Dynamic derivative-based investment strategy for mean–variance asset–liability management with stochastic volatility

This paper considers the derivative-based optimal investment strategies for an asset–liability management (ALM) problem under the mean–variance criterion in the presence of stochastic volatility. Specifically, an asset–liability manager is allowed to invest not only in a risk-free bond and a stock, but also in a derivative, whose price depends on the underlying price of the stock and its volatility. By solving a system of two backward stochastic differential equations, we derive the explicit expressions of the efficient strategies and the corresponding efficient frontiers in two cases, with and without the derivative asset. Moreover, we consider the special case of an optimal investment problem with no liability commitment, which is also not studied in the literature. We also provide some numerical examples to illustrate our results and find that the efficient frontier of the case with the derivative is always better than that of the case without the derivative. Moreover, under the same variance, the expectation of the case with the derivative can reach up to as twice as that of the case without the derivative in some situations.