Abstract
It has been found that the surface of implied volatility has appeared in financial market embrace volatility “Smile” and volatility “Smirk” through the long-term observation. Compared to the conventional Black-Scholes option pricing models, it has been proved to provide more accurate results by stochastic volatility model in terms of the implied volatility, while the classic stochastic volatility model fails to capture the term structure phenomenon of volatility “Smirk.” More attempts have been made to correct for American put option price with incorporating a fast-scale stochastic volatility and a slow-scale stochastic volatility in this paper. Given that the combination in the process of multiscale volatility may lead to a high-dimensional differential equation, an asymptotic approximation method is employed to reduce the dimension in this paper. The numerical results of finite difference show that the multiscale volatility model can offer accurate explanations of the behavior of American put option price.
Highlights
Compared to the European option, the biggest difference is that American option can be exercised any time before its maturity date
The American option can be valued using an analytic approximation approach called the Barone-Adesi and Whaley method (BAW method) when the underlying asset is driven by a stochastic process with constant volatility [4]
P1,0ðS, z, TÞ = P0,1ðS, z, TÞ = 0: Here, we propose a series of theorems correspondingly based on the assumptions above to assist for the pricing of American option
Summary
Compared to the European option, the biggest difference is that American option can be exercised any time before its maturity date. The American option can be valued using an analytic approximation approach called the Barone-Adesi and Whaley method (BAW method) when the underlying asset is driven by a stochastic process with constant volatility [4]. The significant leptokurtic feature of the underlying asset process found from lots of empirical evidences indicates that the volatility should be randomly distributed rather than a constant, and the BAW model can hardly be applied if a stochastic process drives the volatility. To solve this problem, many researchers currently resort for the stochastic volatility models. The multiscale stochastic volatility model has been proposed to deal with the fast data and slow data frequency of the underlying asset
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