2.10 American Exercise Policies and Valuation We now extend our pricing framework to include a family of securities,called "American,"for which there is discretion regarding the timing of cash flows. Given an adapted process X,each finite-valued stopping time r generates a dividend process 6x.r defined by =0,t,and x.=X.In this context,a finite-valued stopping time is an exercise policy,determining the time at which to accept payment.Any exercise policy r is constrained by T<T,for some expiration time T<T.(In what follows,we might take T to be a stopping time,which is useful for the case of certain knockout options.) We say that (X,T)defines an American security.The exercise policy is selected by the holder of the security.Once exercised,the security has no remaining cash flows.A standard example is an American put option on a security with price process p.The American put gives the holder of the option the right,but not the obligation,to sell the underlying security for a fixed exercise price at any time before a given expiration time T.If the option has an exercise price K and expiration time T<T,then X:=(K-p), t≤T,and X:=0,t>T. We will suppose that,in addition to an American security(X,T),there are securities with an arbitrage-free dividend-price process(0,S)that generates complete markets.The assumption of complete markets will dramatically simplify our analysis since it implies,for any exercise policy r,that the dividend process x is redundant given (6,S).For notational convenience, we assume that 0<T<T. Let r be a state-price density associated with (S).From Proposition 2.9,given any exercise policy r,the American security's dividend process x.has an associated cum-dividend price process,say V,which,in the absence of arbitrage,satisfies g=1E,(,X,),t≤r πt This value does not depend on which state-price density is chosen because, with complete markets,state-price densities are identical up to a positive scaling. We consider the optimal stopping problem W三max , (13) T∈T(O) 21
2.10 American Exercise Policies and Valuation We now extend our pricing framework to include a family of securities, called “American,” for which there is discretion regarding the timing of cash flows. Given an adapted process X, each finite-valued stopping time τ generates a dividend process δX,τ defined by δX,τ t = 0, t 6= τ , and δX,τ τ = Xτ . In this context, a finite-valued stopping time is an exercise policy, determining the time at which to accept payment. Any exercise policy τ is constrained by τ ≤ τ , for some expiration time τ ≤ T. (In what follows, we might take τ to be a stopping time, which is useful for the case of certain knockout options.) We say that (X, τ ) defines an American security. The exercise policy is selected by the holder of the security. Once exercised, the security has no remaining cash flows. A standard example is an American put option on a security with price process p. The American put gives the holder of the option the right, but not the obligation, to sell the underlying security for a fixed exercise price at any time before a given expiration time τ . If the option has an exercise price K and expiration time τ<T, then Xt = (K − pt)+, t ≤ τ , and Xt = 0, t > τ . We will suppose that, in addition to an American security (X, τ ), there are securities with an arbitrage-free dividend-price process (δ, S) that generates complete markets. The assumption of complete markets will dramatically simplify our analysis since it implies, for any exercise policy τ , that the dividend process δX,τ is redundant given (δ, S). For notational convenience, we assume that 0 < τ < T. Let π be a state-price density associated with (δ, S). From Proposition 2.9, given any exercise policy τ , the American security’s dividend process δX,τ has an associated cum-dividend price process, say V τ , which, in the absence of arbitrage, satisfies V τ t = 1 πt Et (πτXτ ), t ≤ τ. This value does not depend on which state-price density is chosen because, with complete markets, state-price densities are identical up to a positive scaling. We consider the optimal stopping problem V ∗ 0 ≡ max τ∈T (0) V τ 0 , (13) 21
where,for any time t <7,we let T(t)denote the set of stopping times bounded below by t and above by T.A solution to (13)is called a rational exercise policy for the American security X,in the sense that it maximizes the initial arbitrage-free value of the resulting claim.Merton 1973 was the first to attack American option valuation systematically using this arbitrage- based viewpoint. We claim that,in the absence of arbitrage,the actual initial price Vo for the American security must be V.In order to see this,suppose first that Vo>Vo.Then one could buy the American security,adopt for it a rational exercise policy T,and also undertake a trading strategy replicatingx Since Vo=E(mX)/To,this replication involves an initial payoff of Vo,and the net effect is a total initial dividend of Vo-Vo>0 