A Comparison of constant and stochastic volatility in Merton’s portfolio optimization problem

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2018
Öztürk, Ozan
Merton's Portfolio Problem is a dynamic portfolio choice problem, which assumes asset returns and covariances are constant. There is well documented evidence that, stock returns and volatilities are correlated. Therefore, stochastic volatility models in dynamic portfolio problems can give better results. The work [J. Liu, Portfolio selection in stochastic environments, Review of Financial Studies, 20(1), 2007] developed a general dynamic portfolio model that allows the parameters of the model to depend on an external process X; this general model includes Merton's portfolio problem with Heston stochastic volatility (Merton H) and constant volatility as special cases. Liu's solution involves substituting solutions of a specific form into the Hamilton Jacobi Bellman (HJB) equation associated with the problem and reducing it first to a simpler Partial Differential Equation (PDE), and then reducing this PDE into a sequence of Ordinary Differential Equations (ODE). In this thesis we give the details of these reductions. We then use the explicit solutions provided by Liu for the Merton H model to see the effect of replacing stochastic volatility with constant volatility in Merton's problem. We find that, a ratio(sensitivity to stochastic volatility ratio) depending on mean reversion rate, risk aversion and Sharpe ratio is the most important parameter in this respect. When the value of this ratio is small, incorporating stochastic volatility into the model has little effect on the optimal portfolio. When it is large (when Sharpe ratio is high and the investor has low risk aversion) taking stochastic volatility into consideration is meaningful.

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Citation Formats
O. Öztürk, “A Comparison of constant and stochastic volatility in Merton’s portfolio optimization problem,” M.S. - Master of Science, Middle East Technical University, 2018.