The PDEs and Numerical Scheme for Derivatives under Uncertainty Volatility

Yulian Fan

doi:10.1155/2019/1268301

Abstract

We use the stochastic differential equations (SDE) driven by G-Brownian motion to describe the basic assets (such as stocks) price processes with volatility uncertainty. We give the estimation method of the SDE’s parameters. Then, by the nonlinear Feynman-Kac formula, we get the partial differential equations satisfied by the derivatives. At last, we give a numerical scheme to solve the nonlinear partial differential equations.

Highlights

The financial asset pricing theory is usually under the framework of a stochastic model: a set of future scenarios (Ω, F) and a probability measure P on these outcomes
There are many circumstances in financial decision making where the decision-maker or risk manager is not able to attribute a precise probability to future outcomes
Various case studies have indicated the importance of model uncertainty in financial market, even in macroeconomics, and some spectacular failures in risk management of derivatives have emphasized the consequences of neglecting model uncertainty

Summary

Introduction

The financial asset pricing theory is usually under the framework of a stochastic model: a set of future scenarios (Ω, F) and a probability measure P on these outcomes. When the uncertainty comes from the volatility coefficient, the probabilities {Pθ}θ∈Θ are mutually singular This type of uncertainty was initially studied by [14] for the superhedging of European type contingent claims whose payoffs depend only on the basic asset’s terminal value. We use the stochastic differential equations driven by G-Brownian motion to describe the stock price processes with volatility uncertainty. We give the discretization scheme of the corresponding nonlinear partial differential equations This procedure can be used to pricing derivatives with some kind of path-dependent payoffs and it is practically applicable.

The Stock Price Process and the Estimation of the Parameters

The Nonlinear Feynman-Kac Formula

Solving the PDE

Conclusion