and zero dividends after time 0,which defines an arbitrage.Thus the absence of arbitrage easily leads to the conclusion that Vo >v.It remains to show that the absence of arbitrage also implies the opposite inequality Vo <V. Suppose that Vo>V.One could sell the American security at time 0 for Vo.We will show that for an initial investment of Vo,one can "super- replicate"the payoff at exercise demanded by the holder of the American security,regardless of the exercise policy used.Specifically,a super-replicating trading strategy for (X,T,6,S)is a trading strategy 0 involving only the securities with dividend-price process (S)that has the properties: (a)说-0for0<t<元，and (b)9≥X:for all t≤T, where ve is the cum-dividend market value of 0 at time t.Regardless of the exercise policy r used by the holder of the security,the payment of X,demanded at time r is dominated by the market value ve of a super- replicating strategy 0.(In effect,one modifies 0 by liquidating the portfolio at time r,so that the actual trading strategy o associated with the arbitrage is defined by pt=0 for t <T and p=0 for t >T.)Now,suppose 0 is super-replicating,with V=Vo.If,indeed,Vo >Vo then the strategy of selling the American security and adopting a super-replicating strategy, liquidating at exercise,effectively defines an arbitrage. This notion of arbitrage for American securities,an extension of the def- inition of arbitrage used earlier,is reasonable because a super-replicating strategy does not depend on the exercise policy adopted by the holder (or 22
where, for any time t ≤ τ, we let T (t) denote the set of stopping times bounded below by t and above by τ . A solution to (13) is called a rational exercise policy for the American security X, in the sense that it maximizes the initial arbitrage-free value of the resulting claim. Merton [1973] was the first to attack American option valuation systematically using this arbitragebased viewpoint. We claim that, in the absence of arbitrage, the actual initial price V0 for the American security must be V ∗ 0 . In order to see this, suppose first that V ∗ 0 > V0. Then one could buy the American security, adopt for it a rational exercise policy τ , and also undertake a trading strategy replicating −δX,τ . Since V ∗ 0 = E(πτXτ )/π0, this replication involves an initial payoff of V ∗ 0 , and the net effect is a total initial dividend of V ∗ 0 − V0 > 0 and zero dividends after time 0, which defines an arbitrage. Thus the absence of arbitrage easily leads to the conclusion that V0 ≥ V ∗ 0 . It remains to show that the absence of arbitrage also implies the opposite inequality V0 ≤ V ∗ 0 . Suppose that V0 > V ∗ 0 . One could sell the American security at time 0 for V0. We will show that for an initial investment of V ∗ 0 , one can “superreplicate” the payoff at exercise demanded by the holder of the American security, regardless of the exercise policy used. Specifically, a super-replicating trading strategy for (X, τ, δ, S) is a trading strategy θ involving only the securities with dividend-price process (δ, S) that has the properties: (a) δθ t = 0 for 0 <t< τ , and (b) V θ t ≥ Xt for all t ≤ τ , where V θ t is the cum-dividend market value of θ at time t. Regardless of the exercise policy τ used by the holder of the security, the payment of Xτ demanded at time τ is dominated by the market value V θ t of a superreplicating strategy θ. (In effect, one modifies θ by liquidating the portfolio θτ at time τ , so that the actual trading strategy ϕ associated with the arbitrage is defined by ϕt = θt for t<τ and ϕt = 0 for t ≥ τ .) Now, suppose θ is super-replicating, with V θ 0 = V ∗ 0 . If, indeed, V0 > V ∗ 0 then the strategy of selling the American security and adopting a super-replicating strategy, liquidating at exercise, effectively defines an arbitrage. This notion of arbitrage for American securities, an extension of the definition of arbitrage used earlier, is reasonable because a super-replicating strategy does not depend on the exercise policy adopted by the holder (or 22
sequence of holders over time)of the American security.It would be unrea- sonable to call a strategy involving a short position in the American security an "arbitrage"if,in carrying it out,one requires knowledge of the exercise policy for the American security that will be adopted by other agents that hold the security over time,who may after all act "irrationally." The approach to American security valuation given here is similar to the continuous-time treatments of Bensoussan [1984 and Karatzas [1988,who do not formally connect the valuation of American securities with the absence of arbitrage,but rather deal with the similar notion of"fair price." Proposition.Given (X,T,6,S),suppose (8,S)is arbitrage free and gener- ates complete markets.Then there is a super-replicating trading strategy 0 for (X,T,6,S)with the initial value vo=Vo. In order to construct a super-replicating strategy with the