The Forward PDE for American Puts in the Dupire Model

Transcription

1 The Forward PDE for American Puts in the Dupire Model Peter Carr Ali Hirsa Courant Institute Morgan Stanley New York University 750 Seventh Avenue 51 Mercer Street New York, NY (1) pcarr@nyc.rr.com ali.hirsa@morganstanley.com Current Version: March 17, 003 File Reference: amerputfwdpde7.tex Abstract We derive the forward PDE for American put options in the Dupire model. We thank Peter Friz, Pedro Judice, Bob Kohn, Peter Laurence, Jeremy Staum, and Arun Verma for their comments. This version is very preinary. The usual disclaimer applies.

2 I Introduction Throughout this note, we assume the standard model of perfect capital markets, continuous trading, and no arbitrage opportunities. Let t 0 be the valuation date and let T >t 0 be the longest future date considered. We assume a deterministic interest rate r(t),t [t 0, T ] which is strictly positive so that early exercise is always a possibility. We also assume that the underlying stock pays dividends continuously over time at a level proportional to the stock price. The deterministic dividend yield q(t) has the property that q(t) r(t) in a neighbourhood of t = T so that the exercise boundary terminates in the strike. Let S t be the stock price at time t [t 0, T ] and we suppose that S t0 = S 0, a positive constant. For t>t 0, we assume that the stock price solves the following stochastic differential equation: ds t =[µ t q(t)]s t dt + a(s t,t)dw t, for all t [t 0, T ]. (1) Here, µ t denotes the expected rate of return per unit time and is an arbitrary adapted continuous stochastic process. In contrast, the absolute volatility is determined by a function a(s, t) of just the stock price and time. We assume that a(s, t) > 0 for S>0. The term dw t denotes increments of a standard Wiener process defined on the time set [t 0, T ] and on a complete probability space (Ω, F,P). As a consequence, no arbitrage implies the existence of a unique risk-neutral measure Q equivalent to the statistical measure under which the stock price process is the following diffusion: ds t =[r(t) q(t)]s t dt + a(s t,t)db t, for all t [t 0, T ], () where B is a standard Brownian motion under Q. Suppose we further assume that the owner of the American put chooses his exercise policy so as to maximize the market value of the American put. Then, the contemporaneous price of the underlying stock is the single source of uncertainty which is relevant for pricing the American put. Let P (S, t; K, T) denote the value function defined on the whole domain S>0,t [t 0,T],K > 0,T [t, T ]. This function maps its four dimensional domain R 4 into the nonnegative real line R +. In our setting, P is strictly decreasing in the backwards variables S and t and it is strictly increasing in the forward variables K and T. Since 1

3 the put is always worth at least its exercise value, the following fundamental inequality holds: P (S, t; K, T) (K S) +, S > 0,t [t 0,T],K >0,T [t, T ]. (3) By definition, this lower bound is attained in the so-called stopping region S. More precisely, the stopping region is the subset of the domain for which: P (S, t; K, T) =(K S) +. (4) The complement of the stopping region is the continuation region. On the continuation region C, the put is optimally held alive and hence we have: P (S, t; K, T) > (K S) +. (5) An overview of this document is as follows. In the next section, we review the well known backward boundary value problem which governs how P varies in S and t at each (K, T). In the following section, we reverse the roles and try to determine a novel forward boundary value problem which seeks to determine how P varies in K and T at each (S, t). The final section summarizes and suggests avenues for future research. II Backward Boundary Value Problem Recall that in our setting, the American put value is strictly declining in S: P (S, t; K, T) < 0. (6) S Hence, at each t [t 0, T ] and for each K>0,T [t, T ], there exists a unique deterministic positive critical stock price S(t; K, T) which is defined as the highest stock price equating the continuation value to the stopping value K S. The function S(t; K, T) maps its three dimensional domain R 3 into the nonnegative real line R +. The function splits the four dimensional domain of the American put value function into two

