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

Download
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.

Suggestions

Application of stochastic volatility models with jumps to BIST options
Rahiminejat, Monireh; Sezer, Ali Devin; Department of Financial Mathematics (2017)
This thesis gives a derivation of call and put option pricing formulas under stochastic volatility models with jumps; the precise model is a combination of Merton and Heston models. The derivation is based on the computation of the characteristic function of the underlying process. We use the derived formulas to fit the model to options written on two stocks in the BIST30 index covering the first two months of 2017. The fit is done by minimizing a weighted $L_2$ distance between the observed prices and the ...
An Application of the Black Litterman model in Borsa İstanbul using analysts’ forecasts as views
Adaş, Cansu; Güner, Zehra Nuray; Danışoğlu, Seza; Department of Financial Mathematics (2016)
The optimal number of stocks to include in a portfolio in order to achieve the maximum diversification benefit has been one of the issues in which investors have focused on since Markowitz introduced fundamentals of the Modern Portfolio Theory. Each stock included in an investor's portfolio decreases the portfolio risk, while increasing the transaction costs incurred by the investor to create this portfolio. In this thesis, the size of a well-diversified portfolio consisting of stocks included consistently ...
Robust conditional value–at–risk under parallelpipe uncertainty: an application to portfolio optimization
Kara, Güray; Weber, Gerhard Wilhelm; Department of Financial Mathematics (2016)
In markets with high uncertainties, the trade–off between maximizing expected return and minimizing the risk is one of the main challenges in modeling and decision making. Since investors mostly shape their invested amounts towards certain assets and their risk version level according to their returns; scientists and practitioners has done studies on this subject since the beginning of the stock markets’ establishment. Developments and inventions in the mathematical optimization provide a wide range of solu...
Recent developments in portfolio optimization via dynamic programming
Omole, Oluwakayode John; Yolcu Okur, Yeliz; Wilhelm Weber, Gerhard; Department of Financial Mathematics (2015)
Optimal control is one of the benchmark methods used to handle portfolio optimization problems. The main goal in optimal control is to obtain a control process that optimizes the objective functional. In this thesis, we investigate optimal control problems for diffusion and jump-diffusion processes. Consequently, we present and prove concepts such as the Dynamic Programming principle, Hamilton-Jacobi-Bellman Equation and Verification Theorem. As an application of our results, we study optimization problems ...
A stochastic programming approach to multicriteria portfolio optimization
Sakar, Ceren Tuncer; Köksalan, Mustafa Murat (2013-10-01)
We study a stochastic programming approach to multicriteria multi-period portfolio optimization problem. We use a Single Index Model to estimate the returns of stocks from a market-representative index and a random walk model to generate scenarios on the possible values of the index return. We consider expected return, Conditional Value at Risk and liquidity as our criteria. With stocks from Istanbul Stock Exchange, we make computational studies for the two and three-criteria cases. We demonstrate the trade...
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.