Valuation of an American Put Catastrophe Insurance Futures Option: A Martingale Approach

The goal of this paper is to develop an arbitrage free valuation formula for an American put option on a catastrophe insurance futures contract. This contract (denoted CATS) was introduced in December 1992 by the Chicago Board of Trade. The option buyer’s valuation problem is formulated as an optimal stopping problem within a continuous trading, arbitrage-free and complete financial market. The problem is then analyzed vis Karatzas’ unified equivalent martigale measure framework. His framework is a bit more elaborate than the Harrison-Kreps and Harrison-Pliska seminal models in which the key economic idea of the absence of arbitrage opportunities is linked to the probabilistic concept of a martingale. Specifically, Karatzas’ framework enables market participants to consume as well as invest. This permits a unified approach to the problem of: option pricing, consumption and investment, and equilibrium in a financial market. We extend Karatzas’ framework to a futures option setting by replacing his risky stock with a CATS future contract whose underlying asset is the loss-ratio index of the pool of insurers that comprise the index. This is a nontrivial extension since the loss-ratio index is a non-marketed "asset", i.e., a cashflow determined by the loss claims of insured victims of catastrophes. Thus, determining the market price of risk is difficult. In this setting the CATS future option is a redundant asset and, thus, valued via the usual replication argument. In addition to extending Karatzas’ framework, we obtain two main financial results: 1) a representation of the arbitrage-free put value function as the exception (under the risk-neutral probability measure) of the present value of the option payoff at the optimal exercise tie; and 2) a decomposition (via the Riesz Decomposition Theorem) of the American put value function into the corresponding European option price and the early exercise premium. Finally, do demonstrate the method of obtaining explicit solutions for the value function and the optimal exercise (stopping) boundary, the stopping problem (for the conventional American put equity option) is reformulated (invoking the Feynman-Kac theorem) as a free-boundary (Stefan) problem. Several difficulties that hinder the numerical analysis of this problem are briefly discussed. Reinsurance Research