On the regularity of the free boundary in the parabolic obstacle problem. Application to American options

This paper is devoted to local regularity results on the free boundary of the one-dimensional parabolic obstacle problem with variable coefficients. We give an energy criterion and a density criterion for characterising the subsets of the free boundary which are Hölder continuous in time with exponent 1/2. Our results apply in the theory of American options. As an illustration, we apply these results to the generalised Black–Scholes model of a complete market which rules out arbitrage if the volatility and the interest rate do not depend on time. In this case we prove that the exercise boundary of the American put and call options are Hölder continuous with exponent 1/2 in time for every time.