Chapter 6 Properties of American Derivative Securities 6.1 The properties Definition 6.1 An American derivative security is a sequence of non-negative random variables fG k g n k=0 such that each G k is F k -measurable. The owner of an American derivative security can exercise at any time k , and if he does, he receives the payment G k . (a) The value V k of the security at time k is V k = max  1 + r k f IE 1 + r , G  jF k ; where the maximum is over all stopping times  satisfying   k almost surely. (b) The discounted value process f1 + r ,k V k g n k=0 is the smallest supermartingale whichsatisfies V k  G k ; 8k; almost surely. (c) Any stopping time  which satisfies V 0 = f IE 1 + r , G   is an optimal exercise time. In particular  4 = minfk; V k = G k g is an optimal exercise time. (d) The hedging portfolio is given by  k ! 1 ;::: ;! k = V k+1 ! 1 ;::: ;! k ;H, V k+1 ! 1 ;::: ;! k ;T S k+1 ! 1 ;::: ;! k ;H, S k+1 ! 1 ;::: ;! k ;T ;k =0;1;::: ;n , 1: 85 86 (e) Suppose for some k and ! ,wehave V k !=G k ! . Then the owner of the derivative security should exercise it. If he does not, then the seller of the security can immediately consume V k ! , 1 1+ r f IEV k+1 jF k ! and still maintain the hedge. 6.2 Proofs of the Properties Let fG k g n k=0 be a sequence of non-negative random variables such that each G k is F k -measurable. Define T k to be the set of all stopping times  satisfying k    n almost surely. Define also V k 4 =1+r k max  2T k f IE 1 + r , G  jF k  : Lemma 2.18 V k  G k for every k . Proof: Take  2 T k to be the constant k . Lemma 2.19 The process f1 + r ,k V k g n k=0 is a supermartingale. Proof: Let   attain the maximum in the definition of V k+1 , i.e., 1 + r ,k+1 V k+1 = f IE h 1 + r ,  G   jF k+1 i : Because   is also in T k ,wehave f IE 1 + r ,k+1 V k+1 jF k  = f IE h f IE 1 + r ,  G   jF k+1 jF k i = f IE 1 + r ,  G   jF k   max  2T k f IE 1 + r , G  jF k  = 1 + r ,k V k : Lemma 2.20 If fY k g n k=0 is another process satisfying Y k  G k ;k =0;1;::: ;n; a.s., and f1 + r ,k Y k g n k=0 is a supermartingale, then Y k  V k ;k =0;1;::: ;n; a.s. CHAPTER 6. Properties of American Derivative Securities 87 Proof: The optional sampling theorem for the supermartingale f1 + r ,k Y k g n k=0 implies f IE 1 + r , Y  jF k   1 + r ,k Y k ; 8 2 T k : Therefore, V k = 1 + r k max  2T k f IE 1 + r , G  jF k   1 + r k max  2T k f IE 1 + r , Y  jF k   1 + r ,k 1 + r k Y k = Y k : Lemma 2.21 Define C k = V k , 1 1+r f IEV k+1 jF k  = 1 + r k n 1 + r ,k V k , f IE 1 + r ,k+1 V k+1 jF k  o : Since f1 + r ,k V k g n k=0 is a supermartingale, C k must be non-negative almost surely. Define  k ! 1 ;::: ;! k = V k+1 ! 1 ;::: ;! k ;H, V k+1 ! 1 ;::: ;! k ;T S k+1 ! 1 ;::: ;! k ;H, S k+1 ! 1 ;::: ;! k ;T : Set X 0 = V 0 and define recursively X k+1 = k S k+1 +1+rX k , C k ,  k S k : Then X k = V k 8k: Proof: We proceed by induction on k . The induction hypothesis is that X k = V k for some k 2f0;1;::: ;n , 1g , i.e., for each fixed ! 1 ;::: ;! k  we have X k ! 1 ;::: ;! k =V k ! 1 ;::: ;! k : We need to show that X k+1 ! 1 ;::: ;! k ;H= V k+1 ! 1 ;::: ;! k ;H; X k+1 ! 1 ;::: ;! k ;T=V k+1 ! 1 ;::: ;! k ;T: We prove the first equality; the proof of the second is similar. Note first that V k ! 1 ;::: ;! k  , C k ! 1 ;::: ;! k  = 1 1+ r f IEV k+1 jF k ! 1 ;::: ;! k  = 1 1+ r ~pV k+1 ! 1 ;::: ;! k ;H+ ~qV k+1 ! 1 ;::: ;! k ;T : 88 Since ! 1 ;::: ;! k  will be fixed for the rest of the proof, we will suppress these symbols. For example, the last equation can be written simply as V k , C k = 1 1+r ~pV k+1 H + ~qV k+1 T  : We compute X k+1 H  =  k S k+1 H  + 1 + rX k , C k ,  k S k  = V k+1 H  , V k+1 T  S k+1 H  , S k+1 T  S k+1 H  , 1 + rS k  +1 + rV k , C k  = V k+1 H  , V k+1 T  u , dS k uS k , 1 + rS k  +~pV k+1 H + ~qV k+1 T  = V k+1 H  , V k+1 T  ~q +~pV k+1 H + ~qV k+1 T  = V k+1 H : 6.3 Compound European Derivative Securities In order to derive the optimal stopping time for an American derivative security, it will be useful to study compound European derivative securities, which are also interesting in their own right. A compound European derivative security consists of n +1 different simple European derivative securities (with the same underlying stock) expiring at times 0; 1;::: ;n ; the security that expires at time j has payoff C j . Thus a compound European derivative security is specified by the process fC j g n j =0 , where each C j is F j -measurable, i.e., the process fC j g n j =0 is adapted to the filtration fF k g n k=0 . Hedging a short position (one payment). Here is how we can hedge a short position in the j ’th European derivative security. The value of European derivative security j at time k is given by V j  k =1+r k f IE1 + r ,j C j jF k ;k=0;::: ;j; and the hedging portfolio for that security is given by  j  k ! 