4 four dimensional subsets. If the subset has S S(t; K, T) then it is called the continuation region, while if the subset has S>S(t; K, T), then is called the stopping region: if S S(t; K, T), then P (S, t; K, T) =(K S) + (7) and if S>S(t; K, T), then P (S, t; K, T) > (K S) +. (8) For each (K, T), we may define the exercise boundary as the time path of critical stock prices, S(t; K, T),t [t 0,T]. This boundary is independent of the current stock price S 0 and is a smooth, nondecreasing function of time t whose terminal it is K. P The partial derivatives,, P, and P t S S (PDE) on C: exist and satisfy the following partial differential equation P(S, t; K, T) + a (S, t) P (S, t; K, T) t S P(S, t; K, T) +[r(t) q(t)]s r(t)p (S, t; K, T) =0. (9) S For each (K, T), the American put value function P (S, t; K, T) and the exercise boundary S(t; K, T) jointly solve a backward free boundary problem (FBP), consisting of the backward PDE (9) and the following boundary conditions: P (S, T ; K, T) =(K S) +, S > 0,T [t 0, T ],K >0 (10) S t [t 0,T 0 ],T [t, T ],K >0 (11) S S(t;K,T ) t [t 0,T 0 ],T [t, T ],K >0 (1) P(S,t;K,T ) = 1, S S(t;K,T ) S t [t 0,T 0 ],T [t, T ],K >0. (13) The first three boundary conditions force the American put value to its exercise value along the boundaries. The value matching condition (1) and (7) imply that the put price is continuous across the exercise boundary. Furthermore, the high contact condition (13) and (7) further imply that the slope in S is continuous. Equations (1) and (13) are jointly referred to as the smooth fit conditions. 3

5 Since the left and right hand its of the put value are equal at the exercise boundary, we may write: P (S(t; K, T),t; K, T) =K S(t; K, T), t [t 0,T 0 ],T [t, T ],K >0. (14) Since the left and right deltas are equal at the exercise boundary, we can define the delta at the boundary: P(S(t; K, T),t; K, T) S = 1, t [t 0,T 0 ],T [t, T ],K >0. (15) III Forward Propagation of American Put Values By definition, a forward PDE involves the maturity derivative of P or possibly a higher order version of it. Suppose we assume that the partial derivatives, P T, P tt, P ST, and 3 P S T (9) w.r.t. T leads to the following PDE on C for the value of a calendar spread: where P T (S, t; K, T) P (S, t; K, T). T all exist. Then differentiating P T (S, t; K, T) + a (S, t) t S P T (S, t; K, T) +[r(t) q(t)]s P T (S, t; K, T) r(t)p T (S, t; K, T), =0. (16) S To obtain boundary conditions, consider the cost at time t of a calendar spread of puts: [ ] P (S, t; K, T + T ) P (S, t; K, T) P (S, t; K, T). (17) T T 0 T A long position in a calendar spread requires writing the American put with strike K and maturity T. We assume that the party who owns this put chooses an exercise policy so as to maximize its market value. The calendar spread involves two exercise boundaries, one for each American put. For now, we only consider levels of the 4 variables S,t,K,T such that both puts are not optimally exercised at t. Let τ b be the first passage time of the stock price to the exercise boundary associated with the American put with strike K and maturity T. If the boundary is not hit by T, then we set τ b =. Let τ τ b T be the earlier of this first passage time and the maturity date. Suppose that the calendar spread is sold at τ. Obviously, there is a single payoff to this strategy. We claim that if τ b T, then the payoff at τ = T is 1(τ b T )δ(s T K) a (K,T ). On the other hand, if τ b T, then we claim that the payoff at τ b is zero. 4

6 To see why, first suppose τ b T and S T >K. Then the short put in the calendar spread expires worthless at T, while the long put is out-of-the-money (OTM) at time T, but with T years left to maturity. Due to the continuity of the process, this OTM put has has a value at T which is o( T ). Hence, as T 0, the value of the long put at T goes to zero faster than the holdings of hence the product goes to zero. Now suppose τ b T and S T 1 T go to infinity, and = K. Then the short put in the calendar spread still expires worthless at T, while the long put is now at-the-money (ATM) at T, with T years left to maturity. Due to the continuity of the process, this ATM put has a time T value which is O( T ). Now, as T 0, the time T value of the long put goes to zero slower than the holdings of 1 T go to infinity, and hence the product goes to infinity. The exact magnitude of the infinite payoff is the same as for the European case, i.e. δ(s T K) a (K,T ). Now suppose τ b <T. At time t = τ b, S = S(t; K, T) and we claim that the value of the calendar spread vanishes, i.e.: P (S(t; K, T),t; K, T) =0. (18) T Note that the left hand side of (17) involves a partial derivative not a total derivative. The reason for (17) is that if we take the total derivative of (8) w.r.t. T, the envelope theorem lets us ignore the dependence of S(t; K, T) ont. Alternatively, if one is unaware of the implications of the envelope theorem, then total differentiation of (8) w.r.t. T implies: P (S(t; K, T),t; K, T) S(t; K, T)+ P (S(t; K, T),t; K, T) = S(t; K, T). (19) S T T T Substituting the high contact condition (13) in (19) implies (18). We conclude that the only nonzero payoff from a calendar spread of American puts occurs only at T and is 1(τ b T )δ(s T K) a (K,T ). The above probabilistic arguments lead to the following boundary conditions for a calendar spread of American puts: P T (S, T ; K, T) = a (K,T ) δ(k S), S > 0,T [t 0, T ],K >0 (0) T (S, t; K, T) =0, S t [t 0,T 0 ],T [t, T ],K >0 (1) T (S, t; K, T) =0, S S(t;K,T ) t [t 0,T 0 ],T [t, T ],K >0. () 5

