The discounted portfolio value of a selffinancing strategy in discrete time was given by. δ tj 1 (s tj s tj 1 ) (9.1) j=1

Transcription

1 Chapter 9 The isk Neutral Pricing Measure for the Black-Scholes Model The discounted portfolio value of a selffinancing strategy in discrete time was given by v tk = v 0 + k δ tj (s tj s tj ) (9.) where v tk = e rt k Vtk is the discounted portfolio value at time t k = k t, k = 0,,..., N T = T/ t and s tk = e rt k Stk denotes the discounted asset price. Suppose that the price process S tk is given by a geometric Brownian motion such that s tk = e rt 2 k (µ S 0 e 2 )t k+x tk = S 0 e xt k +( 2 /2)t k =: s(x tk, t k ) (9.2) Suppose further that we have some european option with payoff H or discounted payoff h = e rt H where h = h(s tn ) or more generally h = h(s t0,..., S tn ), N = N T, and suppose that there is a replicating strategy such that δ tk = δ tk (S t,..., S tk ) (9.3) h = v tn = v 0 + Since v 0 = V 0, the price of the option, is a number, we can write v 0 = E v 0 = E h N δ tk (s tk s tk ) (9.4) k= N E δ tk (s tk s tk ) (9.5) k= where the expectation in (9.5) can be choosen arbitrarily. Let us consider (9.5) for the Wiener measure, dw ({x t } 0<t T ) = N T lim Π t 0 k= p t (x (k ) t, x k t ) dx k t (9.6) 65

2 66 Chapter 9 Because of Theorem 4. we have E W δk (s k s k ) = δ k (x t,..., x tk ) ( s(x tk, t k ) s(x tk, t k ) ) Π k p tj t j (x tj, x tj ) dx tj k ( = δ k (x t,..., x tk ) s(x, t tk k) p (x, x ) dx s(x tk tk tk tk tk tk k )), t k = ( δ k (x t,..., x tk ) E W s(xtk, t k ) x tk s(xtk, t k ) k k Π p tj t j (x tj, x tj ) dx tj ) k Π (9.7) p tj t j (x tj, x tj ) dx tj where we introduced the conditional expectation E W f({xs } 0 s T ) x t := f({x s }) dw ({x s } t<s T ) (9.8) with the obvious definition (t = N t t) dw ({x t } t<s T ) := lim N T Π t 0 j=n t+ p t (x (j ) t, x j t ) dx j t (9.9) In particular, if the function f in (9.8) depends only on a single x s, f = f(x s ), then f(x s ) if t s E W f(xs ) x t = (9.0) f(x s ) p s t (x t, x s ) dx s if t < s Let us compute the conditional expectation E W s(xtk, t k ) x tk in the last line of (9.7). We have, using t = t k t k E W s(xtk, t k ) x tk = s(x tk, t k ) p tk t k (x tk, x tk ) dx tk = t S 0 e xt k +( 2 /2)t k e (x tk x tk ) 2 2 t dx tk = S 0 e xt k +( 2 /2)t k = S 0 e xt k +( 2 /2)t k e 2 2 (t k t k ) e t y e y2 2 = s(x tk, t k ) e ()(t k t k ) (9.) Now, suppose the factor e ()(t k t k ) in (9.) would be absent. Then the round brackets in the third line of (9.7) would be zero for all k and the price of the option v 0 would be

3 Chapter 9 67 given by the expectation of the discounted payoff. Thus, we would be able to compute the price without knowing the hedging strategy, provided that there is a replicating strategy. Now, this factor is not absent but we can ask the following question: Is there some measure d W such that s(xtk, t k ) x tk = s(xtk, t k ) (9.2) If this is the case then we can compute the price v 0 of the option with discounted payoff h by taking the expectation value with respect to d W, v 0 = h (9.3) since the round brackets in the third line of (9.7) all vanish. There is the following Theorem 9.: Let s tk = s(x tk, t k ) be a discounted geometric Brownian motion given by (9.2). Define the kernels p t (x, y) = p µ,r, t (x, y) by Then: p t (x, y) := t e (x y t)2 2t (9.4) a) The kernels p satisfy p s (x, y) p t (y, z) = p s+t (x, z) (9.5) and p s(x, y) = for all x. That is, the measure is well defined. d W ({x t } 0<t T ) := lim N T Π t 0 k= p t (x (k ) t, x k t ) dx k t (9.6) b) The price process (9.2) is a martingale with respect to d W. That is, s(xt, t ) x t := s(x t, t ) d W ({x s } t<s T ) = s(x t, t) t < t (9.7) The measure d W is called an equivalenartingale measure (with respect to the price process s t ).

