A General Theory of Option Pricing

We present a generic formalism for option pricing which does not require specifying the stochastic process of the underlying asset price, undergoing a Markovian stochastic behavior. We first derive a consistency condition that the risk neutral density function to maturity must satisfy in order to guarantee no arbitrage. As an example, we show that when the underlying asset price undergoes a continuous stochastic process with deterministic time dependent standard deviation the formalism produces the Black–Scholes–Merton pricing formula. We provide data from the market to prove that the price of European options is independent of the term structure of the volatility prior to maturity. Based on this observation we offer a method to calculate the density function to maturity that satisfies the consistency condition we derived. In the general case the price of European options depends only on the moments of the price return of the underlying asset. When the underlying asset undergoes a continuous time process then only moments up to second order contribute to the European option price. In this case any set of option prices on three strikes with the same maturity contains the information to determine the whole volatility smile for this maturity. Using a great amount of data from the option markets we show that our formalism generates European option prices that match the markets prices very accurately in all asset classes: currencies, equities, interest rates and commodities. Finally, using bootstrapping method with market data of the whole term structure we determine the probability transfer density function from inception to maturity, thus allowing the calculation of path dependent options. Comparison of the results of the model to the market shows a very high level of accuracy.