View source for Ho–Lee model

In [[financial mathematics]], the '''Ho–Lee model''' is a [[short rate model]] widely used in the pricing of [[bond option]]s, [[swaptions]] and other [[interest rate derivatives]], and in modeling future [[interest rate]]s.<ref name="Veronesi"/>{{rp|381}}  It was developed in 1986 by [[Thomas Ho (finance)|Thomas Ho]] and [[Sang Bin Lee]]. It was the first [[Arbitrage#Arbitrage-free|arbitrage free]] model of interest rates. 

Under this model, the short rate follows a [[gaussian|normal process]]:
:<math>dr_t = \theta_t\, dt + \sigma\, dW_t</math>

The model can be calibrated to market data by implying the form of <math>\theta_t</math> from market prices, meaning that it can exactly return the price of bonds comprising the [[yield curve]]. This calibration, and subsequent valuation of [[bond option]]s, [[swaption]]s and other [[interest rate derivative]]s, is typically performed via a [[Binomial options pricing model|binomial]] [[Lattice model (finance)|lattice based model]]. [[Closed form]] valuations of [[bond (finance)|bonds]], and "[[Black model|Black-like]]" bond option formulae are also available.<ref>Graeme West, (2010). [http://www.finmod.co.za/ird.pdf ''Interest Rate Derivatives''], Financial Modelling Agency.</ref> 

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 model|Black–Derman–Toy]] ([[lognormal]] and mean reverting) and [[Hull–White model|Hull–White]] (mean reverting with lognormal variant available) are often preferred.<ref name="Veronesi">Pietro Veronesi (2010). ''Fixed Income Securities: Valuation, Risk, and Risk Management''. [[John Wiley & Sons|Wiley]]. ISBN 0-470-10910-6</ref>{{rp|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'''
{{reflist}}
'''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==
*[http://simonbenninga.com/wiener/ho-lee.pdf Valuation and Hedging of Interest Rates Derivatives with the Ho-Lee Model], Markus Leippold and Zvi Wiener, [[Wharton School]] 
*[http://www.columbia.edu/~mh2078/lattice_models.pdf Term Structure Lattice Models], Martin Haugh, [[Columbia University]]                                
'''Online tools'''
*[http://www.thomasho.com/mainpages/?download=&act=model&file=274 Binomial Tree – Excel implementation], thomasho.com
*[http://www.lombok.demon.co.uk/financial/HoLee.html Binomial Tree – Java Applet], Dr. S.H. Man 

{{Bond market}}
{{Stochastic processes}}

[[Category:Finance theories]]
[[Category:Mathematical finance]]
[[Category:Fixed income analysis]]
[[Category:Short-rate models]]


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