Ho–Lee model: Difference between revisions

From Wikipedia, the free encyclopedia

Content deleted Content added

Inline

Revision as of 09:40, 7 February 2013

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

^ ^a ^b 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

v t e Bond market
Bond Debenture Fixed income
Types of bonds by issuer	Agency bond Corporate bond Senior debt Subordinated debt Distressed debt Government bond Infrastructure bond Municipal bond Global bond
Types of bonds by payout	Accrual bond Auction rate security Commercial paper Consol Convertible bond Contingent Reverse Exchangeable bond Extendible bond Fixed rate bond Floating rate note Inverse Inflation-indexed High-yield debt Lottery bond Perpetual bond Zero-coupon bond
Bond options	Callable bond Convertible bond Embedded option Exchangeable bond Extendible bond Puttable bond
Bond valuation	Clean price Convexity Coupon Credit spread Current yield Dirty price Duration I-spread Mortgage yield Nominal yield Option-adjusted spread Risk-free bond Weighted-average life Yield curve Yield spread Yield to maturity Z-spread
Securitized products	Asset-backed security Collateralized debt obligation Collateralized mortgage obligation Commercial mortgage-backed security Mortgage-backed security
Institutions	Commercial Mortgage Securities Association (CMSA) International Capital Market Association (ICMA) Securities Industry and Financial Markets Association (SIFMA)

v t e Stochastic processes
Discrete time	Bernoulli process Branching process Chinese restaurant process Galton–Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding Biased Maximal entropy
Continuous time	Additive process Bessel process Birth–death process pure birth Brownian motion Bridge Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Dyson Brownian motion Empirical process Feller process Fleming–Viot process Gamma process Geometric process Hawkes process Hunt process Interacting particle systems Itô diffusion Itô process Jump diffusion Jump process Lévy process Local time Markov additive process McKean–Vlasov process Ornstein–Uhlenbeck process Poisson process Compound Non-homogeneous Schramm–Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage
Both	Branching process Gaussian process Hidden Markov model (HMM) Markov process Martingale Differences Local Sub- Super- Random dynamical system Regenerative process Renewal process Stochastic chains with memory of variable length White noise
Fields and other	Dirichlet process Gaussian random field Gibbs measure Hopfield model Ising model Potts model Boolean network Markov random field Percolation Pitman–Yor process Point process Cox Poisson Random field Random graph
Time series models	Autoregressive conditional heteroskedasticity (ARCH) model Autoregressive integrated moving average (ARIMA) model Autoregressive (AR) model Autoregressive–moving-average (ARMA) model Generalized autoregressive conditional heteroskedasticity (GARCH) model Moving-average (MA) model
Financial models	Binomial options pricing model Black–Derman–Toy Black–Karasinski Black–Scholes Chan–Karolyi–Longstaff–Sanders (CKLS) Chen Constant elasticity of variance (CEV) Cox–Ingersoll–Ross (CIR) Garman–Kohlhagen Heath–Jarrow–Morton (HJM) Heston Ho–Lee Hull–White Korn-Kreer-Lenssen LIBOR market Rendleman–Bartter SABR volatility Vašíček Wilkie
Actuarial models	Bühlmann Cramér–Lundberg Risk process Sparre–Anderson
Queueing models	Bulk Fluid Generalized queueing network M/G/1 M/M/1 M/M/c
Properties	Càdlàg paths Continuous Continuous paths Ergodic Exchangeable Feller-continuous Gauss–Markov Markov Mixing Piecewise-deterministic Predictable Progressively measurable Self-similar Stationary Time-reversible
Limit theorems	Central limit theorem Donsker's theorem Doob's martingale convergence theorems Ergodic theorem Fisher–Tippett–Gnedenko theorem Large deviation principle Law of large numbers (weak/strong) Law of the iterated logarithm Maximal ergodic theorem Sanov's theorem Zero–one laws (Blumenthal, Borel–Cantelli, Engelbert–Schmidt, Hewitt–Savage, Kolmogorov, Lévy)
Inequalities	Burkholder–Davis–Gundy Doob's martingale Doob's upcrossing Kunita–Watanabe Marcinkiewicz–Zygmund
Tools	Cameron–Martin formula Convergence of random variables Doléans-Dade exponential Doob decomposition theorem Doob–Meyer decomposition theorem Doob's optional stopping theorem Dynkin's formula Feynman–Kac formula Filtration Girsanov theorem Infinitesimal generator Itô integral Itô's lemma Karhunen–Loève theorem Kolmogorov continuity theorem Kolmogorov extension theorem Lévy–Prokhorov metric Malliavin calculus Martingale representation theorem Optional stopping theorem Prokhorov's theorem Quadratic variation Reflection principle Skorokhod integral Skorokhod's representation theorem Skorokhod space Snell envelope Stochastic differential equation Tanaka Stopping time Stratonovich integral Uniform integrability Usual hypotheses Wiener space Classical Abstract
Disciplines	Actuarial mathematics Control theory Econometrics Ergodic theory Extreme value theory (EVT) Large deviations theory Mathematical finance Mathematical statistics Probability theory Queueing theory Renewal theory Ruin theory Signal processing Statistics Stochastic analysis Time series analysis Machine learning
List of topics Category

This economics-related article is a stub. You can help Wikipedia by expanding it.

Retrieved from "https://en.wikipedia.org/enwiki/w/index.php?title=Ho–Lee_model&oldid=537028036"

Categories:

Hidden category:

All stub articles

@@ Line 6: / Line 6: @@
 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.
+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.
-{{economics-stub}}
 ==References==
 '''Notes'''
@@ Line 14: / Line 14: @@
 * 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]]
@@ Line 28: / Line 29: @@
 [[Category:Fixed income analysis]]
 [[Category:Short-rate models]]
+{{economics-stub}}
 [[ja:ホー・リー・モデル]]