Catastrophe options with stochastic interest rates and compound Poisson losses☆
Introduction
Although catastrophe derivatives have come into the limelight in recent years, little research has been published on the pricing and hedging issues associated with these complex instruments. Cox and Pedersen (2000) examine the pricing of catastrophe risk bonds, and briefly discuss the theory of equilibrium pricing and its relationship to the standard arbitrage-free valuation framework. Dassios and Jang (2003) use the Cox process, also known as the doubly stochastic Poisson process, to model the claim arrival process for catastrophic events; then they apply the model to the pricing of stop-loss catastrophe reinsurance contracts and catastrophe insurance derivatives. Gründl and Schmeiser (2002) analyze double trigger reinsurance contracts – a new class of contracts that has emerged in the area of “alternative risk transfer” – using several simplified modeling assumptions. In this paper, we focus on developing a coherent model for pricing and hedging catastrophe equity put options that are linked to both losses and share value of the issuing company.
In 1996 the first catastrophe equity put option or CatEPut1 was issued on behalf of RLI Corp., giving RLI the right to issue up to US$ 50 million of cumulative convertible preferred shares (Punter, 2001). In general, the CatEPut option gives the owner the right to issue convertible (preferred) shares at a fixed price, much like a regular put option; however, that right is only exercisable if the accumulated losses, of the purchaser of protection, exceed a critical coverage limit during the life time of the option. Such a contract, signed at time t, is a special form of a double trigger option and has a payoff at maturity ofwhere denotes the share value and denotes the total losses of the insured over the time period . The parameter is the trigger level of losses above which the CatEPut becomes in-the-money, while K represents the strike price at which the issuer is obligated to purchase unit shares if losses exceed .
In the event of a catastrophe, the share value of any insurance company that experiences a loss will also experience a downward jump. Consequently, it is likely that the embedded put option will end in-the-money and that the reinsurance company will be required to purchase shares at an unfavorable price. As such, it is prudent to develop a model which jointly describes the dynamics of the share value process and losses. Cox et al. (2004) were the first to introduce such a model for pricing catastrophe linked financial options. They assumed that the asset price process is driven by a geometric Brownian motion with additional downward jumps of a prespecified size in the event of a catastrophe. Since the life time of such an option can be 5 years or more, we generalize the results of Cox et al. (2004) to a stochastic interest rate environment. Moreover, their model assumes that only the event of a catastrophe affects the share value price while the size of the catastrophe itself is irrelevant. Such an assumption is a good first step; however, we propose that the loss sizes themselves should play a role, and therefore assume that the losses follow a compound Poisson process. Furthermore, we assume that the drop in asset price depends on the total loss level, rather than on only the total number of losses. When the losses, contingent on a catastrophe occurring, are a predetermined size and when the interest rates are constant, then our results reduce to those of Cox et al. (2004). In Section 2, we describe our joint asset, loss and interest rate model, in the real world, or statistical, measure, and then introduce a specific measure change to a particular risk-neutral measure.
The pricing of options in the presence of stochastic interest rates can generally be difficult. However, in Section 3 we make use of the forward-neutral measure to simplify the joint dynamics of interest rates and asset prices. This allows us to derive closed form formulae for the price of the CatEPut similar to those found by Cox et al. (2004). To understand the role that stochastic interest rates play, we conduct numerical experiments on the pricing equation in Section 4. The results indicate that stochastic interest rates can significantly affect prices for longer termed options. Our model also allows us to investigate the role that the additional stochasticity in the loss sizes themselves play. We find that the additional variance can either increase or decrease the value of the option depending on the size of the trigger level. This effect is explained by observing that the probability of exercising the option behaves in an analogous manner.
Pricing is only one aspect of the problem. It is also necessary to develop hedging strategies: This analysis is carried out in Section 5. Since the loss process itself is a non-tradable risk factor, we do not attempt to hedge the CatEPut directly. Instead we utilize a Delta, Gamma, Rho hedging scheme, which protects against changes in asset prices and interest rates. Since the asset price is correlated to losses, through the downward jumps in the asset price, it is possible to partially mitigate the tail risks through this scheme. By using simulation methods, we not only obtain the profit and loss distribution based on the prescribed hedging scheme but also illustrate the effects of model misspecifications.
Section snippets
The modeling assumptions
Let denote the share value price process; let denote the loss process of the insured; also let denote the risk-free short rate process (or the force of interest). In addition, let denote the natural filtration generated by these three processes, and let represent the probability space with statistical probability measure . It is natural to assume that the interest rate process is stochastically independent of the loss process2
Pricing the catastrophe put
Under the risk-neutral measure, the value of CatEPut contracts can be obtained via discounted expectations. Letting denote the value of the option at time t, which was signed at time , and matures at time , we haveIf interest rates are deterministic,3
Predetermined loss
The effects of stochastic interest rates can be measured by comparing our results with those of Cox et al. (2004). In addition to assuming constant interest rates, they assume that only the event of a loss decreases the share value of the insurance company, implying that the sizes of the claims are irrelevant. This assumption can be mimicked in our model by assuming that the loss size conditional on a loss is fixed at and that the trigger level is an integer multiple of the loss size, i.e.
Dynamically hedging the catastrophe put
Typically, when a reinsurer sells a CatEPut option to an insurance company, they will simultaneously hedge their position to avoid taking on very large losses. In this section, we will illustrate how to hedge against moderate movements in the asset price level and show how to hedge simultaneously against interest rate changes. In complete markets, asset price movements are hedged through Delta–Gamma hedging techniques that measure the sensitivity of the option’s price to asset price movements
Conclusions
In all, we have extended the analysis of Cox et al. (2004) to include further realism by introducing stochastic interest rates and stochastic claim sizes. Through the framework of a jump-diffusion model, we illustrate how catastrophic losses and an idiosyncratic diffusive component, together with correlated interest rate dynamics, affect option prices. Consequently, we successfully obtained closed form formulae for the price and various hedging parameters of the CatEPut option. Through
Acknowledgements
The authors thank Sheldon X. Lin and an anonymous referee for useful comments and suggestions which ultimately enhanced the presentation of the paper.
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This work was supported in part by the Natural Sciences and Engineering Research Council of Canada.