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Doubly Stochastic Point Processes in Reinsurance and the Pricing of Catastrophe Insurance Derivatives

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Doubly Stochastic Point Processes in Reinsurance and the Pricing of Catastrophe Insurance Derivatives Doubly Stochastic Point Processes in Reinsurance and the Pricing of Catastrophe Insurance Derivatives by Ji-Wook Jang Thesis submitted for the degree of Doctor of Philosophy Department of Statistics, The London School of Economics and Political Science, The University of London April 1998 UMI Number: U115577 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. Disscrrlation Publishing UMI U115577 Published by ProQuest LLC 2014. Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 7562 OF p o l it ic a l «S4,,csÇâ Contents Page Acknowledgements Abstract Introduction 1 1. Doubly Stochastic Poisson Process, ShotNoise Process and Aggregated Process 11 1.1 Doubly stochastic Poisson process 11 1.2 Shot noise process with time dependent parameters 14 1.3 Time homogeneous shot noise process 21 1.4 Aggregated process 25 2. The Cox Process with Shot Noise Intensity 31 2.1 The distribution of number of points in a fixed time interval 31 2.1.1 Time homogeneous case 31 2.1.2 Time dependent parameters 39 2.2 The distribution of the increments of 46 2.2.1 Time homogeneous case 46 2.2.2 Time dependent parameters 51 2.3 The distribution of the intensity at point times 57 2.4 The distribution of interarrival times 62 2.4.1 The distribution of interarrival time between points 62 2.4.2 Mean of interarrival time between points 63 2.4.3 Variance of interarrival time between points 64 3. Insurance Applications 66 3.1 The Esscher transform and change of probability measure 66 3.2 Pricing of a stop-loss reinsurance contract for catastrophic events 70 3.2.1 Constant claim sizes 70 3.2.2 Random claim sizes 83 Contents (continued) 3. Insurance Applications (continued) 3.3 Pricing of catastrophe insurance derivatives 85 3.3.1 Catastrophe insurance futures 85 3.3.2 Catastrophe insurance options on futures 87 3.4 Numerical examples 88 3.5 More equivalent martingale probability measures and the distribution of the total amount of claims 91 3.5.1 Where A, is time homogeneous 96 3.5.2 Where the parameters of are time dependent 98 4. Parametric Estimation 100 4.1 Approximation of as an Omstein-Uhlenbeck process 100 4.2 Where the times of catastrophes and claims are known: A direct approach 113 5. State Estimation 119 5.1 Transformations, approximations and pricing: The Kalman-Bucy filter 119 5.1.1 Transformations and approximations 119 5.1.2 The Kalman-Bucy filter and the distribution of 125 5.1.3 Pricing of a stop-loss reinsurance contract using the Kalman-Bucy filter 129 5.2 Where the times of catastrophes and claims are known 137 5.3 Where the number of claims in a fixed time interval is known 143 6. Conclusions 148 Appendix 149 References 190 Acknowledgements I would like to thank the London School of Economics and Political Science for giving me the opportunity to undertake this research work. In particular I am gratefiil to my supervisor Dr. Angelos Dassios for his patient guidance and invaluable suggestions on this dissertation. I am grateful to the Association of British Insurers for awarding me their scholarship as without their financial support I would not have been able to complete this work. I am grateful to all my family and friends in Seoul and Melbourne for their encouragement and faith in me. I give my special thanks to my wife Josephine for her support and love throughout the course of my studies and my baby daughter Amelia Yonju for providing a lovable and joyful distraction from the world of theorem and proofs. Abstract This dissertation presents pricing models for stop-loss reinsurance contracts for catastrophic events and for catastrophe insurance derivatives. We use doubly stochastic Poisson process or the Cox process for the claim arrival process for catastrophic events. The shot noise process is able to measure the frequency, magnitude and time period needed to determine the effect of the catastrophe. This process is used for the claim intensity function within the Cox process. The Cox process with shot noise intensity is examined by piecewise deterministic Markov process theory. We apply the Cox process incorporating the shot noise process as its intensity to price stop-loss catastrophe reinsurance contracts and catastrophe insurance derivatives. In order to calculate fair prices for reinsurance contracts and catastrophe insurance derivatives we need to assume that there is an absence of arbitrage opportunities in the market. This can be achieved by using an equivalent martingale probability measure in our pricing models. The Esscher transform is used to change probability measure. The dissertation also shows how to estimate the parameters of claim