Options on dividend-paying stocks

Chapter 26 Options on dividend-paying stocks 26.1 American option with convex payoff function Theorem 1.64 Consider the stock price process dS t=rtStdt + tS t dB t; where r and  are processes and rt  0; 0  t  T; a.s. This stock pays no dividends. Let hx be a convex function of x  0 , and assume h0 = 0 .(E.g., hx=x,K + ). An American contingent claim paying hS t if exercised at time t does not need to be exercised before expiration, i.e., waiting until expiration to decide whether to exercise entails no loss of value. Proof: For 0    1 and x  0 ,wehave hx=h1 , 0 + x  1 , h0 + hx = hx: Let T be the time of expiration of the contingent claim. For 0  t  T , 0   t  T  = exp  , Z T t ru du   1 and S T   0 ,so h   t  T  S T     t  T  hS T : (*) Consider a European contingent claim paying hS T  at time T . The value of this claim at time t 2 0;T is X t=tIE  1 T hST     F t  : 263 264 - 6                     . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . r r r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x; hx x h hx hx Figure 26.1: Convex payoff function Therefore, X t  t = 1  t IE   t  T  hS T      F t   1  t IE  h   t  T  S T       F t  (by (*))  1  t h   t IE  S T   T      F t  (Jensen’s inequality) = 1  t h   t S t  t  ( S  is a martingale) = 1  t hS t: This shows that the value X t of the European contingent claim dominates the intrinsic value hS t of the American claim. In fact, except in degenerate cases, the inequality X t  hS t; 0  t  T; is strict, i.e., the American claim should not be exercised prior to expiration. 26.2 Dividend paying stock Let r and  be constant, let  be a “dividend coefficient” satisfying 0 1: CHAPTER 26. Options on dividend paying stocks 265 Let T0 be an expiration time, and let t 1 2 0;T be the time of dividend payment. The stock price is given by S t=  S0 expfr , 1 2  2 t + Btg; 0  t  t 1 ; 1 ,  S t 1  expfr , 1 2  2 t , t 1 +Bt,Bt 1 g; t 1 tT: Consider an American call on this stock. At times t 2 t 1 ;T , it is not optimal to exercise, so the value of the call is given by the usual Black-Scholes formula v t; x= xN d + T , t; x , Ke ,rT ,t N d , T , t; x; t 1 tT; where d  T , t; x= 1  p T ,t  log x K +T ,tr   2 =2  : At time t 1 , immediately after payment of the dividend, the value of the call is v t 1 ; 1 ,  S t 1 : At time t 1 , immediately before payment of the dividend, the value of the call is wt 1 ;St 1 ; where wt 1 ;x = max  x , K  + ;vt 1 ;1 ,  x  : Theorem 2.65 For 0  t  t 1 , the value of the American call is wt; S t ,where wt; x=IE t;x h e ,rt 1 ,t wt 1 ;St 1  i : This function satisfies the usual Black-Scholes equation ,rw + w t + rxw x + 1 2  2 x 2 w xx =0; 0tt 1 ;x0; (where w = wt; x ) with terminal condition wt 1 ;x = max  x , K  + ;vt 1 ;1 ,  x  ;x0; and boundary condition wt; 0 = 0; 0  t  T: The hedging portfolio is t=  w x t; S t; 0  t  t 1 ; v x t; S t; t 1 tT: Proof: We only need to show that an American contingent claim with payoff wt 1 ;St 1  at time t 1 need not be exercised before time t 1 . According to Theorem 1.64, it suffices to prove 1. wt 1 ; 0 = 0 , 266 2. wt 1 ;x is convex in x . Since v t 1 ; 0 = 0 ,wehaveimmediatelythat wt 1 ; 0 = max  0 , K  + ;vt 1 ;1 ,  0  =0: To prove that wt 1 ;x is convex in x , we need to show that v t 1 ; 1 ,  x is convex is x . Obviously, x , K  + is convex in x , and the maximum of two convex functions is convex. The proof of the convexity of v t 1 ; 1 ,  x in x is left as a homework problem. 26.3 Hedging at time t 1 Let x = S t 1  . Case I: v t 1 ; 1 ,  x  x , K  + . The option need not be exercised at time t 1 (should not be exercised if the inequality is strict). We have wt 1 ;x= vt 1 ;1 ,  x; t 1 =w x t 1 ;x= 1,v x t 1 ;1 ,  x=1,t 1 +; where t 1 + = lim tt 1 t is the number of shares of stock held by the hedge immediately after payment of the dividend. The post-dividend position can be achieved by reinvesting in stock the dividends received on the stock held in the hedge. Indeed, t 1 + = 1 1 ,  t 1 =t 1 +  1, t 1  =t 1 + t 1 S t 1  1 ,  S t 1  = # of shares held when dividend is paid + dividends received price per share when dividend is reinvested Case II: v t 1 ; 1 ,  x  x , K  + . The owner of the option should exercise before the dividend payment at time t 1 and receive x , K  . The hedge has been constructed so the seller of the option has x , K before the dividend payment at time t 1 . If the option is not exercised, its value drops from x , K to v t 1 ; 1 ,  x , and the seller of the option can pocket the difference and continue the hedge. . Chapter 26 Options on dividend-paying stocks 26.1 American option with convex payoff function Theorem 1.64 Consider the stock price process. expiration. 26.2 Dividend paying stock Let r and  be constant, let  be a “dividend coefficient” satisfying 0 1: CHAPTER 26. Options on dividend paying stocks