Valuation of American put options under a modified 4/2 stochastic volatility model

In this paper, we study the valuation of American put options based on the 4/2 stochastic volatility model that incorporates multiscale double mean-reverting (DMR) volatility. The option price problem is transformed into a partial differential equation (PDE) problem with free boundary, which in turn leads to PDE problems for a few terms of asymptotic expansions of the option price and free boundary. The approximate American put price is decomposed into the sum of the corresponding European put price and the early exercise premium. The chosen modification of the 4/2 stochastic volatility allows for a step-by-step approach to the option price starting from the Black–Scholes price of the corresponding European option, making it easier to approximate the price of an American put option. We check the accuracy of the resultant approximate option price and free boundary by using the least squares Monte Carlo simulation method and investigate the impact of the Heston and 3/2 factors of the volatility on the option price and free boundary. We calibrate our model to real market data and benchmark it against the widely used Heston model and the 3/2 model. We also conduct a sensitivity analysis to show how small changes in model parameters influence the American put option premium and early exercise boundary, and discuss limiting scenarios when the 3/2 term vanishes or volatility becomes deterministic. In addition, two specific results are provided. We derive a semi-analytic solution for the approximate option price and free boundary when an American put option is near expiration. We also study the pricing of an American put option without an expiration date and obtain a closed-form analytic formula for the approximate option price and free boundary.