. Denominated in units of T -maturity bond, the value of the stock is F t; T = St Bt; 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 ;0tTg 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 S0 Z A S T T dIP; 8A 2FT : 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. Changeofnum´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 Ft 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 changeof measure from IP T to IP S makes 1 F t a martingale, i.e., it changes the mean return to zero, but the changeof 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 fST K g = S 0 Z fS T K g S T S 0 T dIP , KB0;T Z fSTK g 1 B 0;TT dIP = S 0IP S fS T Kg,KB0;TIP T fST Kg =S0IP S fF T Kg,KB0;TIP T fFT Kg =S0IP S 1 F T 1 K , KB0;TIP T fFT 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 S0 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 Tlog S 0 KB0;T = IP S W S T p T 1 F p T log S 0 KB0;T + 1 2 F p T = N 1 ; where 1 = 1 F p T log S 0 KB0;T + 1 2 2 F T : Similarly, F T = S0 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 Tlog KB0;T S0 = IP T W T T p T 1 F p T log KB0;T S0 + 1 2 2 F T = IP T ,W T T p T 1 F p T log S 0 KB0;T , 1 2 2 F T = N 2 ; where 2 = 1 F p T log S 0 KB0;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 0N 1 , KB0;TN 2 : CHAPTER 33. Changeofnum´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=StN 1 t , KBt; 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 tN 1 t , KBt; 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 , tS 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+ Xt, 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+St dN 1 t + dS t dN 1 t , KN 2 t dB t; T , KdBt; T dN 2 t , KBt; T dN 2 t: (3.2) If X 0 = V 0 , will X t=Vt for 0 t T ? Formulas (3.1) and (3.2) are difficult to compare, so we simplify them by a changeof num´eraire. This change is justified by the following theorem. Theorem 3.73 Changes ofnum´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= Xt, P k,1 i=1 i tS 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 tS 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= Xt Nt ; c S i t= S i t Nt ; 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. Changeofnum´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=tdF t+ b Xt,t d1 1 =tdF t: (3.1’) In the new num´eraire, the option value formula V t=N 1 tS t , KBt; T N 2 t becomes b V t= Vt Bt; T = N 1 tF t , KN 2 t; and d b V = N 1 t dF t+Ft 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 changeof measure from IPT to IPS makes F 1t a martingale, i.e., it F changes the mean return to zero, but the changeof 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 1 t 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 t N 1 t , KB t; T N 2 t ; where F t 1 2 T , t ; log K + 2 F 1 t = F T ,t 1 F t , 1 2 T , t : log K 2 F 2 t = p T ,t p1 F This formula also suggests a hedge: at each time t, hold KN 2 t bonds N 1 t shares of stock... 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, 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 t Si 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 t S 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 B t; 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;... 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 1 t 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 ; B t; T B t; T = 1: B t; T The portfolio value evolves according to b b dX t = t dF t + X t , t d 1 = 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