On the Existence of a Unique Optimal Threshold Value for the Early Exercise of Call Options

In the case of early exercise of an American-style call option, we consider the issue of the existence of a “treshhold value,” namely a boundary which, once exceeded, early exercise is optimal for all values of the underlying asset which exceed that value. We discuss optimal exercise thresholds for call options for two-period models, under alternate contexts: geometric Brownian motion vs. mean-reverting Ornstein-Uhlenbeck process, with and without seasonality, and with time-dependent strike prices. We show that, other than the case of geometric Brownian motion without seasonality, there may exist multiple exercise thresholds. Department of Civil and Environmental Engineering, 77 Massachessets Avenue, Building 1-290, Massachussets Institute of Technology, Cambridge, MA 02139, [email protected] †Department of Finance, McCombs School of Business, University of Texas at Austin, 1 University Station, B6600, Austin, TX 78712-1179, [email protected] ‡MSIS Department, McCombs School of Business, University of Texas at Austin, 1 University Station, B6500, Austin, TX 78712-1175, [email protected] On the Existence of a Unique Optimal Threshold Value for the Early Exercise of Call Options We are interested in describing early exercise thresholds of a call option, in a two period model, with respect to the price of the underlying asset. The option may be exercised at times t1 and t2. We consider four types of stochastic processes, which, under the risk-neutral measure, are given by: (i) Geometric Brownian Motion dSt/St = (r−δ)dt +σdWt (ii) Geometric Brownian Motion with Seasonality St = ftDt , dDt/Dt = (r−δ)dt +σdWt (iii) Mean reverting process St = exp(Xt), dXt = κ(ξ−Xt)dt +σdWt (iv) Mean reverting process with Seasonality St = ft exp(Xt), dXt = κ(ξ−Xt)dt +σdWt where r,δ,σ, ft ,κ, are non-negative. In addition, we consider the following payoffs from immediate exercise at times t1, t2: Payoff at time t2: Call with strike K2, ht2(S) = max(0,(S−K2)) Payoff at time t1: Call with strike K1, ht1(S) = max(0,(S−K1)) and, in general, allow K1 6= K2. From the payoff at the terminal date, t2, it is obvious that there exists a unique threshold, namely S = K2, such that exercise is optimal for St2 > S , and is not optimal for St2 < S . We will show that on date t1 it is possible, in some of the cases, to have multiple exercise regions.1 In particular, we will show that the optimal exercise policy has at most one threshold in case 1We point out that the two period example is particularly simple, since the continuation value at time t1 is the value of a European option, for which we have closed form expressions for all four cases.