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Arbitrage-free catastrophe reinsurance valuation for compound dynamic contagion claims

Jiwook Jang, Patrick J. Laub, Tak Kuen Siu, Hongbiao Zhao

TL;DR

This work develops an arbitrage-free framework for pricing catastrophe stop-loss reinsurance under a dynamic contagion model of aggregate losses. It constructs a compound dynamic contagion process (CDCP) with time-inhomogeneous extension to capture both conventional and emerging catastrophe risks, and employs the Esscher transform to define an equivalent martingale measure for risk-neutral pricing. The method yields closed-form tilts in a special exponential/gamma setting and uses Monte Carlo simulation to compute arbitrage-free premiums, with sensitivity analyses over the Esscher parameters and retention level. The results demonstrate that the arbitrage-free (gross) premiums exceed net premiums and show how parameter choices reflect market conditions, enabling practitioners to price catastrophe risk in the presence of climate, cyber, and pandemic risks. The approach offers a versatile tool for pricing catastrophe derivatives and can be extended to broader insurance-derivative applications.

Abstract

In this paper, we consider catastrophe stop-loss reinsurance valuation for a reinsurance company with dynamic contagion claims. To deal with conventional and emerging catastrophic events, we propose the use of a compound dynamic contagion process for the catastrophic component of the liability. Under the premise that there is an absence of arbitrage opportunity in the market, we obtain arbitrage-free premiums for these contacts. To this end, the Esscher transform is adopted to specify an equivalent martingale probability measure. We show that reinsurers have various ways of levying the security loading on the net premiums to quantify the catastrophic liability in light of the growing challenges posed by emerging risks arising from climate change, cyberattacks, and pandemics. We numerically compare arbitrage-free catastrophe stop-loss reinsurance premiums via the Monte Carlo simulation method. Sensitivity analyzes are performed by changing the Esscher parameters and the retention level.

Arbitrage-free catastrophe reinsurance valuation for compound dynamic contagion claims

TL;DR

This work develops an arbitrage-free framework for pricing catastrophe stop-loss reinsurance under a dynamic contagion model of aggregate losses. It constructs a compound dynamic contagion process (CDCP) with time-inhomogeneous extension to capture both conventional and emerging catastrophe risks, and employs the Esscher transform to define an equivalent martingale measure for risk-neutral pricing. The method yields closed-form tilts in a special exponential/gamma setting and uses Monte Carlo simulation to compute arbitrage-free premiums, with sensitivity analyses over the Esscher parameters and retention level. The results demonstrate that the arbitrage-free (gross) premiums exceed net premiums and show how parameter choices reflect market conditions, enabling practitioners to price catastrophe risk in the presence of climate, cyber, and pandemic risks. The approach offers a versatile tool for pricing catastrophe derivatives and can be extended to broader insurance-derivative applications.

Abstract

In this paper, we consider catastrophe stop-loss reinsurance valuation for a reinsurance company with dynamic contagion claims. To deal with conventional and emerging catastrophic events, we propose the use of a compound dynamic contagion process for the catastrophic component of the liability. Under the premise that there is an absence of arbitrage opportunity in the market, we obtain arbitrage-free premiums for these contacts. To this end, the Esscher transform is adopted to specify an equivalent martingale probability measure. We show that reinsurers have various ways of levying the security loading on the net premiums to quantify the catastrophic liability in light of the growing challenges posed by emerging risks arising from climate change, cyberattacks, and pandemics. We numerically compare arbitrage-free catastrophe stop-loss reinsurance premiums via the Monte Carlo simulation method. Sensitivity analyzes are performed by changing the Esscher parameters and the retention level.

Paper Structure

This paper contains 16 sections, 9 theorems, 93 equations, 1 figure, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Dynamics for catastrophe losses
  3. Dynamic contagion process
  4. Compound dynamic contagion process
  5. Time-inhomogeneous compound dynamic contagion process
  6. Insurance market and no-arbitrage
  7. Equivalent martingale measures
  8. Special Case: Exponential/Gamma Distributions for Jump Sizes
  9. Numerical Results
  10. Pricing of a stop-loss reinsurance contract
  11. Sensitivity Analysis for the Esscher parameters
  12. Changing theta
  13. Changing psi
  14. Changing nu
  15. Conclusion
  16. ...and 1 more sections

Key Result

Proposition 2.1

The expectation of $\lambda_{t}$ conditional on $\lambda_0$ under the real-world probability measure $\mathbb{P}$ is given by where $\kappa := \delta - \mu_{G}$. Under the stationary condition $\kappa >0$, the asymptotic first moment of $\lambda_{t}$ under $\mathbb{P}$ are given by The expectation of $N_{t}$ conditional on $N_0=0$ and $\lambda_0$ under $\mathbb{P}$ is given by The expectation o

Figures (1)

  • Figure 1: A collection of 25 sample paths of $C_t$ and the corresponding $\lambda_t$ for the compound dynamic contagion process under the original measure $\mathbb{P}$ (the left column) and the tilted measure $\mathbb{P}^{\ast}$ (the right column) given the constants outlined above.

Theorems & Definitions (23)

  • Definition 2.1: Dynamic contagion process
  • Definition 2.2: Dynamic contagion claims
  • Proposition 2.1: Expectations of $\lambda_t$, $N_t$, and $C_t$ for time-homogeneous CDCP
  • Proposition 2.2: Expectations of $\lambda_t$, $N_t$ and $C_t$ for time-inhomogeneous CDCP
  • proof
  • Definition 3.1
  • Definition 3.2: Arbitrage reinsurance strategy
  • Definition 3.3: Equivalent martingale measure
  • Definition 3.4: Esscher transform
  • Theorem 3.1
  • ...and 13 more