On the American Option Problem

According to the theory of modern finance (see e.g. [10]) the arbitrage-free price of the American put option with a strike price K coincides with the value function V of the optimal stopping problem with the gain function G = (K x)+ . The optimal stopping time in this problem is the first time when the price process (geometric Brownian motion) falls below the value of a timedependent boundary b . When the option’s expiration date T is finite, the mathematical problem of finding V and b is inherently two-dimensional and therefore analytically more difficult (for infinite T the problem is one-dimensional and b is constant). The first mathematical analysis of the problem is due to McKean [13] who considered a ’discounted’ American call with the gain function G = e t(x K)+ and derived a free-boundary problem for V and b . He further expressed V in terms of b so that b itself solves a countable system of nonlinear integral equations (p. 39 in [13]). The approach of expressing V in terms of b was in line with the ideas coming from earlier work of Kolodner [12] on free-boundary problems in mathematical physics (such as Stefan’s ice melting problem). The existence and uniqueness of a solution to the system for b derived by McKean was left open in [13]. McKean’s work was taken further by van Moerbeke [21] who derived a single non-linear integral equation for b (pp. 145-146 in [21]). The connection to the physical problem is obtained by introducing the auxiliary function e V = @(V G)=@t so that the ’smooth-fit condition’ from the optimal stopping problem translates into the ’condition of heat balance’ (i.e. the law of conservation of energy) in the physical problem. A motivation for the latter may be seen from the fact that in the mathematical physics literature at the time it was realized that the existence and local uniqueness of a solution to such nonlinear integral equations can be proved by applying the contraction principle (fixed point theorem) firstly for a small time interval and then extending it to any interval of time by induction (see [14] and [5]). Applying this method van Moerbeke has proved the existence and local uniqueness of a solution to the integral equations of a general optimal stopping problem (see Sections 3.1 and 3.2 in [21]) while the proof of the same claim in the context of the discounted American call [13] is ’merely indicated’ (see Section 4.4 in [21]). One of the technical difficulties