On the Asymptotic Behavior of the Optimal Exercise Price Near Expiry of an American Put Option under Stochastic Volatility

Abstract

The behavior of the optimal exercise price of American puts near expiry has been well studied under the Black–Scholes model as a result of a series of publications. However, the behavior of the optimal exercise price under a stochastic volatility model, such as the Heston model, has not been reported at all. Adopting the method of matched asymptotic expansions, this paper addresses the asymptotic behavior of American put options on a dividend-paying underlying with stochastic volatility near expiry. Through our analyses, we are able to show that the option price will be quite different from that evaluated under the Black–Scholes model, while the leading-order term of the optimal exercise price remains almost the same as the constant volatility case if the spot volatility is given the same value as the constant volatility in the Black–Scholes model. Results from numerical experiments also suggest that our analytical formulae derived from the asymptotic analysis are quite reasonable approximations for options with remaining times to expiry in the order of days or weeks.