7 Assuming that the free boundary is given, the solution to this backward boundary value problem exists and is unique. Now recall that a butterfly spread of European puts maturing at T and with strikes K K, K, and K + K has a payoff at T of δ(s T K). Thus, the payoff of a calendar spread of American puts is that of a (K,T ) down-and-out butterfly spreads of European puts. The lower knockout boundary is the early exercise boundary of the American put with strike K and maturity T. Let g(s, t; K, T) be the value of such a down-and-out butterfly spread. 1 The notation reflects the fact that the butterfly spread value is also the Green s function for the stock price process which absorbs at the first passage time to the free boundary. Under either interpretation, g(s, t; K, T) solves the following backward PDE: g(s, t; K, T) t + a (S, t) g(s, t; K, T) g(s, t; K, T) +[r(t) q(t)]s r(t)g(s, t; K, T) =0. (3) S S The Green s function also satisfies the following boundary conditions: g(s, T ; K, T) =δ(k S), S > 0,T [t 0, T ],K >0 (4) S t [t 0,T 0 ],T [t, T ],K >0 (5) S S(t;K,T ) t [t 0,T 0 ],T [t, T ],K >0. (6) A comparison of the backward BVP for the calendar spread with the backward BVP for the Green s function leads to the claim that: P T (S, t; K, T) = a (K, T) g(s, t; K, T). (7) Hence, assuming a 0: g(s, t; K, T) = P (S, t; K, T). (8) a (K, T) T Now, g(s, t; K, T) solves the following forward PDE: g(s, t; K, T) = T [ a ] (K, T) g(s, t; K, T) K [(r(t ) q(t ))Kg(S, t; K, T)] r(t )g(s, t; K, T), K (9) 1 Note that an American put on the absorbing process has the same value as an American put on the free process. 6

8 subject to some boundary conditions. The initial condition is: g(s, t; K, t) =δ(k S), S,K > 0,t [t 0, T ]. (30) A Dirichlet lower boundary condition is: An upper boundary condition is: K S(t;K,T ) g(s, t; K, T) =0, S > 0,t [t 0, T ],T [t, T ]. (31) K g(s, t; K, T) =0, S > 0,t [t 0, T ],T [t, T ]. (3) Recall we assumed that r(t ) > 0. Since (9) differs from the Kolmogorov forward PDE only by a potential term where r(t ) is nonnegative, the solution to this forward boundary value problem exists and is unique. Substituting (8) in (9) implies that the calendar spread of American puts satisfies the following forward PDE: [ ] T a (K, T) P T (S, t; K, T) = K P T (S, t; K, T) [ ] (r(t ) q(t ))K K a (K, T) P T (S, t; K, T) r(t ) a (K, T) P T (S, t; K, T). (33) Using the product rule: 4 a 3 (K, T) T a(k, T)P T (S, t; K, T)+ a (K, T) T P T (S, t; K, T) = K P T (S, t; K, T) (r(t ) q(t )) a (K, T) P T (S, t; K, T) 4 +(r(t ) q(t ))K a 3 (K, T) K a(k, T)P T (S, t; K, T) (r(t ) q(t ))K a (K, T) K P T (S, t; K, T) r(t ) a (K, T) P T (S, t; K, T). (34) Multiplying both sides by a (K) implies that: P T (S, t; K, T) T = a (K, T) P T (S, t; K, T) (r(t ) q(t ))K P T (S, t; K, T) (35) [ K K ] + a(k, T) T a(k, T)+(r(T ) q(t ))K a (K, T) (r(t ) q(t )) P T (S, t; K, T). a(k, T) 7