4 68 Chapter 9 Proof: a) Let p t (x, y) be the kernel (4.3). Then p t (x, y) = p t ( x t, y) ( = p t x, y + t) ( = p t x, y + ) 2 2 (9.8) such that, with Lemma 4., p s (x, y) p t (y, z) = p s ( x = p s+t ( x which proves part (a). Part (b) is obtained as (9.), s(xt, t ) x t = s(x t, t ) p t t(x t, x t ) dx t = (t t) s, y) ( p t y, z + t) s, z + t) = p s+t (x, z) (9.9) S 0 e x t +( 2 /2)t e (x t x t (t 2 t) ) 2(t t) dx t = S 0 e xt+( 2 /2)t e 2 2 (t t) e (x t (t t) = S 0 e xt+( 2 /2)t e 2 2 (t t) e t t y e y2 2 = S 0 e xt+( 2 /2)t xt+ (t t)) ( x t x t (t t)) 2 e 2(t t) dx t = s(x t, t) (9.20) This proves the theorem. In chapter 5 where we approximated the Black-Scholes model with a suitable Binomial model, we were able to prove the following pricing formula for some non path dependent option with payoff H = T ), see Theorem 5.2: V BS 0 = e rt H ( 2 (r S 0 e ) 2 )T + T x e x2 2 dx (9.2) Let us rederive (9.2) by using the equivalenartingale measure. Suppose we have a european option with discounted payoff h(s T ) = e rt T ) and that the namics of S is given by the Black-Scholes model with drift µ and volatility, S T = S 0 e x T +(µ 2 2 )T (9.22) Observe that the drift parameter µ does not show up in the pricing formula (9.2). This was actually a quite fundamental result of chapter 5. Here we will come up with the same conclusion:

5 Chapter 9 69 Let δ be the replicating strategy. Then N T v T = h(s T ) = v 0 + δ tk (s tk s tk ) (9.23) k= We fix some t = t k and take the expectation with respect to d W ({x s } t<s T ). This gives and we get h(st ) x t = v0 + k δ tj (s tj s tj ) = v tk = v t = e rt V t (9.24) V t = e rt h(st ) x t = e r(t t) H ( S 0 e ) x T +(µ 2 2 )T p T t (x t, x T ) dx T = e r(t t) = e r(t t) = e r(t t) = e r(t t) H ( S 0 e ) x T +(µ 2 2 )T e (x t x T (T t) ) 2 2(T t) dx T (T t) H ( S t e (x T x t)+()(t t) e 2 2 (T t) e r(t t)) e (x T x t + (T t) ) 2 H ( 2 y+(r S t e 2 )(T t)) e y2 2(T t) (T t) H ( S t e T t y+(r 2 2 )(T t)) e y2 2 2(T t) dx T (T t) (9.25) and, for t = 0, this coincides with (9.2). More generally, there holds the following Theorem 9.2: Let {x t } 0<t T be a Brownian motion and let t := S 0 e xt+(µ 2 2 )t (9.26) S (µ) Let dw be the Wiener measure and d W be the equivalenartingale measure. Then the following equality (µ) t ) = E W (r) t,..., S (r) ) (9.27) holds for all payoffs H = t,..., S tm ). In particular, the theoretical fair value V 0 = e rt (µ) t ) = e rt (r) E W t,..., S (r) ) (9.28)

6 70 Chapter 9 does not depend on the (usually not predictable) drift parameter µ, but depends only on the volatility parameter and the interest rate level r. Proof: From the definition of p we have (t 0 := 0) (µ) t ) = Because of m (µ) t ) m Π t + j t j (x t + j t j ) 2 (tj t j ) e x 2(t j t j ) dx tj S (µ) t = S (µ) t (x t ) = S 0 e xt+(µ 2 = S 0 e = S (r) t 2 )t 2 (xt+ t)+(r 2 )t (x t + t) (9.29) the statement follows from the substitution of variables y tj = x tj + t j.