This article investigates the pricing of options when a need arises to carry a path dependent auxiliary state variable. Examples of such problems include the pricing of interest rate claims in a Heath Jarrow Morton paradigm, where the underlying forward rates follow a Markovian process, and the pricing of equity options, when the underlying asset price follows a GARCH process. In the former case, the primary state variable is the spot interest rate, and the auxiliary state variable is the

more »

... d variance of the current spot rate. In the latter case, the primary state variable is the asset price, and the auxiliary state variable is the path dependent statistic representing local volatility. An efficient algorithm is developed for pricing claims under these type of processes. Illustrative examples are presented that demonstrate the efficiency of the algorithm and conditions are developed that ensure the algorithm will produce accurate prices.