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Binomial, trinomial, and other lattice-based option pricing procedures are extremely useful in dealing with early exercise because solution by backward induction is possible. As one rolls back through the tree, the option value at each node depends on the current price for the underlying and the paths that might be followed going forward. But for many kinds of derivatives, the value at a given point in time depends on both the current asset price and the price path that it has followed up to the current point. Derivatives on interest rates in the Heath-Jarrow-Morton (HJM) model are one example. Another is option pricing when the underlying volatility varies over time according to a GARCH model. The most successful technique for adapting a lattice model to a path-dependent process has been to carry along a set of auxiliary variables that summarize (approximately) the necessary path information at each node. These techniques work, but they are typically very time consuming. In this article, Lin and Ritchken present a way to greatly increase the efficiency of this procedure. Some auxiliary path information must still be carried through the tree, but much less is needed. They demonstrate the sharp increase in performance this makes possible in an HJM application and for an underlying that follows a GARCH process.

On Pricing Derivatives in the Presence of Auxiliary State Variables