Determining Volatility Surfaces And Option Values From An Implied Volatility Smile

Using only the implied volatility smile of a single maturity T and an assumption of path-independence, we analytically determine the risk-neutral stock price process and the local volatility surface up to an arbitrary horizon T′ ≥ T. Our path-independence assumption requires that each positive future stock price St is a function of only time t and the level Wt of the driving standard Brownian motion (SBM) for all t ∈ (0, T′). Using the T-maturity option prices, we identify this stock pricing function and thereby analytically determine the risk-neutral process for stock prices. Our path-independence assumption also implies that local volatility is a function of the stock price and time which can be explicitly represented in terms of the known stock pricing function. Finally, we derive analytic valuation formulae for standard and exotic options which are consistent with the observed T-maturity smile.