Doubly-Binomial Option Pricing with Application to Insurance Derivatives

We generalize the standard lattice approach of Cox, Ross, and Rubinstein (1976) from a fixed sum to a random sum in a subordinated process framework to accommodate pricing of derivatives with random-sum characteristics. The asset price change now is determined by two independent Bernoulli trials on information arrival/non-arrival and price up/down, respectively. The subordination leads to a nonstationary trinomial tree in calendar-time, while a time change to information-time restores the simpler binomial tree that now grows with the intensity of information arrival irrespective of the passage of calendar-time. We apply the model to price the CBOT catastrophe futures call spreads as a binomial sum of binomial prices, which illuminates the information conveyed by the randomness of catastrophe arrival. Numerical results demonstrate that the standard binomial formula that ignores random claim arrival produces largest undervaluation error for out-of-money short-maturity options when a small number of significant catastrophes may strike during the option's maturity.