American options exercise boundary when the volatility changes randomly
The American put option exercise boundary is studied. It is shown that the put option premium in the Black and Scholes model [see \textit{F. Black} and \textit{M. Scholes}, J. Political Economy 81, 637-659 (1973)] is a one-to-one function of the volatility parameter. The paper focuses on the dependence of the optimal exercise boundary on the volatility parameter. A stochastic volatility model for the underlying asset, which generalizes the Black and Scholes model by allowing the volatility itself to be varying [see \textit{J. Hull} and \textit{A. White}, J. Finance 3, 281-300 (1987); \textit{L. Scott}, J. Financial and Quantitative Analysis 22, 419-438 (1987); \textit{J. Wiggins}, J. Financial Economics 19, 351-372 (1987)] is considered. It is proved that (i) the American put price function is continuously differentiable, (ii) there exists an optimal exercise boundary and (iii) the American put price function is strictly convex in the current underlying asset price in the continuation region. In the stochastic volatility model studied in the paper the exercise boundary is a function of the volatility variable. It is proved that the optimal exercise boundary is a decreasing function of the volatility.
scientific article; zbMATH DE number 1289884
scientific article