CVaR HEDGING USING QUANTIZATION‐BASED STOCHASTIC APPROXIMATION ALGORITHM

Abstract

In this paper, we investigate a method based on risk minimization to hedge observable but nontradable source of risk on financial or energy markets. The optimal portfolio strategy is obtained by minimizing dynamically the conditional value‐at‐risk (CVaR) using three main tools: a stochastic approximation algorithm, optimal quantization, and variance reduction techniques (importance sampling and linear control variable), as the quantities of interest are naturally related to rare events. As a first step, we investigate the problem of CVaR regression, which corresponds to a static portfolio strategy where the number of units of each tradable assets is fixed at time 0 and remains unchanged till maturity. We devise a stochastic approximation algorithm and study its a.s. convergence and weak convergence rate. Then, we extend our approach to the dynamic case under the assumption that the process modeling the nontradable source of risk and financial assets prices is Markovian. Finally, we illustrate our approach by considering several portfolios in connection with energy markets.

Multi-level stochastic approximation algorithms

“…[BFP09a,BFP09b]. The risk minimization of a financial portfolio by means of SA has been investigated in [BFP10,Fri14]. For more applications and a complete overview in the theory of stochastic approximation, the reader may refer to [Duf96], [KY03] and [BMP90].…”

Shortfall Risk Minimization in Discrete Time Financial Market Models

“…[16]). The arguments given in [3] relies fundamentally on the Rockafellar & Uryasev's static representation of the CVaR as a convex optimization problem. The optimal strategy is computed by minimizing dynamically the CVaR using three main numerical probabilistic tools: stochastic approximation algorithm, optimal quantization and variance reduction techniques.…”

“…However, one of the main contribution of this paper is to prove that a direct stochastic control approach in two steps, one of which consisting in a dynamic programming argument, is still possible. Unlike [3], we take full advantage of the dynamic programming backward induction to deduce first order conditions and use them to derive a completely tractable algorithm relying on stochastic approximation algorithm and optimal quantization. Let us mention that a dynamic programming principle in a non-Markovian setting has already been established in the context of expected utility maximization in discrete time models in [37].…”

“…The optimal strategy is computed by minimizing dynamically the CVaR using three main numerical probabilistic tools: stochastic approximation algorithm, optimal quantization and variance reduction techniques. From a numerical point of view, the approach developed in [3] relies on devising time-consistent risk measures closely connected to the CVaR, based on a new conditional risk measure (called the F-CVaR, where F is a σ-field representing the information available to investors), for which efficient and robust optimization procedures can be designed.…”

“…In [3], the authors considered the problem of risk minimization to hedge observable but nontradable source of risk by means of a CVaR criteria. In order to prove the existence of an optimal dynamic strategy for the CVaR minimization problem, a non Markovian dynamic programming principle is established (see e.g.…”