desired prop- erty,we will make a short excursion into the theory of optimal stopping.For any process Y in L,the Snell envelope W of Y is defined by W:=器E, 0≤t≤T. It can be shown that,naturally,for any t<T,W:max[Yi,E(W+1) which can be viewed as the Bellman equation for optimal stopping.Thus W:E(W+1),implying that W is a supermartingale,implying that we can decompose W in the form W =Z-A,for some martingale Z and some increasing adapted9 process A with A0=0. In order to prove the above proposition,we define Y by Yi=Ximt,and let W,Z,and A be defined as above.By the definition of complete markets, there is a trading strategy 0 with the property that ·9=0for0<t<T; ·69=Z/m ·9=0fort>7. Property(a)defining a super-replicating strategy is satisfied by this strategy 0.From the fact that Z is a martingale and the definition of a state-price density,the cum-dividend value Vo satisfies πtV=E(πr9)=E(Z)=Z, t≤T (14) More can be said,in that At can be taken to be F-1-measurable. 23
sequence of holders over time) of the American security. It would be unreasonable to call a strategy involving a short position in the American security an “arbitrage” if, in carrying it out, one requires knowledge of the exercise policy for the American security that will be adopted by other agents that hold the security over time, who may after all act “irrationally.” The approach to American security valuation given here is similar to the continuous-time treatments of Bensoussan [1984] and Karatzas [1988], who do not formally connect the valuation of American securities with the absence of arbitrage, but rather deal with the similar notion of “fair price.” Proposition. Given (X, τ, δ, S), suppose (δ, S) is arbitrage free and generates complete markets. Then there is a super-replicating trading strategy θ for (X, τ, δ, S) with the initial value V θ 0 = V ∗ 0 . In order to construct a super-replicating strategy with the desired property, we will make a short excursion into the theory of optimal stopping. For any process Y in L, the Snell envelope W of Y is defined by Wt = max τ∈T (t) Et(Yτ ), 0 ≤ t ≤ τ. It can be shown that, naturally, for any t < τ , Wt = max[ Yt, Et(Wt+1)], which can be viewed as the Bellman equation for optimal stopping. Thus Wt ≥ Et(Wt+1), implying that W is a supermartingale, implying that we can decompose W in the form W = Z − A, for some martingale Z and some increasing adapted9 process A with A0 = 0. In order to prove the above proposition, we define Y by Yt = Xtπt, and let W, Z, and A be defined as above. By the definition of complete markets, there is a trading strategy θ with the property that • δθ t = 0 for 0 <t< τ ; • δθ τ = Zτ/πτ ; • δθ t = 0 for t > τ . Property (a) defining a super-replicating strategy is satisfied by this strategy θ. From the fact that Z is a martingale and the definition of a state-price density, the cum-dividend value V θ satisfies πtV θ t = Et(πτ δθ τ ) = Et(Zτ ) = Zt, t ≤ τ. (14) 9More can be said, in that At can be taken to be Ft−1-measurable. 23
From (14)and the fact that Ao=0,we know that V=Vo because 2o Wo ToVo.Since Zt-At=Wi >Yi for all t,from (14)we also know that =召≥上Y+A)=X+4≥X,t≤元 πt πt πt the last inequality following from the fact that At>0 for all t.Thus the dominance property (b)defining a super-replicating strategy is also satisfied, and 0 is indeed a super-replicating strategy with v=V.This proves the proposition and implies that,unless there is an arbitrage,the initial price Vo of the American security is equal to the market value v associated with a rational exercise policy. The Snell envelope W is also the key to showing that a rational exercise policy is given by the the dynamic-programming solution ro=minft:Wi= Yi.In order to verify this,suppose that T is a rational exercise policy.Then W,=Y.(This can be seen from the fact that W,>Y,and if W.>Y then r cannot be rational.From this fact,any rational exercise policy T has the property that r>70.For any such T,we have E,[Y(r】≤W()=Y(), and the law of iterated expectations implies that EY(r】≤E[Y(ro)小，soo is indeed rational.We have shown the following. Theorem.Given (X,T,6,S),suppose that (6,S)admits no arbitrage and generates completes markets.Let n be a state-price deflator.Let W be the Snell envelope of Xn up to the expiration time T.Then a rational exercise policy for (X,T,6,S)is given by T=minft:W:=nXt.The unique initial cum-dividend arbitrage-free price of the American security is 哈=上Ex)(]: In terms of the equivalent martingale measure Q defined in Section 2.8, we can also write the optimal stopping problem(13)in the form 6=% (15) T∈T(0) An optimal exercise time is To=minft:V=Xi},where V*=Wi/mt is the price of the American option at time t.This representation of the rational- exercise problem is sometimes convenient.For example,let us consider the 24
