Pricing Implied Volatility

Expected future volatility plays a central role in finance theory. Consequently, accurate estimation of this parameter is crucial to meaningful financial decision-making. Researchers generally on the past behavior of asset prices to estimate volatility, relating movements in volatility value with prior volatility and/or variables in the investors' information set. These procedures are by nature backward looking using past behavior to predict the future. An alternative approach is to use reported option prices to infer volatility expectations. Since option value depends critically on the expected future volatility, the volatility expectation of market participants can be recovered by inverting the option valuation formula when such a formula exists. The volatility expectation derived from reported option prices depends on the assumptions used to valuate the option. For example, the Black-Scholes model assumes the asset price follows a Geometric Brownian motion with constant volatility. Consequently the options on the same asset, but with different strike prices and maturity dates should provide the same implied volatility. In practice, however, the implied volatility tens to differ across exercise prices and time to expiration producing the so called " volatility smile ". The failure of the Black-Scholes model to describe the structure of reported option prices is thought to arise from its constant volatility assumption. It has been observed that when stock prices go up volatility goes down and vice versa. If the volatility is allowed to depend on the stock price it becomes stochastic and accounting for stochastic volatility within an option valuation formula is not an easy task. In the special