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Dynamic reinsurance via martingale transport

Beatrice Acciaio, Brandon Garcia Flores, Antonio Marini, Gudmund Pammer

TL;DR

This work develops a dynamic reinsurance framework that targets the terminal surplus distribution by leveraging martingale optimal transport, specifically the Bass martingale, to obtain tractable strategies. By decomposing the surplus into a deterministic drift and a pure-jump martingale, the authors cast the reinsurance design as a MOT problem with a quadratic cost equal to the ceded risk’s $L^2$-norm, achieving a Bass-martingale-like optimizer. The approach yields explicit constructions when matching a fixed terminal law, and reduces to convex quantile optimization under moment or risk constraints, with clear results for variance, VaR/ES, and skewness/kurtosis constraints. The methodology provides a practically implementable, dynamically adaptable reinsurance design with broad applicability to regulatory and risk-management objectives, while connecting dynamic reinsurance to the rich theory of martingale transport.

Abstract

We formulate a dynamic reinsurance problem in which the insurer seeks to control the terminal distribution of its surplus while minimizing the L2-norm of the ceded risk. Using techniques from martingale optimal transport, we show that, under suitable assumptions, the problem admits a tractable solution analogous to the Bass martingale. We first consider the case where the insurer wants to match a given terminal distribution of the surplus process, and then relax this condition by only requiring certain moment or risk-based constraints.

Dynamic reinsurance via martingale transport

TL;DR

This work develops a dynamic reinsurance framework that targets the terminal surplus distribution by leveraging martingale optimal transport, specifically the Bass martingale, to obtain tractable strategies. By decomposing the surplus into a deterministic drift and a pure-jump martingale, the authors cast the reinsurance design as a MOT problem with a quadratic cost equal to the ceded risk’s -norm, achieving a Bass-martingale-like optimizer. The approach yields explicit constructions when matching a fixed terminal law, and reduces to convex quantile optimization under moment or risk constraints, with clear results for variance, VaR/ES, and skewness/kurtosis constraints. The methodology provides a practically implementable, dynamically adaptable reinsurance design with broad applicability to regulatory and risk-management objectives, while connecting dynamic reinsurance to the rich theory of martingale transport.

Abstract

We formulate a dynamic reinsurance problem in which the insurer seeks to control the terminal distribution of its surplus while minimizing the L2-norm of the ceded risk. Using techniques from martingale optimal transport, we show that, under suitable assumptions, the problem admits a tractable solution analogous to the Bass martingale. We first consider the case where the insurer wants to match a given terminal distribution of the surplus process, and then relax this condition by only requiring certain moment or risk-based constraints.
Paper Structure (10 sections, 4 theorems, 43 equations, 6 figures)

This paper contains 10 sections, 4 theorems, 43 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Notations.
  3. Optimal Transport and the Bass Martingale
  4. The reinsurance problem
  5. Characterization of the Optimal Reinsurance Strategies
  6. Matching the law of $U_T^R$ to a prescribed probability distribution
  7. Matching $U_T^R$ to prescribed risk and moment constraints
  8. Variance constraint
  9. Value-at-Risk and Expected Shortfall constraints
  10. Skewness and Kurtosis constraints

Key Result

Theorem 2.1

Let $\mu, \nu \in \mathcal{P}_2(\mathbb R^d)$, with $\mu \ll \lambda$, and let $c(x,y)=|x-y|^2$. Then there exists a unique optimal transport plan $\pi \in \Pi(\mu, \nu)$, given by for some convex function $\varphi : \mathbb R^d \rightarrow \mathbb R \cup \{\infty\}$, so that $\nabla\varphi$ is a transport map from $\mu$ to $\nu$. Additionally, if $T: \mathbb R^d \rightarrow \mathbb R^d$ is a tra

Figures (6)

  • Figure 1: Optimization under variance constraint: $\text{Var}(M_T^R) \leq \text{Var}(M_T)/2$. $M$ is a compensated compound Poisson process with intensity $1$ and the claims size is exponentially distributed with parameter $1$
  • Figure 2: Optimization under Value-at-Risk constraint: $Q_{M_T^R}(0.2) - 0.05 \|M_T^R - M_T \|_{L^2} \geq -0.3$. $M$ is a compensated compound Poisson process with intensity $1$ and the claim size is exponentially distributed with parameter $1$.
  • Figure 3: Optimization under Expected Shortfall constraint: $5\int_0^{0.2} Q_{M_T^R}(u) du - 0.05 \|M_T^R - M_T \|_{L^2} \geq -0.3$. $M$ is a compensated compound Poisson process with intensity $1$ and the claim size is exponentially distributed with parameter $1$.
  • Figure 4: Optimization under Skewness constraint: $\text{Skew}_{\text{sup}}[Q_{M_T^R}] \geq 0.6$. $M$ is a compensated compound Poisson process with intensity $1$ and the claim size is exponentially distributed with parameter $1$.
  • Figure 5: Optimization under Kurtosis constraint: $\text{Kurt}_{0.1, 0.25, v}[X] \leq 2.5$. $M$ is a compensated compound Poisson process with intensity $1$ and the claim size is exponentially distributed with parameter $1$.
  • ...and 1 more figures

Theorems & Definitions (15)

  • Theorem 2.1: Brenier's Theorem, Br87
  • Example 3.1
  • Definition 3.2: Admissible strategy
  • Remark 3.3
  • Theorem 4.1
  • proof
  • Remark 4.2
  • Remark 4.3: Stability of the optimizer
  • Theorem 4.4
  • Remark 4.5
  • ...and 5 more