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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 allS  ≤   b  ⁢   (  t  ){S\leq b(t)}    with a candidate functionb  ⁢   (  t  ){b(t)}   , then thisb  ⁢   (  t  ){b(t)}    must coincide with the American put early exercise boundaryc  ⁢   (  t  ){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 functionb  ⁢   (  t  ){b(t)}    satisfies one particular integral equation (which corresponds to the upper limitS  =   b  ⁢   (  t  ){S=b(t)}   ), then all other integral equations (corresponding to  S ,S  ≤   b  ⁢   (  t  ){S\leq b(t)}   ) will be automatically satisfied by the same functionb  ⁢   (  t  ){b(t)}   .

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