Catastrophe Bonds: Modeling Rare Events and Pricing Risk

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A catastrophe (CAT) bond is a debt instrument designed to transfer extreme event risks from insurers to capital market investors. They’re important for financial institutions, especially insurers and reinsurers, because they offer a way to manage large, low-probability. In this issue, I feature research on CAT bonds, how they're priced, and why they matter more than ever in a world of rising tail risks.

In this issue

• Latest Posts

• A Pricing Model for Earthquake Bonds

  • Findings

• No-arbitrage Model for Pricing CAT Bonds

  • Findings

• Closing Thoughts

• Educational Video

  • Catastrophe bonds: Primed for records and growth

• Volatility Weekly Recap

• Around the Quantosphere

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A Pricing Model for Earthquake Bonds

An earthquake bond is a type of catastrophe bond, in which an insurer, reinsurer, or government, transfers a portion or all of the earthquake risk to investors in return for higher yields. Earthquake bonds are crucial in countries prone to earthquakes. However, pricing them presents challenges.

Reference [1] developed a pricing model for pricing earthquake bonds. The authors modeled the risk-free interest rate using the Cox–Ingersoll–Ross model. They accommodated the variable intensity of events with an inhomogeneous Poisson process, while extreme value theory (EVT) was used to model the maximum strength.

Findings

• Earthquake bonds (EBs) connect insurance mechanisms to capital markets, offering a more sustainable funding solution, though pricing them remains a challenge. 

• The paper proposes zero-coupon and coupon-paying EB pricing models that incorporate varying earthquake event intensity and maximum strength under a risk-neutral framework. 

• The models focus on extreme earthquakes, which simplifies data processing and modeling compared to accounting for continuous earthquake occurrences. 

• The earthquake event intensity is modeled using an inhomogeneous Poisson process, while the maximum strength is handled through extreme value theory (EVT). 

• The models are tested using earthquake data from Indonesia’s National Disaster Management Authority covering 2008 to 2021. 

• Sensitivity analyses show that using variable intensity instead of constant intensity significantly affects EB pricing. 

• The proposed pricing model can help EB issuers set appropriate bond prices based on earthquake risk characteristics. 

• Investors can use the sensitivity findings to select EBs that align with their individual risk tolerance.

In summary, the authors modeled the risk-free interest rate using the Cox–Ingersoll–Ross model. They accommodated the variable intensity of events with an inhomogeneous Poisson process, while extreme value theory (EVT) was used to model the maximum strength.

Reference

[1] Riza Andrian Ibrahim, Sukono, Herlina Napitupulu and Rose Irnawaty Ibrahim, Earthquake Bond Pricing Model Involving the Inconstant Event Intensity and Maximum Strength, Mathematics 2024, 12, 786

No-arbitrage Model for Pricing CAT Bonds

Pricing models for catastrophic risk-linked securities have primarily followed two methodologies: the theory of equilibrium pricing and the no-arbitrage valuation framework.

Reference [2] proposed a pricing approach based on the no-arbitrage framework. It utilizes the CIR stochastic process model for interest rates and the jump-diffusion stochastic process model for losses.

Findings

• This paper explores the concept of CAT bonds and explains how they are modeled using financial mathematics.

• Through a semi-discretization approach, a PIDE and a first-order differential equation were derived.

• A key component, the market price of risk of damage, was unavailable, so a quadratic term was constructed using market ask and bid prices to estimate this variable.

• By utilizing the Euler-Lagrange equation, a Poisson PDE was derived.

• The paper concludes by presenting an approach and numerical results for determining the market price of risk.

We find the stochastic model, equation (1), to be particularly insightful and effective in describing catastrophic losses.

Last year has witnessed numerous hurricanes across Asia, Europe, and America, leading to significant claims for insurers. This paper represents a contribution to advancing risk-sharing practices in the insurance industry.

Reference

[2] S. Pourmohammad Azizi & Abdolsadeh Neisy, Inverse Problems to Estimate Market Price of Risk in Catastrophe Bonds, Mathematical Methods of Statistics, Vol. 33 No. 3 2024

Closing Thoughts

In this issue, I discussed catastrophe bonds and why they matter for investors navigating extreme event risks. The first paper focused on earthquake bonds, which present a challenge to model due to their rare and severe nature. Interestingly, both pricing models in the paper relied on the Cox–Ingersoll–Ross framework for modeling interest rates, a reminder that even in the world of tail-risk instruments, some core quantitative models remain consistent.

Educational Video

Catastrophe bonds: Primed for records and growth

This video features a panel discussion on CAT bonds. At the time, the market was in its usual hurricane season slowdown, but the panel looked back at a record-breaking first half of the year and shared expectations for a busy pipeline ahead. They touched on investor sentiment, sponsor motivations, and why issuance was likely to accelerate.

Watch the full video to hear the industry experts discuss the state of the catastrophe bond market, the appetite of investors for new cat bond investments, and the key role the cat bond market has developed as a source of reinsurance and risk transfer protection.

Volatility Weekly Recap

The figure below shows the term structures for the VIX futures (in colour) and the spot VIX (in grey).

You’ve probably read and heard plenty of market news last week, so I won’t repeat any of it here. I’ll just leave you with a snapshot of the volatility market: both spot and futures are in extremely steep backwardation, and spot volatilities are trading well above futures.

Around the Quantosphere

• World First: New Catastrophe Bond ETF Hits The Market (etf)

• Hedge fund winners and losers in March selloff (hedgeweek)

• Are pure high-frequency trading firms quietly dying? (efinancialcareers)

• Hedge funds hit with steepest margin calls since 2020 Covid crisis (ft)

• Biggest multistrategy hedge funds slowed hiring last year, still added 550 PMs (efinancialcareers)

Disclaimer

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