At QuantMinds Intenational, Professor David Gershon addressed one of the most fundamental questions in the field of options since after the publication of the Black Scholes model: the volatility smile. Prof. Gershon is the founder of SuperDerivatives and the inventor of its benchmark models for option pricing.
In my lecture and paper I derive the volatility smile without any assumption on the stochastic process of the underlying asset. The derivation is independent of the asset class and indeed generates a volatility smiles that precisely match option prices in the market in FX, rates, commodities and equities.
Utilizing no arbitrage conditions, I derive a consistency condition on the probability density function to maturity that is independent of the stochastic process of the underlying asset. Using years of empirical evidence that that the Volatility Smile is independent of the term structure of volatility, a method to solve the consistency condition is proposed. The density function to maturity depends on 3 quantities only, which are related to moments of the price return and its correlation. Hence the prices of options of 3 strikes determine the whole volatility smile.
In the model testing process, enormous amounts of data is taken from the market. For example, in the past 3 years there was never a case that the model was out of the bid/ask price of major stocks and indices like Google, Footsie, SPX.
At the end of my talk, I will derive the conditional probability transfer density via bootstrapping from the term structure of the Vanilla Market. The method is validated by comparing the results of this calculation to the prices of path dependent options from the market confirms. The conclusion is that we finally found the explanation to the volatility smile we observe in the options market.