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9. Change of Numeraire 鄭凱允 9.1 Introduction • A numeraire is the unit of account in which other assets are denominated and change the numeraire by changing to the currency of a country. • In this chapter, we will work within the d-dimensional market model by independent Brownian motion W (t ) (W1 (t ),...,Wd (t )),0 t T , There is an adapted interest rate process R(t) ,0 t T. This can be used to create a money market account whose price per share at time t is 0 R (u ) du . M (t ) e t We also define the discount process 0 R ( u ) du D (t ) e t 1 . M (t ) There are m primary assets in the model of this chapter, and their prices satisfy equation (5.4.6), dSi (t ) i (t ) Si (t ) dt Si (t ) ij (t ) dW j (t ), j 1 d i 1,..., m. (9.1.1) p We assume there is a unique risk-neutral measure (i.e., there is a unique d-dimensional process (t ) (1 (t ),..., d (t )) satisfying the market price of risk equations (5.4.18)). See Appendix p By Theorem 5.4.1, under , the Brownian motions j (t ) W (t ) t (u )du, W j j 0 j 1,..., d , Are independent of one another. Under p , the discounted asset prices D(t ) Si (t ) are martingales. if we were to denominate the ith asset in terms of the money market account, its price would be Si (t ) / M (t ) D(t ) Si (t ) . In other words, at time t, the ith asset is worth D(t ) Si (t ) shares of the money market account. We say the measure is risk-neutral for the money p market account numeraire. 9.2 Numeraire The numeraires we consider in this chapter are: • Domestic money market account associated risk-neutral measure by in section 9.1 p • Foreign money market account associated risk-neutral measure by f in section 9.3 p • A zero-coupon bond maturing at time T associated riskT neutral measure by . It is called the T-forward p measure and is used in Section 9,4 Theorem 9.2.1 (Stochastic representation of assets). Let N be a strictly positive price process for a non-dividendpaying asset, either primary or derivative, in the multidimensional market model of Section 9.1. then there exists a vector volatility process v(t ) (v1 (t ),..., vd (t )) such that dN (t ) R (t ) N (t )dt N (t )v (t ) dW (t ). (9.2.1) This equation is equivalent to each of the equations d ( D(t ) N (t )) D(t ) N (t )v(t ) dW (t ), (u ) 1 t v(u ) 2 du}. D(t ) N (t ) N (0) exp{ v(u ) dW 0 2 0 t (u ) t ( R(u ) 1 v(u ) 2 )du} N (t ) N (0) exp{ v(u ) dW 0 0 2 t (9.2.2) (9.2.3) (9.2.4) p Proof: Under the risk-neutral measure , the discounted price process D(t ) N (t ) must be a martingale. According to the Martingale Representation Theorem, Theorem 5.4.2, d ( D (t ) N (t )) j (t )dW j (u ) (t ) dW (u ) j 1 d For some adapted d-dimensional process (t ) (1 (t ),..., d (t )). Because N(t) is strictly positive, we can define the vector v(t ) (v1 (t ),..., vd (t )) by j (t ) v j (t ) . D (t ) N (t ) Then d ( D(t ) N (t )) D(t ) N (t )v(t ) dW (u ), which is (9.2.2). The solution to (9.2.2) is (9.2.3), as we now show. Define (u ) 1 t v(u ) 2 du X (t ) v(u ) dW 0 2 0 t 1 d t 2 v j (u )dW j (u ) v j (u )du , 0 2 j 1 0 j 1 t d so that 1 2 dX (t ) v(t ) dW (t ) v(t ) dt 2 1 d 2 v j (t )dW j (t ) v j (t ) dt. 2 j 1 j 1 d Let f ( x) N (0)e x , and complete 1 df ( X (t )) f '( X (t ))dX (t ) f "( X (t ))dX (t )dX (t ) 2 f ( X (t ))v(t ) dW (t ). We see that f(X(t)) solves (9.2.2), f(X(t)) has the desired initial condition f(X(0))=N(0), and f(X(t)) is the right-hand side of (9.2.3). Form (9.2.3), we have immediately that (9.2.4) holds. Applying the Ito-Doeblin formula to (9.2.4), we obtain (9.2.1). According to Theorem 5.4.1, we can use the volatility vector of N(t) to change the measure. Define (jN ) (t ) t v (u )du W (t ), j 1,..., d , W j j 0 (9.2.5) And a new probability measure (N ) P ( A) 1 D(T ) N (T )d P N (0) A for all A F. (9.2.6) D(T ) N (T ) We see from (9.2.3) that is the random variable Z(T) N (0) appearing in (5.4.1) of the multidimensional Girsanov Theorem if we replace j (t ) by v (t ) for j = 1,…,m. here we are using the probability measure