A General Computational Method for Calibration Based on Di(cid:11)erential Trees

An ordinary option pricing problem begins with a set of input parameters and computes the option value that precludes profitable arbitrage. The inverse problem, often known as model calibration, is also familiar, as in computing the implied volatility from an option's observed market price. Calibration is substantially more complicated than pricing an option once, however, since it normally requires solving a non-linear equation by numerical search. Standard efficient search techniques, like Newton-Raphson, typically involve derivatives of the pricing function with respect to the parameters that are being calibrated. These are often approximated using finite differences but that multiplies the number of times the valuation equation must be solved. This article presents a simple calibration technique for lattice models, based on computing both the option value and the necessary derivatives in the same pass through the tree. Examples of its use in calibrating an interest rate model to the initial term structure, in computing an option-adjusted spread on a callable bond, and in calculating the implied volatility for an American option all show it to be highly efficient.