Merton's portfolio optimization problem in a Black and Scholes market with non‐Gaussian stochastic volatility of Ornstein‐Uhlenbeck type

Fred Espen Benth · Kenneth Hvistendahl Karlsen · Kristin Reikvam

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Stochastic processes and financial applications Financial Risk and Volatility Modeling Risk and Portfolio Optimization

Abstract

We study Merton's classical portfolio optimization problem for an investor who can trade in a risk‐free bond and a stock. The goal of the investor is to allocate money so that her expected utility from terminal wealth is maximized. The special feature of the problem studied in this paper is the inclusion of stochastic volatility in the dynamics of the risky asset. The model we use is driven by a superposition of non‐Gaussian Ornstein‐Uhlenbeck processes and it was recently proposed and intensively investigated for real market data by Barndorff‐Nielsen and Shephard (2001) . Using the dynamic programming method, explicit trading strategies and expressions for the value function via Feynman‐Kac formulas are derived and verified for power utilities. Some numerical examples are also presented.

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References

1971Optimum consumption and portfolio rules in a continuous-time model 1969Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case 1983Stochastic Differential Equations and Diffusion Processes 2001Non-Gaussian Ornstein–Uhlenbeck-based Models and Some of Their Uses in Financial Economics 1998The Variance Gamma Process and Option Pricing 1990The Variance Gamma (V.G.) Model for Share Market Returns 1997Processes of normal inverse Gaussian type 1995Hyperbolic Distributions in Finance

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2008Optimal investment and consumption in a Black–Scholes market with Lévy-driven stochastic coefficients 2010UTILITY MAXIMIZATION IN AFFINE STOCHASTIC VOLATILITY MODELS 2005Optimal Consumption-Investment Problems in Incomplete Markets with Stochastic Coefficients 2016Pricing variance and volatility swaps for Barndorff-Nielsen and Shephard process driven financial markets 2016GENERALIZED BN–S STOCHASTIC VOLATILITY MODEL FOR OPTION PRICING

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