On Pricing Derivatives in the Presence of Auxiliary State Variables

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 accrued 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. This article is concerned with the efficient pricing of contingent claims when a need arises to carry a path dependent auxiliary state variable. These problems are often encountered when pricing specialized derivative claims such as average rate options, where the auxiliary state variable is the average price to date, or lookback options, where the auxiliary state variable might be the maximum price to date. There are other problems, however, where the auxiliary variable arises more naturally in the underlying dynamics of the process rather than being defined by the nature of the financial claim. For example, consider pricing interest rate claims under the assumption that volatilities of forward rates belong to a one factor Heath Jarrow Morton (1992) (hereafter HJM) family, where bond prices can be represented by a low dimensional Markovian system. In such models, the primary underlying state variable is the short rate and the second, auxiliary state variable, is a path statistic that corresponds to the accrued variance of the spot rate. As a second example, consider the pricing of an equity derivative when the underlying dynamics are given by a GARCH process. Here the primary state variable is the stock price and the second auxiliary state variable is a path statistic, representing the local volatility. Several authors have established algorithms for implementing option models with auxiliary state variables. Most methods rely on a lattice scheme for the primary state variable, and the auxiliary state variable is represented by an array of values at each node in the tree. Examples of this approach include Ritchken, Sankarasubramanian, and Vijh (1993), Hull and White (1993a), and Barraquand and Pudet (1996).1 In most applications, the number of possible values of the auxiliary variable at a node equals the number of distinct paths to the node. In this case, it is not possible to track every possible value of the auxiliary variable, and instead a grid of representative values is established.2 Option pricing then proceeds at each point in the grid, for each node in the lattice. Typically, it is necessary to interpolate option prices at successor points when solving backwards through the tree. Li, Ritchken, and Sankarasubramanian (1995) (hereafter LRS) apply this approach to pricing Markovian interest rate models in the HJM family, where volatility varies with the level of rates, and Ritchken and Trevor (1999) apply similar ideas to price options under GARCH processes, where volatility is persistent and path dependent. For the most part, formal proofs of convergence of prices in these papers have been absent, but illustrations of convergence of prices have been provided, as the lattice (for the primary variable) and the grid (for the auxiliary variable) have been refined. In this article we focus primarily on the efficient implementation of the low dimensional Markovian HJM models that were described by Ritchken and Sankarasubramanian (1995). They identified the family of volatility structures for forward rates that permitted a HJM models to be represented by a two state Markovian system, thereby allowing American claims to be priced, This is not the only approach. Chiarella and El-Hassan (1997), for example, look at such a problem and evaluate prices as path integrals using fast Fourier transform techniques. For example, it might be possible to identify the maximum and minimum value of the auxiliary state variable to each node. Then this range can be split up into buckets of equal width.