Option Pricing with Polynomial Chaos Expansion Stochastic Bridge Interpolators and Signed Path Dependence

Abstract

Recent empirical studies in Equity markets show evidence that, while asset log-returns are largely uncorrelated, it is possible to predict with some accuracy their future sign. Such prediction is made over a given forecast horizon based solely on the observed sign of the cumulative log-return over a lookback horizon. This manuscript proposes a methodology to study the impact of such findings on option pricing by embedding into the risk premium the effects of signed path dependence. This is achieved by devising a model-free empirical risk-neutral distribution based on Polynomial Chaos Expansions coupled with stochastic bridge interpolators that includes information from the entire set of observable European call option prices under all available strikes and maturities for a given underlying asset. Under the real-world measure we propose a price dynamics model that is compatible with an asset price process that is largely uncorrelated but still exhibits signed path dependence. The risk premium behaviour is subsequently inferred non-parametrically via a stochastic bridge interpolation that couples the risk neutral Polynomial Chaos Expansion result with the signed path dependence mixture binomial tree dynamic model to obtain a dynamic stochastic model for the implied risk premium process.