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A Martingale approach to continuous Portfolio Optimization under CVaR like constraints

Jérôme Lelong, Véronique Maume-Deschamps, William Thevenot

TL;DR

This work addresses continuous-time portfolio optimization under an explicit DCVaR constraint in a complete market, applying a martingale approach to convert the dynamic problem into a static terminal-wealth problem. By representing DCVaR via a CVaR formulation and utilizing the state-price density, the authors derive a convex, constrained optimization over terminal payoffs, yielding a tractable, piecewise-structured optimal policy. They further specialize to deterministic multi-asset Black–Scholes settings, obtaining semi-explicit wealth and policy expressions in terms of normal probabilities and presenting a gradient-based method to optimize the auxiliary parameter $\alpha$. Numerical experiments illustrate the three-valued final wealth distribution, the effect of the capital-at-risk bound $K$, and the appearance of two regimes in the efficient frontier, highlighting both practical insights and limitations such as unbounded borrowing and sensitivity to $B$ in the tail distribution.

Abstract

We study a continuous-time portfolio optimization problem under an explicit constraint on the Deviation Conditional Value-at-Risk (DCVaR), defined as the difference between the CVaR and the expected terminal wealth. While the mean-CVaR framework has been widely explored, its time-inconsistency complicates the use of dynamic programming. We follow the martingale approach in a complete market setting, as in Gao et al. [4], and extend it by retaining an explicit DCVaR constraint in the problem formulation. The optimal terminal wealth is obtained by solving a convex constrained minimization problem. This leads to a tractable and interpretable characterization of the optimal strategy.

A Martingale approach to continuous Portfolio Optimization under CVaR like constraints

TL;DR

This work addresses continuous-time portfolio optimization under an explicit DCVaR constraint in a complete market, applying a martingale approach to convert the dynamic problem into a static terminal-wealth problem. By representing DCVaR via a CVaR formulation and utilizing the state-price density, the authors derive a convex, constrained optimization over terminal payoffs, yielding a tractable, piecewise-structured optimal policy. They further specialize to deterministic multi-asset Black–Scholes settings, obtaining semi-explicit wealth and policy expressions in terms of normal probabilities and presenting a gradient-based method to optimize the auxiliary parameter . Numerical experiments illustrate the three-valued final wealth distribution, the effect of the capital-at-risk bound , and the appearance of two regimes in the efficient frontier, highlighting both practical insights and limitations such as unbounded borrowing and sensitivity to in the tail distribution.

Abstract

We study a continuous-time portfolio optimization problem under an explicit constraint on the Deviation Conditional Value-at-Risk (DCVaR), defined as the difference between the CVaR and the expected terminal wealth. While the mean-CVaR framework has been widely explored, its time-inconsistency complicates the use of dynamic programming. We follow the martingale approach in a complete market setting, as in Gao et al. [4], and extend it by retaining an explicit DCVaR constraint in the problem formulation. The optimal terminal wealth is obtained by solving a convex constrained minimization problem. This leads to a tractable and interpretable characterization of the optimal strategy.

Paper Structure

This paper contains 15 sections, 11 theorems, 108 equations, 15 figures.

Table of Contents

  1. Problem settings
  2. Resolution
  3. Optimal terminal wealth formulation
  4. Lagrange relaxation of problem
  5. Lagrange multipliers values and uniqueness
  6. Market setting with a deterministic multi dimensional Black Scholes model
  7. Asymptotic behavior when $B$ is large
  8. Procedure to find the optimal $\alpha$
  9. Numerical Experiments
  10. A simple example
  11. Efficent frontier visualisation
  12. Summary and limitations
  13. Proof of Proposition \ref{['prop:sol_lagg_relaxation_pb']}
  14. Proof of Theorem \ref{['th_lagrange_mul_behavior']}
  15. Auxiliary Lemmas

Key Result

Proposition 2.1

Let $M\in L^2(\Omega,\mathcal{F}_T,\mathbb{P})$. Then there exist a unique ${\mathbb F}$‑predictable processes $\bigl(\beta^0_t\bigr)_{0\le t\le T}$ and $\bigl(\beta_t\bigr)_{0\le t\le T}$ such that the self‑financing portfolio value process $(V_t)_{0\le t\le T}$ defined by satisfies

Figures (15)

  • Figure 4: Optimal expected terminal value $\textrm{E}[V^*_T]$ and $-\alpha^*$ as functions of $B$.
  • Figure 5: ${\mathbb P}_B$ and ${\mathbb P}_\alpha$ as functions of $B$ for $B\in[200,10\,000]$.
  • Figure 6: Optimal expected terminal value $\textrm{E}[V^*_T]$ and $-\alpha^*$ as functions of $K$ for the previous case with four risky assets and $B=500$.
  • Figure : Evolution of asset prices $S_t$.
  • Figure : Evolution of asset prices $S_t$.
  • ...and 10 more figures

Theorems & Definitions (14)

  • Proposition 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Lemma 3.2
  • Proposition 3.3
  • Theorem 3.4
  • Proposition 3.5
  • Remark 3.6
  • Remark 4.1: On the regime change
  • Lemma B.1
  • ...and 4 more