- Short rate model
In the context of
interest ratederivatives, a short rate model is a mathematical modelthat describes the future evolution of interest rates by describing the future evolution of the short rate.
The short rate
The short rate, usually written "r""t" is the (annualized) interest rate at which an entity can borrow money for an infinitesimally short period of time from time "t". Specifying the current short rate does not specify the entire
yield curve. However no-arbitrage arguments show that, under some fairly relaxed technical conditions, if we model the evolution of "r""t" as a stochastic process under a risk-neutral measure"Q" then the price at time "t" of a zero-coupon bond maturing at time "T" is given by
where is the natural filtration for the process. Thus specifying a model for the short rate specifies future bond prices. This means that instantaneous forward rates are also specified by the usual formula
And its third equivalent, the yields are given as well.
Particular short-rate models
Throughout this section represents a standard
Brownian motionand its differential.
Rendleman-Bartter modelmodels the short rate as
Vasicek modelmodels the short rate as
Ho-Lee modelmodels the short rate as
Hull-White model(also called the extended Vasicek model sometimes) posits . In many presentations one or more of the parameters and are not time-dependent. The process is called an Ornstein-Uhlenbeck process.
Black-Karasinski modela variable "X""t" is assumed to follow an Ornstein-Uhlenbeck process and "r""t" is assumed to follow .
Besides the above one-factor models, there are also multi-factor models of the short rate, among them the best known are Longstaff and Schwartz two factor model and Chen three factor model (also called "stochastic mean and stochastic volatility model"):
Longstaff-Schwartz modelsupposes the short rate dynamics is given by the following two equations: , .
Chen modelmodels the short rate, also called stochastic mean and stochastic volatility of the short rate, is given by : , , .
Other interest rate models
The other major framework for interest rate modelling is the
Heath-Jarrow-Morton framework(HJM). Unlike the short rate models described above, this class of models is generally non-Markovian. This makes general HJM models computationally intractable for most purposes. The great advantage of HJM models is that they give an analytical description of the entire yield curve, rather than just the short rate. For some purposes (e.g., valuation of mortgage backed securities), this can be a big simplification. The Cox-Ingersoll-Ross and Hull-White models in one or more dimensions can both be straightforwardly expressed in the HJM framework. Other short rate models do not have any simple dual HJM representation.
The HJM framework with multiple sources of randomness, including as it does the
Brace-Gatarek-Musiela modeland market models, is often preferred for models of higher dimension.
* cite book | author = Jessica James and Nick Webber | year = 2000 | title = Interest Rate Modelling
publisher = Wiley Finance | id = ISBN 0-471-97523-0
*cite book | title = Interest Rate Models - An Introduction | author = Andrew J.G. Cairns | publisher =
Princeton University Press| year = 2004 | id = ISBN 0-691-11894-9
Wikimedia Foundation. 2010.
Look at other dictionaries:
Short rate — may refer to: Short rate (old short rate) cancellation (insurance) Penalty method of calculating return premium of an insurance policy Short rate (90% pro rata) cancellation (insurance) Penalty method of calculating return premium of an insurance … Wikipedia
Short-term effects of alcohol — on the human body can take many forms. The drug alcohol, specifically ethanol, is a central nervous system depressant with a range of side effects. The amount and circumstances of consumption play a large part in determining the extent of… … Wikipedia
short story — short story, adj. a piece of prose fiction, usually under 10,000 words. [1885 90] * * * Brief fictional prose narrative. It usually presents a single significant episode or scene involving a limited number of characters. The form encourages… … Universalium
Short (finance) — Schematic representation of short selling in two steps. The short seller borrows shares and immediately sells them. He then waits, hoping for the stock price to decrease, when the seller can profit by purchasing the shares to return to the lender … Wikipedia
Short Scion — infobox Aircraft name = S.16 Scion/Scion II type = Light transport landplane/floatplane manufacturer = Short Brothers Pobjoy Airmotors Ltd. caption = designer = Arthur Gouge first flight = 18 August 1933 introduced = retired = produced = number… … Wikipedia
Rate of return — In finance, rate of return (ROR), also known as return on investment (ROI), rate of profit or sometimes just return, is the ratio of money gained or lost (whether realized or unrealized) on an investment relative to the amount of money invested.… … Wikipedia
Model aircraft — A die cast Boeing 747 400 model. Model aircraft are flying or non flying models of existing or imaginary aircraft using a variety of materials including plastic, diecast metal, polystyrene, balsa wood, foam and fibreglass. Flying designs range… … Wikipedia
Model rocket — A typical model rocket during launch A model rocket is a small rocket that is commonly advertised as being able to be launched by anybody, to, in general, low altitudes (usually to around 100–500 m (300–1500 ft) for a 30 g (1 oz.) model) and … Wikipedia
Model (art) — For non artistic human models, see Model (person). Art model posing in a French painting school Art models are models who pose for photographers, painters, sculptors, and other artists as part of their work of art. Art models who pose in the nude … Wikipedia
Short 330 — Infobox Aircraft name= Short 330 (SD3 30) type = Transport airplane manufacturer=Short Brothers caption=A Short 330 of Mississippi Valley Airlines at Minneapolis Saint Paul International Airport in 1985 first flight= 22 August 1974… … Wikipedia