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CV@R penalized portfolio optimization with biased stochastic mirror descent

Manon Costa, Sébastien Gadat, Lorick Huang

TL;DR

This article uses a Stochastic biased Mirror Descent to find optimal resource allocation for a portfolio whose underlying assets cannot be generated exactly and may only be approximated with a numerical scheme that satisfies suitable error bounds, under a risk management constraint.

Abstract

This article studies and solves the problem of optimal portfolio allocation with CV@R penalty when dealing with imperfectly simulated financial assets. We use a Stochastic biased Mirror Descent to find optimal resource allocation for a portfolio whose underlying assets cannot be generated exactly and may only be approximated with a numerical scheme that satisfies suitable error bounds, under a risk management constraint. We establish almost sure asymptotic properties as well as the rate of convergence for the averaged algorithm. We then focus on the optimal tuning of the overall procedure to obtain an optimized numerical cost. Our results are then illustrated numerically on simulated as well as real data sets.

CV@R penalized portfolio optimization with biased stochastic mirror descent

TL;DR

This article uses a Stochastic biased Mirror Descent to find optimal resource allocation for a portfolio whose underlying assets cannot be generated exactly and may only be approximated with a numerical scheme that satisfies suitable error bounds, under a risk management constraint.

Abstract

This article studies and solves the problem of optimal portfolio allocation with CV@R penalty when dealing with imperfectly simulated financial assets. We use a Stochastic biased Mirror Descent to find optimal resource allocation for a portfolio whose underlying assets cannot be generated exactly and may only be approximated with a numerical scheme that satisfies suitable error bounds, under a risk management constraint. We establish almost sure asymptotic properties as well as the rate of convergence for the averaged algorithm. We then focus on the optimal tuning of the overall procedure to obtain an optimized numerical cost. Our results are then illustrated numerically on simulated as well as real data sets.
Paper Structure (46 sections, 15 theorems, 224 equations, 7 figures, 1 table, 2 algorithms)

This paper contains 46 sections, 15 theorems, 224 equations, 7 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. CV@R$_\alpha$ constrained mean value optimization
  3. Financial model and constrained optimization
  4. Biased stochastic Mirror Descent
  5. Deterministic case
  6. Stochastic Mirror Descent (SMD) using biased simulations
  7. Portfolio hedging under $\text{CV@R}_\alpha$ constraint
  8. Description of the portfolio dynamics
  9. Simulation of the portfolio
  10. Correlated Brownian motion
  11. Geometric Brownian Motion simulations
  12. Interest rate simulation
  13. Overall simulation algorithm
  14. Stochastic Mirror Descent with sampling approximation
  15. SMD at fixed time horizon $n$
  16. ...and 31 more sections

Key Result

Proposition 2.1

The collection of the two convex problems $(\mathcal{P}_M)_{M >0}$ and $(\mathcal{Q}_\lambda)_{\lambda>0}$ are equivalent. More precisely, for any $M>0$ that defines a feasible constraint, a solution $u_M^\star$ exists such that: Moreover, $\lambda_M^\star$ is a decreasing function of $M$. Oppositely, any solution $v_\lambda$ of $\mathcal{Q}_{\lambda}$ solves $\mathcal{P}_{M}$ with $M=\text{CV@R}

Figures (7)

  • Figure 1: Time evolution of the return of the discretized trajectories associated to the assets of the synthetic portfolio (CIR + GBM).
  • Figure 2: Evolution of the weights of the assets with the number of iterations of the SMD (left) and of the PSGD (right) on $p_\lambda$ with $\lambda=0.9$ and $\alpha=0.05$. The composition of optimal portfolio may be different as represented above with SMD and PSGD.
  • Figure 3: Evolution of the CV@R and ER of the optimal portfolio with the number of iterations of the SMD (orange) and of the PSGD (blue) on $p_\lambda$ with $\lambda=0.9$ and $\alpha=0.05$.
  • Figure 4: Evolution of the V@R estimation (left) of the portfolio with the computational time of the SMD and of the MC-MD $p_\lambda$ with $\lambda=0.95$ and $\alpha=0.05$. The composition of optimal portfolio (right) converges to similar values as represented on the right with SMD and MC-MD but with a different cost in computational time.
  • Figure 5: Composition of the optimal portfolio when $\lambda$ increases. Left: 3 assets, Right: 8 assets. ER and CV@R are indicated in the legend box of each subfigure.
  • ...and 2 more figures

Theorems & Definitions (25)

  • Proposition 2.1
  • Theorem 1: Proposition 1 in Lan:2012
  • Remark 1
  • Theorem 2: Almost sure convergence of the biased SMD
  • Theorem 3: Finite-time guarantees
  • Corollary 2.1
  • Proposition 3.1
  • Proposition 3.2
  • proof : Proof of Proposition \ref{['prop:equivalence']}
  • Lemma B.1: Three points lemma
  • ...and 15 more