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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^[2] and Sang Bin Lee.^[3]

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.^[4]

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

[edit]

Notes

^ ^a ^b Pietro Veronesi (2010). Fixed Income Securities: Valuation, Risk, and Risk Management. Wiley. ISBN 0-470-10910-6
^ Thomas S.Y. Ho Ph.D, thcdecisions.com
^ Sang Bin Lee, shanghai.nyu.edu
^ Graeme West, (2010). Interest Rate Derivatives Archived 2012-04-17 at the Wayback Machine, 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

[edit]

Valuation and Hedging of Interest Rates Derivatives with the Ho-Lee Model, Markus Leippold and Zvi Wiener, Wharton School
Term Structure Lattice Models Archived 2012-01-23 at the Wayback Machine, Martin Haugh, Columbia University

Online tools

Binomial Tree – Excel implementation^{[permanent dead link‍]}, thomasho.com

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+{{Short description|Short-rate model in financial mathematics}}
-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]].
+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<ref>[https://www.thcdecisions.com/thc/team.asp Thomas S.Y. Ho Ph.D], thcdecisions.com</ref> and Sang Bin Lee.<ref>[https://shanghai.nyu.edu/academics/faculty/directory/sang-bin-lee Sang Bin Lee], shanghai.nyu.edu</ref>
 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 expression|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>
+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 expression|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''] {{Webarchive|url=https://web.archive.org/web/20120417044831/http://www.finmod.co.za/ird.pdf |date=2012-04-17 }}, 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 (finance)|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.
@@ Line 17: / Line 19: @@
 ==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]]
+*[http://www.columbia.edu/~mh2078/lattice_models.pdf Term Structure Lattice Models] {{Webarchive|url=https://web.archive.org/web/20120123193422/http://www.columbia.edu/~mh2078/lattice_models.pdf |date=2012-01-23 }}, Martin Haugh, [[Columbia University]]
 '''Online tools'''
-*[http://www.thomasho.com/mainpages/?download=&act=model&file=274 Binomial Tree – Excel implementation], thomasho.com
+*[http://www.thomasho.com/mainpages/?download=&act=model&file=274 Binomial Tree – Excel implementation]{{Dead link|date=August 2024 |bot=InternetArchiveBot |fix-attempted=yes }}, thomasho.com
 {{Bond market}}