Abstract
Natural catastrophes lead to problems of insurance and reinsurance industry. Classic insurance mechanisms are often inadequate for dealing with consequences of catastrophic events. Therefore, new financial instruments, including catastrophe bonds (cat bonds), were developed. In this paper we price the catastrophe bonds with a generalized payoff structure, assuming that the bondholder’s payoff depends on an underlying asset driven by a stochastic jump-diffusion process. Simultaneously, the risk-free spot interest rate has also a stochastic form and is described by the multi-factor Cox–Ingersoll–Ross model. We assume the possibility of correlation between the Brownian part of the underlying asset and the components of the interest rate model. Using stochastic methods, we prove the valuation formula, which can be applied to the cat bonds with various payoff functions. We use adaptive Monte Carlo simulations to analyze the numerical properties of the obtained pricing formula for various settings, including some similar to the practical cases.
Highlights
Nowadays overwhelming risks caused by natural catastrophes, like hurricanes, floods and earthquakes, lead to severe problems of insurance and reinsurance industry
Increasing number of natural catastrophes leads to problems of insurance and reinsurance industry
Since classic insurance mechanisms could be inadequate for dealing with consequences of extreme catastrophic events, even a single catastrophe could result in bankruptcy or insolvency of insurers and reinsurers
Summary
Nowadays overwhelming risks caused by natural catastrophes, like hurricanes, floods and earthquakes, lead to severe problems of insurance and reinsurance industry. The arbitrage method for cat bonds pricing is used by Vaugirard (2003) He addresses the problem of non completeness of the market, caused by catastrophic risk, and non-traded insurance-linked underlyings in the Merton’s manner (see Merton 1976). Since we use stochastic models of the spot interest rate and the underlying asset, stochastic analysis methods play the key role in derivation and proof of the cat bond pricing formula. Since the mentioned above risk-free interest rate is modeled by the multi-factor affine process, the underlying asset is defined by the stochastic jump-diffusion and the Brownian parts of both processes can be correlated, the derivation and proof of the valuation formula required application of stochastic analysis. Special attention is paid to the influence of the parameters of the underlying asset like correlation coefficients (which are important properties of the model considered in this paper) on the numerically evaluated price
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