A note on the nonlinear Volterra integral equation for the early exercise boundary

Abstract

In this paper, we study the nonlinear Volterra integral equation satisfied by the early exercise boundary of the American put option in the one-dimensional diffusion model for a stock price with constant interest rate and constant dividend yield and with a local volatility depending on the current time t and the current stock price S. In the classical Black–Sholes model for a stock price, Theorem 4.3 of [S. D. Jacka, Optimal stopping and the American put, Math. Finance 1 1991, 2, 1–14] states that if the family of integral equations (parametrized by the variable S) holds for all S≤b⁢(t) with a candidate function b⁢(t), then this b⁢(t) must coincide with the American put early exercise boundary c⁢(t). We generalize Peskir’s result [G. Peskir, On the American option problem, Math. Finance 15 2005, 1, 169–181] to state that if the candidate function b⁢(t) satisfies one particular integral equation (which corresponds to the upper limit S=b⁢(t)), then all other integral equations (corresponding to S, S≤b⁢(t)) will be automatically satisfied by the same function b⁢(t).