Pricing Interest-Rate Derivatives with Piecewise Multilinear Interpolations and Transition Parameters

Abstract

Interest-rate derivatives like swaptions are challenging to price because they depend on the term structure of interest rates, which potentially has many degrees of freedom, requiring modeling numerous underlying stochastic factors (often at least three for the level and more if stochastic volatility is allowed). In this article, the authors show how the valuation problem can be streamlined in an affine framework by, effectively, digitizing the derivative product’s terminal density. They assume standard affine dynamics for the state vector X, then divide the state space at option expiration into discrete regions and approximate the payoff in each one, as a constant in the simplest case. They then compute the transition probabilities to move from the initial price to each of the terminal regions, using truncated conditional Laplace transforms. With the probabilities and the set of discrete payoffs, the expected value of the payoff to the derivative product under the forward measure can be computed, which is then discounted to compute the initial value. Examples with two- and three-factor affine term structures show that the procedure is accurate and relatively fast.