1 ;::: ;! k = V j k+1 ! 1 ;::: ;! k ;H, V j k+1 ! 1 ;::: ;! k ;T S j k+1 ! 1 ;::: ;! k ;H, S j k+1 ! 1 ;::: ;! k ;T ;k =0;::: ;j , 1: Thus, starting with wealth V j  0 , and using the portfolio  j  0 ;::: ; j j,1  , we can ensure that at time j we have wealth C j . Hedging a short position (all payments). Superpose the hedges for the individual payments. In other words, start with wealth V 0 = P n j =0 V j  0 . At each time k 2f0;1;::: ;n , 1g ,firstmakethe payment C k and then use the portfolio  k = k k+1 + k k+2 + :::+ k n CHAPTER 6. Properties of American Derivative Securities 89 corresponding to all future payments. At the final time n , after making the final payment C n ,we will have exactly zero wealth. Suppose you own a compound European derivative security fC j g n j =0 . Compute V 0 = n X j =0 V j  0 = f IE 2 4 n X j =0 1 + r ,j C j 3 5 and the hedging portfolio is f k g n,1 k=0 . You can borrow V 0 and consume it immediately. This leaves you with wealth X 0 = ,V 0 . In each period k , receive the payment C k and then use the portfolio , k . At the final time n , after receiving the last payment C n , your wealth will reach zero, i.e., you will no longer have a debt. 6.4 Optimal Exercise of American Derivative Security In this section we derive the optimal exercise time for the owner of an American derivative security. Let fG k g n k=0 be an American derivative security. Let  be the stopping time the owner plans to use. (We assume that each G k is non-negative, so we may assume without loss of generality that the owner stops at expiration – time n – if not before). Using the stopping time  ,inperiod j the owner will receive the payment C j = I f =j g G j : In other words, once he chooses a stopping time, the owner has effectively converted the American derivative security into a compound European derivative security, whose value is V   0 = f IE 2 4 n X j =0 1 + r ,j C j 3 5 = f IE 2 4 n X j =0 1 + r ,j I f =j g G j 3 5 = f IE 1 + r , G  : The owner of the American derivative security can borrow this amount of money immediately, if he chooses, and invest in the market so as to exaclty pay off his debt as the payments fC j g n j =0 are received. Thus, his optimal behavior is to use a stopping time  which maximizes V   0 . Lemma 4.22 V   0 is maximized by the stopping time   = minfk ; V k = G k g: Proof: Recall the definition V 0 4 = max  2T 0 f IE 1 + r , G   = max  2T 0 V   0 90 Let  0 be a stoppingtime whichmaximizes V   0 , i.e., V 0 = f IE h 1 + r , 0 G  0 i : Because f1 + r ,k V k g n k=0 is a supermartingale, we have from the optional sampling theorem and the inequality V k  G k ,the following: V 0  f IE h 1 + r , 0 V  0 jF 0 i = f IE h 1 + r , 0 V  0 i  f IE h 1 + r , 0 G  0 i = V 0 : Therefore, V 0 = f IE h 1 + r , 0 V  0 i = f IE h 1 + r , 0 G  0 i ; and V  0 = G  0 ; a.s. We have just shown that if  0 attains the maximum in the formula V 0 = max  2T 0 f IE 1 + r , G   ; (4.1) then V  0 = G  0 ; a.s. But we have defined   = minfk ; V k = G k g; andsowemusthave     0  n almost surely. The optional sampling theorem implies 1 + r ,  G   = 1 + r ,  V    f IE h 1 + r , 0 V  0 jF   i = f IE h 1 + r , 0 G  0 jF   i : Taking expectations on both sides, we obtain f IE h 1 + r ,  G   i  f IE h 1 + r , 0 G  0 i = V 0 : It follows that   also attains the maximum in (4.1), and is therefore an optimal exercise time for the American derivative security. . Chapter 6 Properties of American Derivative Securities 6.1 The properties Definition 6.1 An American derivative security is a sequence of non-negative. supermartingale, then Y k  V k ;k =0;1;::: ;n; a.s. CHAPTER 6. Properties of American Derivative Securities 87 Proof: The optional sampling theorem for the supermartingale