Abstract
This paper considers a mean-variance portfolio selection problem when the stock price has a 3/2 stochastic volatility in a complete market. Specifically, we assume that the stock price and the volatility are perfectly negative correlated. By applying a backward stochastic differential equation (BSDE) approach, closed-form expressions for the statically optimal (time-inconsistent) strategy and the value function are derived. Due to time-inconsistency of mean variance criterion, a dynamic formulation of the problem is presented. We obtain the dynamically optimal (time-consistent) strategy explicitly, which is shown to keep the wealth process strictly below the target (expected terminal wealth) before the terminal time. Finally, we provide numerical studies to show the impact of main model parameters on the efficient frontier and illustrate the differences between the two optimal wealth processes.
Highlights
In the last several decades, various stochastic volatility models have been developed in the literature to explain the volatility smile and heavy tails of return distribution as widely observed in the financial market; see, for example, Heston (1993); Hull and White (1987); Lewis (2000); and Stein and Stein (1991)
We consider a dynamic mean-variance portfolio selection problem within the framework developed in Pedersen and Peskir (2017) in a complete market with two primitive assets: a risk-free asset and a stock with 3/2 stochastic volatility
We review the definition of dynamic optimality in problem (4) for the readers’ convenience
Summary
In the last several decades, various stochastic volatility models have been developed in the literature to explain the volatility smile and heavy tails of return distribution as widely observed in the financial market; see, for example, Heston (1993); Hull and White (1987); Lewis (2000); and Stein and Stein (1991). To extend the results to more realistic models with random parameters, on the assumption that the return rate, volatility, and risk premium are all bounded stochastic processes, the backward stochastic differential equation (BSDE) approach is introduced by Lim and Zhou (2002) to solve a mean-variance problem in a complete market. They overcome the timeinconsistency by recomputing the statically optimal (pre-committed) strategy during the investment period, and they can obtain dynamically optimal (time-consistent) strategies by solving infinitely many optimization problems Motivated by these aspects, we consider a dynamic mean-variance portfolio selection problem within the framework developed in Pedersen and Peskir (2017) in a complete market with two primitive assets: a risk-free asset and a stock with 3/2 stochastic volatility.
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