Abstract

The implied volatility surface is a graphical representation depicting Black-Scholes implied volatilities across various option strikes and expirations. Implied volatility serves as a gauge for the market's anticipation of future price volatility in an underlying asset, inferred from current option prices. For a given maturity, the implied volatility curve can exhibit different shapes for distinct strike prices, such as a smile, smirk, or tilt. A smile occurs when implied volatilities are elevated for both out-of-the-money (OTM) and in-the-money (ITM) options compared to at-the-money (ATM) options. Conversely, a smirk or tilt arises when the skew is more pronounced on one side, either higher for OTM or ITM options. The presence of a non-flat implied volatility curve signals that the stock price distribution implied in option prices deviates from the lognormal distribution supporting Black-Scholes option pricing. The paper demonstrates a possible construction of a stock price stochastic process that aligns with observed option prices, by employing a non-linear monotonic transformation of a standard Brownian motion. The paper includes a detailed numerical example that illustrates the implementation of this idea with reference to plain vanilla options and then the  pricing of calendar spread options. Possible extensions are also discussed.