9 Recall that we assumed that the variables S,t,K,T were such that both American puts were not optimally exercised early at t. Now for each S, t, T, one can consider raising the strike price K. The lowest K for which the American put achieves its exercise value K S is termed the critical strike price, denoted K(S, t; T ). At each S and t, we can regard K(S, t; T ) as a function of just T. This function is a smooth increasing function of T beginning at S. For strikes K< K(S, t; T ), the American put of that strike is optimally held alive and hence we have P (S, t; K, T) >K S. In contrast, for strikes K K(S, t; T ), the American put of that strike is optimally exercised and hence we have P (S, t; K, T) =K S. The critical strike price is the inverse of the critical stock price. As a result, the following pair of equations hold: S(t; T, K(S, t; T )) = S K(S(t; T,K),t; T )=K. (36) To complete the specification of the first order forward boundary value problem for P T, we need boundary conditions. The initial condition is: A Dirichlet lower boundary condition is: P T (S, t; K, t) = a (K, t) δ(k S), S,K > 0,t [t 0, T ]. (37) A Dirichlet upper boundary condition is: K 0 P T (S, t; K, T) =0, S > 0,t [t 0, T ],T [t, T ]. (38) P T (S, t; K, T) =0, S > 0,t [t 0, T ],T [t, T ]. (39) K K(S,t;T ) We need a final condition to determine the free boundary. We have that: K K(S,t;T ) Differentiating w.r.t. T implies: K K(S,t;T ) K P (S, t; K, T) =1, S > 0,t [t 0, T ],T [t, T ]. (40) K P T (S, t; K, T) = P (S, t; K, T) K T K(S, t; T ), S > 0,t [t 0, T ],T [t, T ]. (41) 8

10 We propose that his condition be used to determine the free boundary. Future research should focus on alternative conditions which may be easier to implement. If one desires specifying P T (S, t; K, T) in the stopping region, a specification that induces continuity of P T in K at the critical strike price is P T (S, t; K, T) =0forS>0,t [t 0, T ],K > K(S, t; T ),T [t, T ]. To derive a first order forward PDE for the put value rather than for the calendar spread, suppose r, q and a are all independent of T, but a can still can depend on K. Then a(k) = 0 and we can integrate T out T in (35): P(S, t; K, T) T = a (K) P (S, t; K, T) P(S, t; K, T) (r q)k [ K ] K + (r q)k a (K) (r q) P (S, t; K, T)+A(K), (4) a(k) where A(K) is the constant of integration. To determine A(K), we let K. Assuming that: and that: P(S, t; K, T) a (K) = K T K K [ K (r q)k a (K) a(k) P(S, t; K, T) P (S, t; K, T) = K = 0 (43) K K (r q) ] P (S, t; K, T) =0, (44) we conclude that A(K) = 0. Hence, when the evolution parameters are independent of time, the following forward PDE governs American put values on the continuation region: P(S, t; K, T) T = a (K) P (S, t; K, T) (r q)k K If we furthermore have a(k) = σk, then on the continuation region: [ ] P(S, t; K, T) + (r q)k a (K) (r q) P (S, t; K, T). K a(k) (45) P(S, t; K, T) T = σ K P (S, t; K, T) P(S, t; K, T) (r q)k qp(s, t; K, T), (46) K K which is the same as the forward PDE for European puts. Hence, on the continuation region, the forward PDE for American puts in the Black Scholes model is the same as the one for European puts. Carr and Hirsa[1] prove a generalization of this result to when the log price is a Lévy process. 9

11 To complete the specification of the first order forward boundary value problem for P, we need boundary conditions. The initial condition is: P (S, t; K, t) =(K S) +, S,K > 0,t [t 0, T ]. (47) A Dirichlet lower boundary condition is: The smooth fit conditions are: K 0 P (S, t; K, T) =0, S > 0,t [t 0, T ],T [t, T ]. (48) K K(S,t;T ) P (S, t; K, T) = K(S, t; T ) S, S > 0,t [t 0, T ],T [t, T ] (49) P(S,t;K,T ) =1, K K K(S,t;T ) S > 0,t [t 0, T ],T [t, T ]. (50) If one desires specifying P (S, t; K, T) in the stopping region, a specification that induces continuity and differentiability in K at the critical strike price is P (S, t; K, T) =K S for S > 0,t [t 0, T ],K > K(S, t; T ),T [t, T ]. IV Summary and Future Research We developed backward and forward boundary value problems for American puts in the Dupire model. References [1] Carr, P., and A. Hirsa, 003, Why Be Backward, Risk, January, [] Dupire, B., 1994, Pricing with a Smile, Risk, 7, 1, [3] Merton, R., 1973, Theory of Rational Option Pricing, Bell Journal of Economics and Management Science, 4,