American Options

Chapter 25 American Options This and the following chapters form part of the course Stochastic Differential Equations for Fi- nance II. 25.1 Preview of perpetual American put dS = rS dt + S dB Intrinsic value at time t :K,St + : Let L 2 0;K be given. Suppose we exercise the first time the stock price is L or lower. We define  L = minft  0; S t  Lg; v L x=IEe ,r L K , S  L  + =  K , x if x  L , K , LIEe ,r L if xL: Theplanistocomute v L x andthenmaximizeover L to find the optimal exercise price. We need to know the distribution of  L . 25.2 First passage times for Brownian motion: first method (Based on the reflection principle) Let B be a Brownian motion under IP ,let x0 be given, and define  = minft  0; B t=xg:  is called the first passage time to x . We compute the distribution of  . 247 248 K K x Intrinsic value Stock price Figure 25.1: Intrinsic value of perpetual American put Define M t = max 0ut B u: From the first section of Chapter 20 we have IP fM t 2 dm; B t 2 dbg = 22m , b t p 2t exp  , 2m , b 2 2t  dm db; m0;b  m: Therefore, IP fM t  xg = Z 1 x Z m ,1 22m , b t p 2t exp  , 2m , b 2 2t  db dm = Z 1 x 2 p 2t exp  , 2m , b 2 2t      b=m b=,1 dm = Z 1 x 2 p 2t exp  , m 2 2t  dm: We make the change of variable z = m p t in the integral to get = Z 1 x= p t 2 p 2 exp  , z 2 2  dz : Now   tM t  x; CHAPTER 25. American Options 249 so IP f 2 dtg = @ @t IP f  tg dt = @ @t IP fM t  xg dt = " @ @t Z 1 x= p t 2 p 2 exp  , z 2 2  dz  dt = , 2 p 2 exp  , x 2 2t  : @ @t  x p t  dt = x t p 2t exp  , x 2 2t  dt: We also have the Laplace transform formula IEe , = Z 1 0 e ,t IP f 2 dtg = e ,x p 2 ; 0: (See Homework) Reference: Karatzas and Shreve, Brownian Motion and Stochastic Calculus, pp 95-96. 25.3 Drift adjustment Reference: Karatzas/Shreve, Brownian motion and Stochastic Calculus, pp 196–197. For 0  t1 ,define e B t=t + Bt; Z t = expf,B t , 1 2  2 tg; = expf, e B t+ 1 2  2 tg; Define ~ = minft  0; e B t=xg: We fix a finite time T and change the probability measure “only up to T ”. More specifically, with T fixed, define f IP A= Z A ZTdP; A 2FT: Under f IP , the process e B t; 0  t  T , is a (nondrifted) Brownian motion, so f IP f ~ 2 dtg = IP f 2 dtg = x t p 2t exp  , x 2 2t  dt; 0 tT: 250 For 0 tT we have IP f ~  tg = IE h 1 f ~ tg i = f IE  1 f ~ tg 1 Z T   = f IE h 1 f ~ tg expf e B T  , 1 2  2 T g i = f IE  1 f ~ tg f IE  expf e B T  , 1 2  2 T g     F ~ ^ t  = f IE h 1 f ~ tg expf e B ~ ^ t , 1 2  2 ~ ^ tg i = f IE h 1 f~tg expfx , 1 2  2 ~ g i = Z t 0 expfx , 1 2  2 sg f IP f~ 2 dsg = Z t 0 x s p 2s exp  x , 1 2  2 s , x 2 2s  ds = Z t 0 x s p 2s exp  , x , s 2 2s  ds: Therefore, IP f ~ 2 dtg = x t p 2t exp  , x , t 2 2t  dt; 0 tT: Since T is arbitrary, this must in fact be the correct formula for all t0 . 25.4 Drift-adjusted Laplace transform Recall the Laplace transform formula for  = minft  0; B t=xg for nondrifted Brownian motion: IEe , = Z 1 0 x t p 2t exp  ,t , x 2 2t  dt = e ,x p 2 ; 0;x  0: For ~ = minft  0; t + Bt=xg; CHAPTER 25. American Options 251 the Laplace transform is IEe ,~ = Z 1 0 x t p 2t exp  ,t , x , t 2 2t  dt = Z 1 0 x t p 2t exp  ,t , x 2 2t + x , 1 2  2 t  dt = e x Z 1 0 x t p 2t exp  , + 1 2  2 t , x 2 2t  dt = e x,x p 2+ 2 ; 0;x  0; where in the last step we have used the formula for IEe , with  replaced by  + 1 2  2 . If ~ !  1 ,then lim 0 e , ~ !  =1; if ~ ! =1 ,then e , ~ ! =0 for every 0 ,so lim 0 e , ~ !  =0: Therefore, lim 0 e , ~ !  = 1 ~1 : Letting 0 and using the Monotone Convergence Theorem in the Laplace transform formula IEe ,~ = e x,x p 2+ 2 ; we obtain IP f ~1g = e x,x p  2 = e x,xj j : If   0 ,then IP f ~1g =1: If 0 ,then IP f ~1g = e 2x  1: (Recall that x0 ). 