An extension of the Clark–Haussmann formula and applications

ABSTRACT

This work considers a financial market stochastic model where the uncertainty is driven by a multidimensional Brownian motion. The market price of the 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 require the market price of risk process to be bounded. However, in several financial models the boundedness assumption of the market price of risk fails; for example a financial market model with the market price of risk following an Ornstein–Uhlenbeck process. This work extends the Clark–Haussmann representation formula to underlying stochastic processes which fail to satisfy the standard requirements. Our methodology is classical, and it uses a sequence of mollifiers. Our result can be applied to hedging and optimal investment in financial markets with unbounded market price of risk. In particular, the mean variance optimization problem can be addressed within our framework.

KEYWORDS:

• Clark–Haussmann formula
• Ornstein–Uhlenbeck process
• multi-dimensional stock market
• state price density
• mean variance optimization

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

T. A. Pirvu http://orcid.org/0000-0002-5234-3094

Additional information

Funding

This work was supported by Natural Sciences and Engineering Research Council of Canada [371653].