Valuation of American options under time-varying volatility

Abstract

Black and Scholes assumed constant variance rate of the underlying asset. But there are evidences that the volatilities change over time. So it would be appropriate to consider option values when the volatility is time-varying. When volatility is a deterministic function of time, European option price depends only on the average volatility until maturity. But American option prices are not same when volatility follows different path with the same time average. When volatility is decreasing over time, the American option value is higher than the value with constant volatility, and option value with increasing volatility us lower than the option value with constant volatility. These results are due to the early exercise feature of the American options. When the volatility is decreasing, there seem to be much chance of early-exercise.