Valuation of European Options under an Uncertain Market Price of Volatility Risk

TL;DR

This paper introduces a Hamilton-Jacobi-Bellman framework to quantify the impact of parameter uncertainty on European option prices within the Heston model, highlighting nonlinear effects on Delta under volatility risk uncertainty.

Contribution

It develops a novel approach combining HJB equations and finite element methods to evaluate worst and best case option prices considering market volatility risk uncertainty.

Findings

Nonlinear dependence of Delta on uncertainty magnitude

Variation of option prices across different parameter regimes

Effective numerical approximation of HJB equations

Abstract

We propose a model to quantify the effect of parameter uncertainty on the option price in the Heston model. More precisely, we present a Hamilton-Jacobi-Bellman framework which allows us to evaluate best and worst case scenarios under an uncertain market price of volatility risk. For the numerical approximation the Hamilton--Jacobi--Bellman equation is reformulated to enable the solution with a finite element method. A case study with butterfly options exhibits how the dependence of Delta on the magnitude of the uncertainty is nonlinear and highly varied across the parameter regime. Keywords: Uncertain market price, Volatility risk, Hamilton-Jacobi-Bellman equation, Finite element method, Uncertainty quantification