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An optimal transport approach for the multiple quantile hedging problem

Cyril Bénézet, Jean-François Chassagneux, Mohan Yang

TL;DR

The paper addresses pricing and hedging under distributional risk constraints by introducing multiple quantile hedging (MQH), which subsumes super-hedging, quantile hedging, and P&L distribution shaping. It reframes MQH as a non-linear transport problem, establishing Monge and Kantorovich representations and proving no duality gap in the linear setting, which yields a practical stochastic-gradient-based algorithm. Theoretical contributions include a relaxed Monge formulation, Kantorovich relaxations, and dual representations; computational advances derive from semi-discrete OT and SGD methods, enabling stable pricing for path-dependent derivatives under finite quantile constraints. The results demonstrate that MQH prices can be computed efficiently, with numerical examples for quantile hedging and P&L hedging illustrating robustness and accuracy, and the approach offering a flexible framework for risk-managed hedging in both complete and non-linear markets.

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.

An optimal transport approach for the multiple quantile hedging problem

TL;DR

The paper addresses pricing and hedging under distributional risk constraints by introducing multiple quantile hedging (MQH), which subsumes super-hedging, quantile hedging, and P&L distribution shaping. It reframes MQH as a non-linear transport problem, establishing Monge and Kantorovich representations and proving no duality gap in the linear setting, which yields a practical stochastic-gradient-based algorithm. Theoretical contributions include a relaxed Monge formulation, Kantorovich relaxations, and dual representations; computational advances derive from semi-discrete OT and SGD methods, enabling stable pricing for path-dependent derivatives under finite quantile constraints. The results demonstrate that MQH prices can be computed efficiently, with numerical examples for quantile hedging and P&L hedging illustrating robustness and accuracy, and the approach offering a flexible framework for risk-managed hedging in both complete and non-linear markets.

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.
Paper Structure (20 sections, 17 theorems, 130 equations, 1 figure, 3 tables, 2 algorithms)

This paper contains 20 sections, 17 theorems, 130 equations, 1 figure, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notations that will be used throughout the paper:
  3. Multiple quantile hedging problem
  4. Financial setting
  5. The problem and its relaxed Monge formulation
  6. Kantorovitch problems
  7. The MQH problem as a Kantorovitch problem
  8. Linear setting: duality and numerical illustration
  9. Linear setting
  10. Dual representation
  11. Numerical Applications
  12. Application to the quantile hedging problem
  13. Numerical solution for the P&L distribution hedging problem
  14. Explicit solution
  15. Numerical illustration
  16. ...and 5 more sections

Key Result

Theorem 2.1

Under Assumptions ass: driver , for any $\xi \in L^{2}(\mathcal{F}_T,\mathbb P;\mathbb R)$, there exists a unique solution $(Y[\xi],Z[\xi]) \in \mathscr{S}^{2}(\mathbb F,\mathbb P;\mathbb R)\times\mathscr H^2(\mathbb F,\mathbb P;\mathbb R^m)$ to the BSDE with driver $f$ and terminal condition $\xi$, In particular, given the initial wealth $\bar{y}:=Y_0[\xi]\in\mathbb R$ and the strategy $\bar{\nu}

Figures (1)

  • Figure 1: Comparison of two methods: SG-solver & Exact solution benezet2021numericalfollmer1999quantile for Put and Call option, with market parameters $S_0=100$, $r = 0$, $\sigma = 0.2$ and $\beta = 0.1$, strike $K = 100$, terminal time $T = 1$. The ADAM algorithm parameters' value are specified in Appendix \ref{['se adam']}. Additionally, for the Put option (resp. the Call option), the maximal number of iterations is $M_{iter} = 5000$ (resp. $M_{iter}=2500$), while the batch size is $B=256$ (resp. $B=64$).

Theorems & Definitions (30)

  • Theorem 2.1
  • Example 2.1: Different interest rate
  • Example 2.2: Linear setting
  • Lemma 2.1
  • proof
  • Remark 2.1
  • Definition 2.1: Multiple Quantile Hedging
  • Example 2.3: Super-replication problem
  • Example 2.4: Quantile hedging problem
  • Example 2.5: P&L distribution hedging
  • ...and 20 more