Lattice Methods for No-Arbitrage Pricing of Interest Rate Securities

Abstract

Closed-form solutions for derivatives pricing problems yield exact prices nearly instantaneously. But for only a handful of actual derivatives are payoff structures simple enough and the underlying returns processes tractable enough for closed-form valuation. Finite difference approximation of the fundamental partial differential equation is common, although achieving convergence can sometimes be difficult. The binomial model has been a workhorse of derivatives valuation, but it is quite limited, so the trinomial–the equivalent of a backward-looking “explicit” finite-difference scheme–is more common for many practical situations, especially those involving interest rates. One reason is that in order to match market prices for bonds and interest rate derivatives, the approximating lattice must be calibrated to both the observed term structure and the volatility surface. These requirements can impose major computational problems for numerical approximation, such that accurate pricing becomes very time consuming. Daglish proposes a new approach based on approximating the solution to the PDE in the forward direction. The result is a valuation procedure that is as accurate as the existing techniques for a given size of time step, but with a much faster execution time.