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

Marcos Escobar-Anel, Yevhen Havrylenko, Rudi Zagst

TL;DR

This paper develops a dynamic-programming framework to solve Value-at-Risk constrained utility maximization in incomplete markets driven by stochastic volatility (Heston model). It proves that the constrained optimal wealth can be represented as a Vega-neutral contingent claim on the unconstrained optimum, linking unconstrained and constrained strategies via a price function $D^{\mathbb{Q}(\lambda^v)}$ and a policy relation $\pi_c^*(t)=\pi_u^*(t)\, y \frac{D_y^{\mathbb{Q}}(t,y,v)}{D^{\mathbb{Q}}(t,y,v)}$. In the detailed Heston application, the authors construct a continuum of vega-neutral payoffs $\widehat{D}(\cdot;k_{\varepsilon,t},k_{v,t})$ with time-varying strikes to satisfy the VaR constraint, which can be reduced to a single derivative $D(Y^{*},v)$ via a proposition. Numerical studies with average-parameter settings show the constrained allocation can differ from the unconstrained one by about 20% for short horizons and low risk aversion, and the gap shrinks as risk aversion increases or horizon lengthens. The results also demonstrate the feasibility of pricing the auxiliary derivative via Carr–Madan Fourier methods and solving a family of nonlinear equations to obtain the optimal strategy, highlighting the practical relevance for risk management in banks and insurers under 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.

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

TL;DR

This paper develops a dynamic-programming framework to solve Value-at-Risk constrained utility maximization in incomplete markets driven by stochastic volatility (Heston model). It proves that the constrained optimal wealth can be represented as a Vega-neutral contingent claim on the unconstrained optimum, linking unconstrained and constrained strategies via a price function and a policy relation . In the detailed Heston application, the authors construct a continuum of vega-neutral payoffs with time-varying strikes to satisfy the VaR constraint, which can be reduced to a single derivative via a proposition. Numerical studies with average-parameter settings show the constrained allocation can differ from the unconstrained one by about 20% for short horizons and low risk aversion, and the gap shrinks as risk aversion increases or horizon lengthens. The results also demonstrate the feasibility of pricing the auxiliary derivative via Carr–Madan Fourier methods and solving a family of nonlinear equations to obtain the optimal strategy, highlighting the practical relevance for risk management in banks and insurers under 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.
Paper Structure (11 sections, 6 theorems, 152 equations, 4 figures)

This paper contains 11 sections, 6 theorems, 152 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Constrained portfolio optimization problem and its solution
  3. Detailed application of Theorem \ref{['MainTheo']}
  4. Numerical studies
  5. Model parameterization and numerical procedure
  6. Numerical results
  7. Conclusion
  8. Results on unconstrained problem
  9. Proofs of main results
  10. Explicit formulas for the left-hand side of \ref{['eq:SNLE_rho_zero']}
  11. Numerical studies for more turbulent markets

Key Result

Theorem 2.1

\newlabelMainTheo0 $\,$ Assume that Condition KraftCondition holds and that the VaR constraint is feasible in MainControlProb. Let $D(\cdot,\cdot)$, $y_0$, $\lambda^v(\cdot)$ and $\lambda_{\varepsilon}$ be such that $\mathbb{P}_{0, y_0, v_0}\left( D(Y^{y_0, \pi^{\ast}_{u}}(T), v(T)) < K \right) = \ where $D^{\mathbb{Q}(\lambda^{v})}$ is given by eq:def_of_financial_derivative_on_Y, $\overline{U}^{

Figures (4)

  • Figure 1: Comparison of payoff structures of to-be-conjectured $D$ in complete and incomplete markets
  • Figure 1: The impact of VaR-probability threshold on the solution to Problem \ref{['MainControlProb']}
  • Figure 1: The impact of $\rho$, $\sigma$, $\kappa$ on the optimal investment strategies in a more turbulent market
  • Figure 2: The impact of risk aversion and time horizon on the optimal investment strategies

Theorems & Definitions (16)

  • Theorem 2.1: Representation of constrained-problem solution
  • Proof 1
  • Lemma 2.2: Sufficient condition for \ref{['cond:U_D_yy_y']} and \ref{['cond:U_D_yv_y']}
  • Proof 2
  • Proposition 3.1
  • Proof 3
  • Corollary 3.2: Solution to \ref{['MainControlProb']}
  • Proof 4
  • Proposition A.1
  • Proof 5: Proof of Proposition \ref{['prop:unconstrained_problem_solution']}
  • ...and 6 more