In
financial mathematics
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets.
In general, there exist two separate branches of finance that require ...
, 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 rate
An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, ...
s.
It was developed in 1986 by Thomas Ho and Sang Bin Lee.
Under this model, the short rate follows a
normal process:
:
The model can be calibrated to market data by implying the form of
from market prices, meaning that it can exactly return the price of bonds comprising the
yield curve
In finance, the yield curve is a graph which depicts how the yields on debt instruments - such as bonds - vary as a function of their years remaining to maturity. Typically, the graph's horizontal or x-axis is a time line of months or ye ...
. This calibration, and subsequent valuation of
bond options,
swaption
A swaption is an option granting its owner the right but not the obligation to enter into an underlying swap. Although options can be traded on a variety of swaps, the term "swaption" typically refers to options on interest rate swaps.
Types of ...
s and other
interest rate derivative
In finance, an interest rate derivative (IRD) is a derivative whose payments are determined through calculation techniques where the underlying benchmark product is an interest rate, or set of different interest rates. There are a multitude of diff ...
s, is typically performed via a
binomial lattice based model.
Closed form valuations of
bonds, and "
Black-like" bond option formulae are also available.
[Graeme West, (2010)]
''Interest Rate Derivatives''
, Financial Modelling Agency.
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.
[Pietro Veronesi (2010). ''Fixed Income Securities: Valuation, Risk, and Risk Management''. ]Wiley
Wiley may refer to:
Locations
* Wiley, Colorado, a U.S. town
* Wiley, Pleasants County, West Virginia, U.S.
* Wiley-Kaserne, a district of the city of Neu-Ulm, Germany
People
* Wiley (musician), British grime MC, rapper, and producer
* Wiley Mil ...
. The
Kalotay–Williams–Fabozzi model
A short-rate model, in the context of interest rate derivatives, is a mathematical model that describes the future evolution of interest rates by describing the future evolution of the short rate, usually written r_t \,.
The short rate
Under a sho ...
is a
lognormal analogue to the Ho–Lee model, although is less widely used than the latter two.
References
Notes
Primary references
* T.S.Y. Ho, S.B. Lee, ''Term structure movements and pricing interest rate contingent claims'', ''
Journal of Finance
''The Journal of Finance'' is a peer-reviewed academic journal published by Wiley-Blackwell on behalf of the American Finance Association. It was established in 1946 and is considered to be one of the premier finance journals. The editor-in-chief i ...
'' 41, 1986.
* John C. Hull, ''Options, futures, and other derivatives'', 5th edition,
Prentice Hall,
External links
Valuation and Hedging of Interest Rates Derivatives with the Ho-Lee Model Markus Leippold and Zvi Wiener,
Wharton School
The Wharton School of the University of Pennsylvania ( ; also known as Wharton Business School, the Wharton School, Penn Wharton, and Wharton) is the business school of the University of Pennsylvania, a private Ivy League research university in P ...
Term Structure Lattice Models Martin Haugh,
Columbia University
Columbia University (also known as Columbia, and officially as Columbia University in the City of New York) is a private research university in New York City. Established in 1754 as King's College on the grounds of Trinity Church in Manhatt ...
Online tools
Binomial Tree – Excel implementation thomasho.com
Fixed income analysis
Short-rate models
Financial models
{{Econ-theory-stub