Uncertain Volatility Derivative Model Based on the Polynomial Chaos

Abstract

In the modern financial market the derivative pricing considers the use of historical or implied volatility which is actually the forward expectation of uncertainty. The common way of derivative pricing is to use the volatility as constant value in the well known Black Sholes equation. The aim of the current work was to develop a model where the uncertainty of volatility propagates to the derivative pricing and hedging according to the Black-Sholes PDE considering the volatility as stochastic process rather as a constant. A stochastic finite element method using generalized polynomial chaos was used to develop an algorithm of uncertainty propagation solving finally a deterministic problem for derivative pricing. The output of the method leads to derivative price distribution and the results of Monte Carlo Method for the derivative’s distribution were used as the exact solution against those rose from the new algorithm.