Ho–Lee model

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In financial mathematics, the Ho–Lee model is a short rate model widely used in the pricing of bond options, swaptions and other interest rate derivatives, and in modeling future interest rates.^[1]^:381 It was developed in 1986 by Thomas Ho and Sang Bin Lee. It was the first arbitrage free model of interest rates.

Under this model, the short rate follows a normal process:

dr_t = \theta_t\, dt + \sigma\, dW_t

The model can be calibrated to market data by implying the form of $\theta_t$ from market prices, meaning that it can exactly return the price of bonds comprising the yield curve. This calibration, and subsequent valuation of bond options, swaptions and other interest rate derivatives, is typically performed via a binomial lattice based model. Closed form valuations of bonds, and "Black-like" bond option formulae are also available.^[2]

As the model generates a symmetric ("bell shaped") distribution of rates in the future, negative rates are possible. Further, it does not incorporate mean reversion. For both of these reasons, models such as Black–Derman–Toy (lognormal and mean reverting) and Hull–White (mean reverting with lognormal variant available) are often preferred.^[1]^:385 The Kalotay–Williams–Fabozzi model is a lognormal analogue to the Ho–Lee model, although is less widely used than the latter two.

References

Notes

↑ ^1.0 ^1.1 Pietro Veronesi (2010). Fixed Income Securities: Valuation, Risk, and Risk Management. Wiley. ISBN 0-470-10910-6
↑ Graeme West, (2010). Interest Rate Derivatives, Financial Modelling Agency.

Primary references

T.S.Y. Ho, S.B. Lee, Term structure movements and pricing interest rate contingent claims, Journal of Finance 41, 1986. doi:10.2307/2328161
John C. Hull, Options, futures, and other derivatives, 5th edition, Prentice Hall, ISBN 0-13-009056-5

External links

Valuation and Hedging of Interest Rates Derivatives with the Ho-Lee Model, Markus Leippold and Zvi Wiener, Wharton School
Term Structure Lattice Models, Martin Haugh, Columbia University

Online tools

Binomial Tree – Excel implementation, thomasho.com
Binomial Tree – Java Applet, Dr. S.H. Man

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