We consider a ﬁnancial contract that delivers a single cash ﬂow given by the terminal value of a cumulative gains process.
The problem of modelling such an asset and associated derivatives is important, for example, in the determination of optimal insurance claims reserve policies, and in the pricing of reinsurance contracts. In the insurance setting, aggregate claims play the role of cumulative gains, and the terminal cash ﬂow represents the totality of the claims payable for the given accounting period. A similar example arises when we consider the accumulation of losses in a credit portfolio, and value a contract that pays an amount equal to the totality of the losses over a given time interval. An expression for the value process of such an asset is derived as follows. We ﬁx a probability space, together with a pricing measure, and model the terminal cash ﬂow by a random variable; next, we model the cumulative gains process by the product of the terminal cash ﬂow and an independent gamma bridge; ﬁnally, we take the ﬁltration to be that generated by the cumulative gains process.
An explicit expression for the value process is obtained by taking the discounted expectation of the future cash ﬂow, conditional on the relevant market information. The price of an Arrow–Debreu security on the cumulative gains process is determined, and is used to obtain a closed-form expression for the price of a European-style option on the value of the asset at the given intermediate time. The results obtained make use of remarkable properties of the gamma bridge process, and are applicable to a wide variety of ﬁnancial products based on cumulative gains processes such as aggregate claims, credit portfolio losses, deﬁned beneﬁt pension schemes, emissions, and rainfall. (Co-authors: D. C. Brody, Imperial College London, and A.
Macrina, King's College London and ETH Zurich. Downloadable at
www.mth.kcl.ac.uk.