From (14) and the fact that A0 = 0, we know that V θ 0 = V ∗ 0 because Z0 = W0 = π0V ∗ 0 . Since Zt − At = Wt ≥ Yt for all t, from (14) we also know that V θ t = Zt πt ≥ 1 πt (Yt + At) = Xt + At πt ≥ Xt, t ≤ τ, the last inequality following from the fact that At ≥ 0 for all t. Thus the dominance property (b) defining a super-replicating strategy is also satisfied, and θ is indeed a super-replicating strategy with V θ 0 = V ∗ 0 . This proves the proposition and implies that, unless there is an arbitrage, the initial price V0 of the American security is equal to the market value V ∗ 0 associated with a rational exercise policy. The Snell envelope W is also the key to showing that a rational exercise policy is given by the the dynamic-programming solution τ 0 = min{t : Wt = Yt}. In order to verify this, suppose that τ is a rational exercise policy. Then Wτ = Yτ . (This can be seen from the fact that Wτ ≥ Yτ , and if Wτ > Yτ then τ cannot be rational.) From this fact, any rational exercise policy τ has the property that τ ≥ τ 0. For any such τ , we have Eτ 0 [Y (τ )] ≤ W(τ 0 ) = Y (τ 0 ), and the law of iterated expectations implies that E[Y (τ )] ≤ E[Y (τ 0)], so τ 0 is indeed rational. We have shown the following. Theorem. Given (X, τ, δ, S), suppose that (δ, S) admits no arbitrage and generates completes markets. Let π be a state-price deflator. Let W be the Snell envelope of Xπ up to the expiration time τ . Then a rational exercise policy for (X, τ, δ, S) is given by τ 0 = min{t : Wt = πtXt}. The unique initial cum-dividend arbitrage-free price of the American security is V ∗ 0 = 1 π0 E X(τ 0 )π(τ 0 ) . In terms of the equivalent martingale measure Q defined in Section 2.8, we can also write the optimal stopping problem (13) in the form V ∗ 0 = max τ∈T (0) EQ Xτ R0,τ . (15) An optimal exercise time is τ 0 = min{t : V ∗ t = Xt}, where V ∗ t = Wt/πt is the price of the American option at time t. This representation of the rationalexercise problem is sometimes convenient. For example, let us consider the 24
case of an American call option on a security with price process p.We have X:=(p-K)+for some exercise price K.Suppose the underlying security has no dividends before or at the expiration time 7.We suppose positive interest rates,meaning that Rs>1 for all t and s >t.With these assumptions,we will show that it is never optimal to exercise the call option before its expiration date T.This property is sometimes called "no early exercise,”or“better alive than dead." We define the "discounted price process"p*by p=pt/Ro.t.The fact that the underlying security pays dividends only after the expiration time T implies,by Lemma 2.8,that p*is a Q-martingale at least up to the expiration time元.That is,,fort≤s≤T,we have E(p)-pt. With positive interest rates,we have,for any stopping time T <T, 六-K时=(：-)门 E91 (-)】 ≤E9 EQ Ro.r EQ Ro. EQ (P-K) the first inequality by Jensen's inequality,the second by the positivity of interest rates.It follows that T is a rational exercise policy.In typical cases, T is the unique rational exercise policy. If the underlying security pays dividends before expiration,then early exercise of the American call is,in certain cases,optimal.From the fact that the put payoff is increasing in the strike price(as opposed to decreasing for the call option),the second inequality above is reversed for the case of a put option,and one can guess that early exercise of the American put is sometimes optimal. 25
case of an American call option on a security with price process p. We have Xt = (pt − K)+ for some exercise price K. Suppose the underlying security has no dividends before or at the expiration time τ . We suppose positive interest rates, meaning that Rt,s ≥ 1 for all t and s ≥ t. With these assumptions, we will show that it is never optimal to exercise the call option before its expiration date τ . This property is sometimes called “no early exercise,” or “better alive than dead.” We define the “discounted price process” p∗ by p∗ t = pt/R0,t. The fact that the underlying security pays dividends only after the expiration time τ implies , by Lemma 2.8, that p∗ is a Q-martingale at least up to the expiration time τ . That is, for t ≤ s ≤ τ, we have EQ t (p∗ s) = p∗ t . With positive interest rates, we have, for any stopping time τ ≤ τ , EQ 1 R0,τ (pτ − K) + = EQ " p∗ τ − K R0,τ +# = EQ " EQ τ p∗ τ − K R0,τ +!# ≤ EQ " EQ τ p∗ τ − K R0,τ +!# = EQ " p∗ τ − K R0,τ +# ≤ EQ " p∗ τ − K R0,τ +# = EQ 1 R0,τ (pτ − K) + , the first inequality by Jensen’s inequality, the second by the positivity of interest rates. It follows that τ is a rational exercise policy. In typical cases, τ is the unique rational exercise policy. If the underlying security pays dividends before expiration, then early exercise of the American call is, in certain cases, optimal. From the fact that the put payoff is increasing in the strike price (as opposed to decreasing for the call option), the second inequality above is reversed for the case of a put option, and one can guess that early exercise of the American put is sometimes optimal. 25