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A Universal Model for Pricing All Options

Abstract:

The article presents a new formalism to price options which truly reflects the market prices in all asset classes. Using quasi no-arbitrage conditions on the volatility smile and integral representations of the volatility smile allow obtaining the probability density function of the underlying asset without any external parameter and accordingly the calculation of the whole volatility smile. The density function for a given time t is uniquely determined by 3 inputs from the smile at expiry time t (e.g. the prices of 3 strikes). For any t1<t2, given the density functions g(t1,S), g(t2,S) for time t1and t2, we show how to calculate the density function at each underlying asset price s1at time t1for time t2g(t2, S| t1, s1). Deriving the contingent probability function of the underlying asset purely from the vanilla options market allows the calculation of the prices of path dependent options. The new model is compared to enormous amount of data from the market is all asset classes (currencies, interest rates, equities, commodities and energy) including exotic options and matches the data remarkably well in all cases. Last, we implement the dynamic replication approach in the new framework