  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 and
pricing 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, the aggregate
claims play the role of the 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 explicit expression for
the value process is obtained. 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.
The results obtained make use of various 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.
