An Extension of Clark-Haussman Formula and Applications

TL;DR

This paper extends the Clark-Haussmann formula to stochastic models with unbounded market price of risk, enabling better hedging and investment strategies in more realistic financial market scenarios.

Contribution

It introduces a generalized Clark-Haussmann formula applicable to models with unbounded market price of risk, expanding its practical applicability.

Findings

Extended Clark-Haussmann formula for unbounded risk processes

Application to hedging in stock markets with Ornstein-Uhlenbeck risk

Improved modeling of realistic financial risks

Abstract

This work considers a stochastic model in which the uncertainty is driven by a multidimensional Brownian motion. The market price of risk process makes the transition between real world probability measure and risk neutral probability measure. Traditionally, the martingale representation formulas under the risk neutral probability measure requires the market price of risk process to be bounded. However, in several financial models the boundedness assumption of the market price of risk fails. One example is a stock price model with the market price of risk following an Ornstein-Uhlenbeck process. This work extends Clark-Haussmann formula to underlying stochastic processes which fail to satisfy the standard requirements. Our result can be applied to hedging and optimal investment in stock markets with unbounded market price of risk.