Lattice

Lattice methods for no-arbitrage pricing of interest rate

securities

Toby Daglish∗

May 20, 2010

Abstract

We explore calibration of single factor no-arbitrage short rate models to yield and volatility

information. We note that the calculation of Arrow-Debreu prices for interest rate securities is

analogous to solving the Kolmogorov Forward Equation. This insight allows us to implement

implicit methods, which exhibit more rapid convergence than explicit methods. We develop an

algorithm for calibrating a model to match both yield and volatility curves, which is general

across single factor short rate models, and also across finite difference techniques. Numerical

examples confirm that our approach vastly improves computation times for derivative pricing.

The use of short-rate no-arbitrage models for pricing interest rate securities has a strong popularity

amongst academics and practitioners. Being able to calibrate a model to zero coupon bond prices

∗

Senior Lecturer, Victoria University of Wellington. PO Box 600, Wellington, New Zealand 6140. E-mail:

toby.daglish vuw.ac.nz. Phone: 64-4-4635451. The author thanks Graeme Guthrie, Alan White, Mairead de Roiste,

Bethanna Jackson, Vladimir Petkov and Leigh Roberts for their comments, as well as participants at the New Zealand

Finance Colloquium 2010.

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ensures that pricing of derivative securities is consistent with observed interest rates, rather than

being based on some best fit, as would be the case for equilibrium interest rate models.

A number of such no-arbitrage models exist. The best known of these models are Ho and Lee

[1986], Black, Derman, and Toy [1990], Hull and White [1990] and Black and Karasinski [1991].

While closed form solutions are available in some cases, for most implementations, the models are

set up as binomial or trinomial trees, which are calibrated in order to match information on the yield

curve. Hull and White [1993] show that their trinomial tree-building...