LA EDUCACIÓN DE LA CONTRARREFORMA

Recall that American options can be exercised at any date before expiration. Intuitively, then, one may expect that these option must be significantly harder to price than their European counterparts, since determining when to exercise the option in an optimal way becomes part of the problem. It can be proved that the point of optimal exercise can be found by monitoring the price of the underlying stock: when the stock price crosses a certain critical value, the option should be exercised. Naturally, the value of a call option (i.e., an option to buy stock) goes up with the stock price, so this option is exercised when the stock price rises above the critical value, while for a put option (i.e., an option to sell stock),

the stock price has to drop below the critical value for exercise to be optimal. The Black- Scholes equation governs the behavior of an American call (put) only below (respectively, above) the critical price, which has to be determined as part of the solution. This is a very familiar setting indeed, and therefore the equivalence of American option pricing to a moving boundary problem, first established in [68], is not surprising. We proceed with the formulation of this problem for the case of an American call on a dividend-paying stock.

LetC(t, S)denote the value of the call at timetwhen the price of the underlying stock

isS. Let K be the strike price of the call; r be the interest rate; q, the dividend yield; and σ, the volatility of the stock. Then, at any time before expirationT and for any stock price

below the optimal exercise valueB(t),C(t, S) satisfies the Black-Scholes equation (7.1.5) 1

2σ

2_S2_C

SS + (r−q)SCS+Ct−rC=LBS[C] = 0

At expiration, the option will be exercised only if the stock price is larger than the strike price. (Since, otherwise, it is preferable for the investor to buy the stock on the market for

S dollars, rather than buy it for K > S dollars from the writer of the option.2) Therefore

the value of the call at expiration is

C(T, S) = max(0, S−K)

It can be proved that zero is an absorbing barrier for the geometric Brownian motion process; therefore, if the stock price ever hits 0, it stays at 0. But the call is worthless in this case, so we have

C(t,0) = 0

By definition of the optimal exercise price, when the stock price hits B(t), the call is exer- cised, and its value is

C(t, B(t)) =B(t)−K

Combining these equations, we obtain the differential equations and boundary conditions 2_{The call option is called in the money at a given time, if the stock price satisfiesS > K at this time; if}

S =K orS < K, the corresponding terms are at-the-money and out-of-the-money calls, respectively. For put options, the terminology is the same, but the inequalities have to be reversed.

for call-option problem: 1 2σ 2_S2_C SS+ (r−q)SCS+Ct=rC, t < T, 0< S < B(t) C(T, S) = max(0, S−K), C(t,0) = 0, C(t, B(t)) =B(t)−K

As we know from our experience with moving boundary problems, the above formulation is not complete: we need an additional condition on the moving front to be able to solve for B(t). A finance argument, relying on absence of arbitrage opportunities, was used to find this condition in [70]. In [7, Section 5-A.1], integration over a small neighborhood of

B(t)was used to derive the corresponding condition for an American put. P. Bossaerts [15] suggested an alternative approach, based on the fact that the optimal exercise boundary is chosen in such a way as to maximize the value of the call option at any instant of time. Our derivation, based on these ideas, is presented below.

Mathematically speaking, the above argument implies that

B(t) = arg max

f(t) C t, S;f(t)

The maximum is taken over a suitable class of admissible functions f(t). For example, we can consider all continuous functionsf(t) withf(T) =B(T). In the neighborhood of B(t)

f(t) =B(t) +g(t)

withg(t)continuous andg(T) = 0. DenoteC=C t, S;f(t)

=C t, S;B(t) +g(t) . The necessary condition forC to be maximized by B(t) is

∂C ∂ =0 = 0 (7.2.1)

LetU =∂C/∂. By differentiating the partial differential equation, obtain

1 2σ 2_S2_U SS+ (r−q)SUS+Ut−rU =LBS h∂C ∂ i = 0, t < T, 0< S < B(t) +g(t)

Also,

∂C

∂ (t, S) = 0 = ∂C

∂ (t,0)

since the right-hand sides do not depend on . Now find the boundary value of ∂C/∂. By

the chain rule,

dC d t, B(t) +g(t) =CS t, B(t) +g(t) g(t) +∂C ∂

On the other hand, direct differentiation of the boundary condition forC gives

dC d t, B(t) +g(t) = d d B(t) +g(t)−K =g(t) so we have CS t, B(t) +g(t) −1 g(t) = ∂C ∂ Thus for C₀0_,≡ ∂C ∂ =0 we have 1 2σ 2_S2 C₀0_,SS + (r−q)SC₀0_,S+C₀0_,t−rC₀0_,= 0, t < T, 0< S < B(t) C₀0_,(T, S) = 0 C₀0_,(t,0) = 0 C₀0_,(t, B(t)) =1−CS t, B(t) g(t)

For (7.2.1) (i.e., C ≡0) to be the solution of this problem, we need the last condition to

be homogeneous, so we must have

1−CS t, B(t)

g(t) = 0

for any admissible functiong(t). This immediately implies that the desired second condition on the moving front

CS(t, B(t)) = 1 (7.2.2)

American options. If we perform the same analysis for an American putP(t, S)(see below), we get this condition in the form

PS(t, B(t)) =−1 (7.2.3)

Thus for the call option, we have the following formulation 1 2σ 2_S2_C SS+ (r−q)SCS+Ct=rC, t < T, 0< S < B(t) (7.2.4) C(T, S) = max(0, S−K), (7.2.5) C(t,0) = 0, C(t, B(t)) =B(t)−K, CS(t, B(t)) = 1, B(T−) =Kmax 1,r q (7.2.6) The expression (7.2.6) for the optimal exercise price just before expiration was derived in [56]. For the put option, we have

1 2σ 2_S2_P SS+ (r−q)SPS+Pt=rP, t < T, S > B(t) (7.2.7) P(T, S) = max(0, K−S), (7.2.8) P(t,+∞) = 0, P(t, B(t)) =K−B(t), PS(t, B(t)) =−1, B(T−) =Kmin1,r q (7.2.9) The description is analogous to the case of the American call. Note the difference between (7.2.5) and (7.2.8) and the resulting differences in the boundary conditions at the moving end. We also recognize the two option problems, in the notation of Chapter 3, as a finite one for the call and a semi-infinite one for the put. An important symmetry result between these two problems greatly facilitates further analysis.