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8-4 Finite-Expiration American Put 財金所 潘政宏 Consider an American put on a stock whose price is the geometric Brownian motion: dS(t) rS(t)dt S(t)dW(t) (8.3.1) But now the put has a finite expiration time T. Definition 8.4.1 Let 0 t T and x 0 be given. Assume S(x) x. Let F , t u T, (t ) u denote the - algebra generated by the process S( ) as range over [t, u], and let denote the set of the sopping time for the filtration t ,T F , t u T taking values in [t, T] or taking the value . In other (t ) u words, { u} F for every u [t, T]; a stopping time in makes (t ) u t ,T the decision t o stop at a time u [t, T] based only on the path of the stock price between ti mes t and u. The price at time t of the American put expiring at time T is defined to be ~ v(t, x) max E[e (K - S( )) | S(t) x] (8.4.1) - r( - t) t ,T In the event that , we interpret e (K - S( )) to be zero. -r This is the case when the put expires unexercise d. 8.4.1 Analytical Characterization of the Put Price The finite-expiration American put price function v(t,x) satisfies the linear complementarity conditions : v(t,x) (K-x) for all t [0,T], x 0 (8.4.2) 1 2 2 rv(t,x)-v t (t , x) rxvx (t , x) x vxx (t , x) 0 2 for all t [0,T), x 0, and (8.4.3) for each t [0,T) and x 0, equality holds in either (8.4.2) or(8.4.3) (8.4.4) L(T-t) Level L(T-t) depends on the time to expiration T-t L(T) is decreases with increasing T The set {(t , x); 0 t T , x 0} can be divided into two regions, the stopping set S {(t,x);v(t,x) (K-x) } (8.4.5) and the continuation set C {(t,x);v(t,x) (K-x) } (8.4.6) 1 2 2 rv(t , x) - vt (t , x) rxvx (t , x) x vxx (t , x) 2 rK rx 0 rx 0 rK Because v(t,x) K-x for 0 x L(T-t), we also have the left-handderivatives v x (t , x ) 1 on the curve x L(T t ). The put price v(t,x) satisfies the smooth-pasting condition that v x (t , x) is continuous, even at x L(T-t). In other words, v x (t , x ) v x (t , x ) 1 for x L(T t ), 0 t T (8.4.7) The smooth-pasting condition does not hold at t T. Indeed, L(0) K and v(T,x) (K-x) so v x (T , x ) 1, whereas v x (T , x ) 0. Also, v t (T , x) and v xx (t , x) are not continuous along the curve x L(T t) . (8.4.8) Determine the function v(t,x) 1 2 2 rv(t,x)-v t (t , x) rxvx (t , x) x vxx (t , x) 0 ,x L(T-t), 2 v(t,x) K-x, 0 x L(T-t), v x (t , x ) v x (t , x ) 1 for x L(T t ), 0 t T lim v(t , x) 0 x 8.4.2 Probabilistic Characterization of the Put Price Theorem 8.4.2 Let S(u),t u T,be the stock price of(8.3.1) starting at S(t) x and with the stopping set S defined by(8.4.5). Let * min{u [t , T ];(u, S (u)) S} (8.4.10) where we interpret * to be if (u,s(u)) doesn't enter S for any u [t.T]. Then e-ru v(u, S (u)), t u T,is a supermartingale under P, and the stopped process e -r(u * ) v(u, S (u * )), t u T, is a martingale. d [e- ru v(u , S (u ))] v(u, S (u ))de- ru e- ru dv(u , S (u )) e- ru [rv(u, S (u ))du vu (u, S (u ))du vx (u, S (u ))dS (u ) 1 vxx (u, S (u ))dS (u )dS (u )] 2 e- ru [rv(u, S (u )) vu (u, S (u )) rS (u )vx (u, S (u )) 1 2 S 2 (u )vxx (u, S (u ))]du e- ru S (u )vx (u, S (u ))dW (u ) 2 (8.4.11) -ru According to Figure 8.4.1, the du term in (8.4.11) is - e rKΙ {S(u) L(T u)} ~ This is nonpositiv e, and so e v(u, S(u)) is a supermarti ngale under P. - ru In fact, starting from u t and up until time , we have S(u) L(T - u), * so the du term is zero. Therefore, the stopped process e v(u , S (u )), t u T , is a martingale . - r(u * ) * * Corollary 8.4.3 Consider an agent with initial capital X(0) v(0, S(0)), the initial finite - expiration put price. Suppose this agent uses the portfolio process (u ) v (u , S (u )) and consumes cash at rate C(u) rKΙ x {S(u) L(T u)} per unit time. Then X(u) v(u, S(u)) for all times u between u 0 and the time the option is exercised or expires. In particular , S(u) (K - S(u)) for all times u until the option is exercised or expires, so the agent can pay off a short option position regardless of when the option is exercised. Proof 8.4.3 dX t t dS t r X t t S t dt C t dt dS (t ) rS (t )dt S (t )dW (t ) d e rt X t e rt rX t dt dX t e rt t dS t r t S t dt C t dt e rt t S t dW t C t dt d e rt X t d e rt v t , S t and X 0 v 0, S 0 we obtain X t v t , S t Remark 8.4.4 The proofs of Theorem 8.4.2 and Corollary 8.4.3 use the analytic characterization of the American put price captured in Figure 8.4.1 plus the smooth-pasting condition that guarantees that v x (t , x) is ˆ continuous even on the curve x L(T-t) so that the Ito-Doeblin formula can be applied. Here we show that the only function v(t,x) satisfying these conditions is the function v(t,x) defined by(8.4.1) To do this, we first fix t with 0 t T . The supermartingale property for e-rt v(t , S (t ))of Theorem 8.4.2 and Theorem 8.4.2(optional sampling) implies that e-r(t ) v(t , S (t )) E[e-r(T ) v(T , S (T )) | F(t)] e-r(t ) v(t , S (t )) E[e-r(T ) v(T , S (T )) | F(t)] For t ,T , we have t t , whereas T if and T T if . Therefore, for t ,T e-rt v(t , S (t)) E[e-r v( , S ( )){ } e -rT v(T , S (T )){ } | F (t )] E[e-r v( , S ( )) | F (t )] (8.4.12) where, as usual, we interpret e-r v( , S ( )) 0 if . Inequality (8.4.2) and the fact that (K-S(t)) K S (t ) imply that E[e-r v( , S ( )) | F (t )] E[e-r ( K S ( )) | F (t )] E[e-r ( K S ( )) | F (t )] Putting (8.4.12) and (8.4.13) together, we have (8.4.13) e-rt v(t , S (t )) E[e-r ( K S ( )) | F (t )] (8.4.14) Because S(t) is a Markov process, the right-hand side of (8.4.14) is e-rt v(t , x) E[e-r ( K S ( )) | S (t ) x] since (8.4.15) holds for any t,T , we conclude that v(t , x) max E[e-r( t ) ( K S ( )) | S (t ) x] t ,T (8.4.15) (8.4.16) For the reverse inequality, we recall form Theorem 8.4.2 that the stopped process e-r(t * ) v(t * , S (t * )) is a martingale,where * defined by (8.4.10) is such that v( * , S ( * )) K S ( * ) if * . e-r(t * ) v(t * , S (t * )) E[e-r(T * ) v(T * , S (T * )) | F(t)] e-rt v(t , S (t)) E[e-r * v( * , S ( * )){ * } e -rT v(T , S (T )){ * } | F (t )] E[e-r * v( * , S ( * )) | F (t )] Finally, because v( * ,S( * )) K-S( * ) if * v(t , x) E[e-r( * t ) ( K S ( * )) | S (t ) x] (8.4.17) Equality (8.4.17) shows that equality must hold in (8.4.16), and this is (8.4.1).