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

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 nonGaussian 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.

On the explicit evaluation of the Geometric Asian options in stochastic volatility models with jumps

“…The minimal entropy martingale measure for BNS models without and with leverage are studied in [25][26][27][28], Esscher transforms and other equivalent martingale measures are studied in [29]. Portfolio optimization has been studied in [30][31][32]. Paper [33] considers Variance Swaps.…”

Robust portfolio choice with derivative trading under stochastic volatility

“…Benth et al (2003) solve the problem of optimal wealth allocation (the solution is obtained through the Feynman-Kac representation) when the volatility is a weighted sum of non-Gaussian Ornstein-Uhlenbeck processes. Chacko and Viceira (2005) consider an infinite horizon investor who maximizes the utility of consumption when the precision (the reciprocal of volatility) follows a mean-reverting process.…”

Mean–variance portfolio selection based on a generalized BNS stochastic volatility model

“…Here, we remark that portfolio optimization in stochastic factor models has recently gained much attention in the financial literature, and particularly, Merton's classic portfolio optimization problems for the BNS volatility model and its generalized form have been discussed by Benth et al [5], Lindberg [24], Delong and Klüppelberg [12], etc., where power utility functions with exponents between 0 and 1 are considered. So their problems are different from our meanvariance portfolio selection problem (interested readers are also referred to Bielecki et al [7] and Steinbach [35] for discussions on crucial differences between Merton's utility and Markowitz's mean-variance types of models).…”

“…Our market model includes the Barndorff-Nielsen and Shephard (BNS) volatility model suggested in BNS [3] and further studied in, such as, Benth et al [5], Benth and Meyer-Brandis [4] and Lindberg [24] as a special case, and closely relates to the model considered in Delong and Klüppelberg [12]. The volatility level in these models are allowed to have sudden shifts in the upward direction, while decreasing exponentially between such shifts, and moreover, the empirical investigations on exchange rates 3522 W. Dai for real market data in BNS [3] demonstrate that such models fit the empirical auto-correlation and the leptokurtic behaviour of log-return data remarkably well.…”