An optimal transport approach for the multiple quantile hedging problem

ArXiv ID: 2308.01121 “View on arXiv”

Authors: Unknown

Abstract

We consider the multiple quantile hedging problem, which is a class of partial hedging problems containing as special examples the quantile hedging problem (F{“ö”}llmer & Leukert 1999) and the PnL matching problem (introduced in Bouchard & Vu 2012). In complete non-linear markets, we show that the problem can be reformulated as a kind of Monge optimal transport problem. Using this observation, we introduce a Kantorovitch version of the problem and prove that the value of both problems coincide. In the linear case, we thus obtain that the multiple quantile hedging problem can be seen as a semi-discrete optimal transport problem, for which we further introduce the dual problem. We then prove that there is no duality gap, allowing us to design a numerical method based on SGA algorithms to compute the multiple quantile hedging price.

Keywords: Optimal Transport, Quantile Hedging, Non-Linear Markets, Monge Problem, Duality Gap, Derivatives

Complexity vs Empirical Score

  • Math Complexity: 9.2/10
  • Empirical Rigor: 2.5/10
  • Quadrant: Lab Rats
  • Why: The paper relies heavily on advanced mathematical concepts from optimal transport theory, stochastic control, and BSDEs, with extensive theoretical derivations and proofs, while lacking any empirical implementation details, backtests, or datasets.
  flowchart TD
    A["Research Goal<br>Multiple Quantile Hedging<br>in Non-Linear Markets"] --> B{"Data & Inputs"}
    B --> B1["Financial Instruments<br>Option Payoffs"]
    B --> B2["Market Constraints<br>Dynamic Strategies"]
    
    B1 & B2 --> C["Key Methodology<br>Optimal Transport Framework"]
    
    C --> C1["Reformulation<br>Monge vs Kantorovitch<br>Problem Equivalence"]
    C --> C2["Duality Analysis<br>Linear Case<br>No Duality Gap"]
    
    C1 & C2 --> D["Computational Process<br>Stochastic Gradient Ascent Algorithm"]
    
    D --> E["Key Findings & Outcomes"]
    
    E --> E1["Valuation<br>Hedging Price Determination"]
    E --> E2["Theoretical<br>Equivalence Proof"]
    E --> E3["Practical<br>Numerical Method"]