intensity using the likelihood function. In order to estimate the distribution of claim intensity, state estimation is employed as well. Since the claim intensity is not observable we filter it out on the basis of the number of claims, i.e. we employ the Kalman-Bucy filter. We also derive pricing formulae for stop-loss reinsurance contracts for catastrophic events using the distribution of claim intensity that is obtained by the Kalman-Bucy filter. Both estimations are essential in pricing stop-loss reinsurance contracts and catastrophe insurance derivatives. Introduction 1. General The principal aim of this dissertation is practical reinsurance problem solving. Namely, to create models for the pricing of catastrophe reinsurance contracts and catastrophe insurance derivatives. As most of our references ignore the e@ect of interest rates we will do so as well. Insurance companies have traditionally used reinsurance contracts to hedge themselves against losses from catastrophic events. During the last decade, the world has experienced a higher level of catastrophic events both in terms of frequency and severity. Some of the recent catastrophes are Hurricane Andrew (USA 1992) and the Kobe earthquake (Japan 1995) (see Booth (1997)). This has had a marked effect on the reinsurance market. Such events have impacted the profitability and capital bases of reinsurance companies some of which have withdrawn from the market and others have reduced the level of catastrophe cover they are willing to provide. In the early 1990s, some beheved that there was undercapacity provided by the reinsurance market. Some investment banks, particularly US banks, recognised the opportunities that existed in the reinsurance market. Through their large capital bases the investment banks were able to offer alternative reinsurance products. This caused reinsurance companies to reassess their strategies for the type of products offered to the market. The Chicago Board of Trade introduced catastrophe insurance futures and catastrophe insurance options on futures traded on a quarterly basis (Jan-Mar, Apr-June, July-Sep and Oct-Dec) in December 1992. The CBOT devised a loss ratio index as the underlying instrument for catastrophe insurance futures and options contracts. The Insurance Service Office calculates the index from loss data reported by at least 25 selected companies (see CBOT (1994, 1995a, 1995b)). The loss ratio index is the reported losses incurred in a given quarter and reported by the end of the following quarter, , divided by one fourth of the premiums received in the previous year, U, i.e. The value of the insurance futures, at maturityt, is the nominal contract value, US$25,000, times the loss ratio index capped at 2, i.e. = 25,000 X Min The CBOT capped the maximum loss ratio at 200% in order to limit the credit risk from unexpected huge losses and to make the contract look like a non-proportional reinsurance policy. However, to date there has not been an incident where the maximum loss ratio has been reached; the highest estimated loss ratio being 179% for Hurricane Andrew. Therefore ignoring the maximum loss ratio, the value of the catastrophe insurance call options on futures, P,, at maturityt is given by P, = Max[F, -E,0) = [F, - EY = (^25,000 x ^ j A - B)* Y\K where E is the exercise price and B = 25,000 There has been discussion and research into the possibility of using catastrophe insurance futures and options contracts rather than conventional reinsurance contracts (see Lomax & Lowe (1994), Smith (1994), Ryan (1994), Sutherland (1995), Kielholz & Durrer (1997) and Smith, Canelo & Di Dio (1997)). The competitiveness in the reinsurance market emphasises the need for an appropriate pricing model for reinsurance contracts and catastrophe insurance derivatives. It is common practice for most references to ignore the effect of interest rates. We will also adopt this approach. In insurance modelling, the Poisson process has been used as a claim arrival process. Extensive discussion of the Poisson process, from both applied and theoretical viewpoints, can be found in Cramer (1930), Cox & Lewis (1966), Bühlmann (1970), Cinlar (1975), and Medhi (1982). However, there has been a significant volume of literature that questions the appropriateness of the Poisson process in insurance modelling (see Seal (1983) and Beard et a/.(1984)) and more specifically for rainfall modelling (see Smith (1980) and Cox & Isham (1986)). As catastrophic events occur periodically, the assumption that resulting claims occur in terms of the Poisson process is inadequate. Therefore an alternative point process needs to be used to generate the claim arrival process. We will employ a doubly stochastic Poisson process, or the Cox process, (see Cox (1955), Bartlett (1963), Serfozo (1972), Grandell (1976, 1991), Bremaud (1981) and Lando (1994)).
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