Valuation of Catastrophe Equity Puts with Markov-Modulated Poisson Processes

Article excerpt

ABSTRACT

We derive the pricing formula for catastrophe equity put options (CatEPuts) by assuming catastrophic events follow a Markov Modulated Poisson process (MMPP) whose intensity varies according to the change of the Atlantic Multidecadal Oscillation (AMO) signal. U.S. hurricanes events from 1960 to 2007 show that the CatEPuts pricing errors under the MMPP(2) are smaller than the PP by 30 percent to 66 percent. The scenario analysis indicates that the MMPP outperforms the exponential growth pattern (EG) if the hurricane intensity is the AMO signal, whereas the EG may outperform the MMPP if the future climate is warming rapidly.

INTRODUCTION

The increasing number of catastrophe (CAT) events, particularly hurricane activity in the early 1990s, has created large fluctuations in the price and availability of reinsurance and several CAT-linked instruments (e.g., CAT Bonds, CatEPuts, etc). Increases in Atlantic hurricane activity over recent decades are believed to reflect simultaneous increases in tropical Atlantic warmth (e.g., Emanuel, 2005). Some recent studies (e.g., Goldenberg et al., 2001; Landsea, 2005) attribute these increases to a natural climate cycle termed the Atlantic Multidecadal Oscillation (AMO), which is a climate signal measuring the change in the sea surface temperature (SST, and salinity) of the North Atlantic, (1) whereas the other studies suggest that climate change is instead playing the dominant role (Emanuel, 2005; Webster et al., 2005). Therefore, it is crucial to model the dynamic process of hurricane activity properly and price CAT-linked instruments (e.g., CatEPuts) corresponding to the global climate change or the change of the AMO signal for the development of the (re)insurance market.

CatEPuts are a form of options that give the owner the right to sell a specified amount of its stock to investors at a predetermined price if CAT losses surpass a specified trigger. Thus, CatEPuts can provide insurers with additional equity capital precisely when they need funds to cover CAT losses. The first CatEPut was issued on behalf of RLI Corporation in October 1996, giving RLI the right to issue up to $50 million of cumulative convertible preferred shares. In 1997, Horace Mann Educators Corporation and LaSalle Re Educators Corporation also entered into a multi-year $100 million CatEPut, respectively. In 2001, the Trenwick Group contracted the right to issue up to $55 million of cumulative convertible preferred shares to European Reinsurance Company of Zurich, a subsidiary of Swiss Re. The CatEPut was exercised in the next year to add equity to Trenwick's balance sheet. Hence, in practice CatEPuts have provided insurance firms with a useful channel to raise additional capital to hedge against CAT losses.

When pricing CatEPuts, it is prudent to develop a model that depicts the joint dynamics of the share value and losses process. Cox, Fairchild, and Pedersen (2004) assume that the share price process is driven by a geometric Brownian motion with additional downward jumps of a specific size in a CAT event. Their model assumes that only a CAT event affects the stock price, whereas the size of the CAT is irrelevant. Jaimungal and Wang (2006) extend the results of Cox, Fairchild, and Pedersen (2004) to analyze the pricing of CatEPuts under stochastic interest rates with losses generated by a compound Poisson process (PP). In addition to Cox, Fairchild, and Pedersen (2004) and Jaimungal and Wang (2006), others, for example, Cummins and Geman (1995) and Chang, Chang, and Yu (1996) look at CAT futures options, Louberge, Kellezi, and Gilli (1999) investigate a CAT bond with a pure PP, and Vaugirard (2003a, 2003b) and Lee and Yu (2002, 2007) research a CAT bond with a compound PP. (2)

[FIGURE 1 OMITTED]

Figure 1 shows the annual frequency of U.S. hurricanes between 1960 and 2007 relative to the AMO index. (3) If the intensity process of hurricane events stands for a PP, then the frequency of hurricane events should stay at the same level over the years. …