P in place of P in theorem 5.4.1 j With these replacements, Theorem 5.4.1 implies that, under P , (N) (N) (N) W (t ) (W 1 (t ),...,W d (t )) is a d-dimensional the process ) ( N , the Brownian motions Brownian motion. In particular, under P (N ) are independent of one another. The expected (N ) value of an arbitrary random variable X under P can be computed by the formula (N ) X E 1 E[ XD (T ) N (T )]. N (0) (N ) (N) W 1 (t ),...,W d (t ) (9.2.7) More generally, D(t ) N (t ) D(T ) N (T ) E[ | F (t )], N (0) N (0) 0t T is the Radon-Nikodym derivative process Z(t) in the Theorem 5.4.1, and Lemma 5.2.2 implies that for 0 s t T and Y an F(t)-measurable random variable, ( N ) [Y | F ( s)] E 1 E[YD(t ) N (t ) | F ( s )]. D( s ) N ( s ) (9.2.8) Remark 9.2.3. Equation (9.2.9) says that the volatility vector of S ( N ) (t ) is the difference of the volatility vectors of S(t) and N(t). In particular, after the change of numeraire, the price of the numeraire becomes identically 1, N (N ) N (t ) (t ) 1 N (t ) and this has zero volatility vector: dN (N ) (t ) N (N ) ( N ) (t ) 0. (t )[v(t ) v(t )] dW We are saying that volatility vectors subtract. The process N(t) in Theorem 9.2.2 has the stochastic differential representation (9.2.1), which we may rewrite as dN (t ) R(t ) N (t )dt v(t ) N (t )dB N (t ), (9.2.10) where v(u ) Remark 9.2.4. If we take the money market account as the numeraire in Theorem 9.2.2, then we have d(D(t)N(t)) = 0. the volatility vector for the money market account is v(t)=0, and the volatility vector for an asset S ( N ) (t ) denominated in units of money market account is the same as the volatility vector of the asset denominated in units of currency. Discounting an asset using the money market account does not affect its volatility vector. 0 j 1 B (t ) N t d v j (u ) dW u (t ). Remark 9.2.5. Theorem 9.2.2 is a special case of a more general result. Whenever M 1 (t ) and M 2 (t ) are martingales under a measure P, M 2 (0) 1 and M 2 (t ) takes only positive values, then , M1 (t ) / M 2 (t ) is a martingale under the measure P ( M ) defined by 2 P( M 2 ) ( A) M 2 (T )dP. A Proof of Theorem 9.2.2: we have (u ) 1 t (u ) 2 du}, D(t ) S (t ) S (0) exp{ (u ) dW 0 2 0 t (u ) 1 t v(u ) 2 du}, D(t ) N (t ) N (0) exp{ v(u ) dW 0 2 0 t and hence S (N ) t S (0) (u ) 1 t ( (u ) 2 v(u ) 2 )du} (t ) exp{ ( (u ) v(u )) dW 0 N (0) 2 0 To apply the Ito-Doeblin formula to this, we first define (u ) 1 t ( (u ) 2 v(u) 2 )du, X (t ) ( (u ) v(u )) dW 0 2 0 t so that (t ) 1 ( (u ) 2 v(u ) 2 )dt dX (t ) ( (t ) v(t )) dW 2 1 ( j (t ) v j (t ))dW j (t ) ( 2 (t ) v 2 (t ))dt , j j 2 j 1 j 1 d d dX (t )dX (t ) ( j (t ) v j (t )) 2 dt j 1 d ( 2 (t ) 2 j (t )v j (t ) v 2 (t )) dt j j j 1 d (u ) dt 2 (t ) v(t )dt v(u ) dt. 2 2 S (0) x With f ( x) N (0) e , we have S ( N ) (t ) f ( X (t )) and dS ( N ) (t ) df ( X (t )) 1 f "( X )dXdX 2 (N ) 1 2 dt 1 v 2 dt S [( v) dW 2 2 1 1 2 2 dt vdt v dt ] 2 2 f '( X )dX S ( N ) [( v) dW v ( v)dt ] S ( N ) ( v) (vdt dW ) S (N ) (N ) (N ). ( v) dW (N ) Since W (t ) is a d-dimensional Brownian motion under P , the process S ( N ) (t ) is a martingale under this measure. Appendix Use matrix to extensive (9.1.1) dS1 (t ) [ ][ 1 (t ) S1 (t ) S1 (t ) 11 (t ) S1 (t ) 1d (t ) dW1 (t ) ]dt [ ][ ] dSm (t ) m (t ) Sm (t ) Sm (t ) m1 (t ) S m (t ) md (t ) dWd (t ) In order to make risk-free return, we plus (t ) with diffusion term. dS1 (t ) [ dSm (t ) ][ r (t ) S1 (t ) S1 (t ) 11 (t ) S1 (t ) 1d (t ) dW1 (t ) 1 (t )dt ]dt [ ][ ] r (t ) Sm (t ) Sm (t ) m1 (t ) Sm (t ) md (t ) dWd (t ) d (t )dt so that we find (t ) satisfying (5.4.18)

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risk neutral, probability measure, stock price, asset pricing, Brownian Motion, change of measure, money market, risk-neutral measure, interest rate, mathematical finance, martingale measure, the dynamics, fundamental theorem, Option Pricing, forward measure

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posted: | 12/17/2009 |

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