Value-at-Risk constrained portfolios in incomplete markets: a dynamic programming approach to Heston's model

TL;DR

This paper develops a dynamic programming approach to solve a utility maximization problem with Value-at-Risk constraints in a stochastic volatility market, linking constrained and unconstrained solutions via financial derivatives.

Contribution

It introduces a novel representation of the constrained value function using a vega-neutral derivative, connecting constrained and unconstrained optimal strategies in Heston's model.

Findings

Optimal strategies are significantly affected by risk aversion and investment horizon.

A 20% difference in allocations observed between constrained and unconstrained cases.

The approach effectively handles incomplete markets with stochastic volatility.

Abstract

We solve an expected utility-maximization problem with a Value-at-risk constraint on the terminal portfolio value in an incomplete financial market due to stochastic volatility. To derive the optimal investment strategy, we use the dynamic programming approach. We demonstrate that the value function in the constrained problem can be represented as the expected modified utility function of a vega-neutral financial derivative on the optimal terminal wealth in the unconstrained utility-maximization problem. Via the same financial derivative, the optimal wealth and the optimal investment strategy in the constrained problem are linked to the optimal wealth and the optimal investment strategy in the unconstrained problem. In numerical studies, we substantiate the impact of risk aversion levels and investment horizons on the optimal investment strategy. We observe a 20% relative difference between the constrained and unconstrained allocations for average parameters in a low-risk-aversion short-horizon setting.