25.5 First passage times: Second method (Based on martingales) Let 0 be given. Then Y t = expfB t , 1 2  2 tg 252 is a martingale, so Y t ^   is also a martingale. We have 1=Y0 ^   = IEY t ^   = IE expfBt ^   , 1 2  2 t ^  g: = lim t!1 IE expfBt ^   , 1 2  2 t ^  g: We want to take the limit inside the expectation. Since 0  expfB t ^   , 1 2  2 t ^  ge x ; this is justified by the Bounded Convergence Theorem. Therefore, 1=IE lim t!1 expfBt ^   , 1 2  2 t ^  g: There are two possibilities. For those ! for which  !   1 , lim t!1 expfBt ^   , 1 2  2 t ^  g = e x, 1 2  2  : For those ! for which  ! =1 , lim t!1 expfB t ^   , 1 2  2 t ^  g lim t!1 expfx , 1 2  2 tg =0: Therefore, 1=IE lim t!1 expfB t ^   , 1 2  2 t ^  g = IE  e x, 1 2  2  1 1  = IEe x, 1 2  2  ; where we understand e x, 1 2  2  to be zero if  = 1 . Let  = 1 2  2 ,so  = p 2 . We have again derived the Laplace transform formula e ,x p 2 = IEe , ; 0;x  0; for the first passage time for nondrifted Brownian motion. 25.6 Perpetual American put dS = rS dt + S dB S 0 = x S t=xexpfr , 1 2  2 t + Btg = x exp 8        :  2 6 6 6 4  r  ,  2  | z   t + B t 3 7 7 7 5 9    =    ; : CHAPTER 25. American Options 253 Intrinsic value of the put at time t : K , S t + . Let L 2 0;K be given. Define for x  L ,  L = minft  0; S t=Lg = minft  0; t + Bt= 1  log L x g = minft  0; ,t , Bt= 1  log x L g Define v L =K,LIEe ,r L =K,L exp  ,   log x L , 1  log x L p 2r +  2  =K,L  x L  ,   , 1  p 2r+ 2 : We compute the exponent ,   , 1  p 2r +  2 = , r  2 + 1 2 , 1  s 2r +  r  , =2  2 = , r  2 + 1 2 , 1  s 2r + r 2  2 , r +  2 =4 = , r  2 + 1 2 , 1  s r 2  2 + r +  2 =4 = , r  2 + 1 2 , 1  s  r  + =2  2 = , r  2 + 1 2 , 1   r  + =2  = , 2r  2 : Therefore, v L x= 8  : K,x; 0x L; K , L , x L  ,2r= 2 ; x  L: The curves K , L , x L  ,2r= 2 ; are all of the form Cx ,2r= 2 . We want to choose the largest possible constant. The constant is C =K,LL 2r= 2 ; 254 σ 2 -2r/ K K x Stock price K - x (K - L) (x/L) value Figure 25.2: Value of perpetual American put σ 2 -2r/ C 1 x σ 2 -2r/ x C 2 σ 2 -2r/ x C 3 x Stock price value Figure 25.3: Curves. CHAPTER 25. American Options 255 and @C @L = ,L 2r  2 + 2r  2 K , LL 2r  2 ,1 = L 2r  2  ,1+ 2r  2 K ,L 1 L  = L 2r  2  ,  1+ 2r  2  + 2r  2 K L  : We solve ,  1+ 2r  2  + 2r  2 K L =0 to get L = 2rK  2 +2r : Since 0  2r 2 +2r; we have 0 LK: Solution to the perpetual American put pricing problem (see Fig. 25.4): v x= 8  : K,x; 0x L  ; K,L   , x L   ,2r= 2 ; x  L  ; where L  = 2rK  2 +2r : Note that v 0 x=  ,1; 0 xL  ; , 2r  2 K,L  L   2r= 2 x ,2r= 2 ,1 ; xL  : We have lim xL  v 0 x=,2 r  2 K,L   1 L  =,2 r  2  K, 2rK  2 +2r   2 +2r 2rK = ,2 r  2   2 +2r,2r  2 +2r !  2 +2r 2r = ,1 = lim x"L  v 0 x: 256 σ 2 -2r/ K K x Stock price K - x value (K - L ) (x/L ) * * * L Figure 25.4: Solution to perpetual American put. 25.7 Value of the perpetual American put Set  = 2r  2 ; L  = 2rK  2 +2r =   +1 K: If 0  xL  ,then v x= K ,x .If L   x1 ,then v x= K ,L  L    | z  C x , (7.1) = IE x h e ,r K , L   + 1 f1g i ; (7.2) where S 0 = x (7.3)  = minft  0; S t=L  g: (7.4) If 0  xL  ,then ,rvx+ rxv 0 x+ 1 2  2 x 2 v 00 x=,rK,x+ rx,1 = ,rK: If L   x1 ,then ,rvx+rxv 0 x+ 1 2  2 x 2 v 00 x = C,rx , , rxx ,,1 , 1 2  2 x 2  , , 1x , ,2  = Cx , ,r , r , 1 2  2  , , 1 = C , , 1x ,  r , 1 2  2  2r  2  =0: In other words, v solves the linear complementarity problem: (See Fig. 25.5). [...]... the Laplace transform formula for = minft  0; B t = xg for nondrifted Brownian motion: IEe, For   Z1 x x2 dt = e,xp2 ; p exp , t , 2t = 0 t 2t ~ = minft  0; t + B t = xg; 0; x 0: CHAPTER 25 American Options the Laplace transform is 251   Z1 x x , t2 dt p exp , t , 2t 0 t 2t   Z1 x x2 + x , 1 2 t dt p exp , t , 2t = 2 0 t 2t  Z1 x 2 1 2 t , x x p exp , + 2 =e dt 2t t 2t IEe, ~ = 0 . Value of perpetual American put σ 2 -2r/ C 1 x σ 2 -2r/ x C 2 σ 2 -2r/ x C 3 x Stock price value Figure 25.3: Curves. CHAPTER 25. American Options 255 and. 1 x= p t 2 p 2 exp  , z 2 2  dz : Now   tM t  x; CHAPTER 25. American Options 249 so IP f 2 dtg = @ @t IP f  tg dt = @ @t IP fM t  xg dt