Short-rate model

Overview

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

{\displaystyle r_{t}\,}

.

The short rate

Under a short rate model, the stochastic state variable is taken to be the instantaneous spot rate. The short rate,

r

t

{\displaystyle r_{t}\,}

, then, is the (continuously compounded, annualized) interest rate at which an entity can borrow money for an infinitesimally short period of time from time

t

{\displaystyle 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

{\displaystyle r_{t}\,}

as a stochastic process under a risk-neutral measure

Q

{\displaystyle Q}

, then the price at time

t

{\displaystyle t}

of a zero-coupon bond maturing at time

T

{\displaystyle T}

Particular short-rate models

Throughout this section

W

t

{\displaystyle W_{t}\,}

represents a standard Brownian motion under a risk-neutral probability measure and

d

W

t

{\displaystyle dW_{t}\,}

its differential. Where the model is lognormal, a variable

X

t

{\displaystyle X_{t}}

is assumed to follow an Ornstein–Uhlenbeck process and

r

t

{\displaystyle r_{t}\,}

is assumed to follow

r

t

One-factor short-rate models

Following are the one-factor models, where a single stochastic factor – the short rate – determines the future evolution of all interest rates. Other than Rendleman–Bartter and Ho–Lee, which do not capture the mean reversion of interest rates, these models can be thought of as specific cases of Ornstein–Uhlenbeck processes. The Vasicek, Rendleman–Bartter and CIR models are endogenous models and have only a finite number of free parameters and so it is not possible to specify these parameter values in such a way that the model coincides with a few observed market prices ("calibration") of zero coupon bonds or linear products such as forward rate agreements or swaps, typically, or a best fit is done to these linear products to find the endogenous short rate models parameters that are closest to the market prices. This does not allow for fitting options like caps, floors and swaptions as the parameters have been used to fit linear instruments instead. This problem is overcome by allowing the parameters to vary deterministically with time, or by adding a deterministic shift to the endogenous model. In this way, exogenous models such as Ho-Lee and subsequent models, can be calibrated to market data, meaning that these can exactly return the price of bonds comprising the yield curve, and the remaining parameters can be used for options calibration. The implementation is usually via a (binomial) short rate tree or simulation; see Lattice model (finance) § Interest rate derivatives and Monte Carlo methods for option pricing, although some short rate models have closed form solutions for zero coupon bonds, and even caps or floors, easing the calibration task considerably. We list the following endogenous models first.

Merton's model (1973) explains the short rate as

r

t

=

r

0

+ a t + σ

W

t

∗

{\displaystyle r_{t}=r_{0}+at+\sigma W_{t}^{*}}

: where

W

t

∗

{\displaystyle W_{t}^{*}}

is a one-dimensional Brownian motion under the spot martingale measure. In this approach, the short rate follows an arithmetic Brownian motion. The Vasicek model (1977) models the short rate as

d

r

Multi-factor short-rate models

Besides the above one-factor models, there are also multi-factor models of the short rate, among them the best known are the Longstaff and Schwartz two factor model and the Chen three factor model (also called "stochastic mean and stochastic volatility model"). Note that for the purposes of risk management, "to create realistic interest rate simulations", these multi-factor short-rate models are sometimes preferred over One-factor models, as they produce scenarios which are, in general, better "consistent with actual yield curve movements".

The Longstaff–Schwartz model (1992) supposes the short rate dynamics are given by

d

X

t

= (

a

t

− b

X

t

)

d t +

X

t

c

t

d

W

1 t

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 model and market models, is often preferred for models of higher dimension. Models based on Fischer Black's shadow rate are used when interest rates approach the zero lower bound.

See also

Fixed-income attribution

Further reading

Martin Baxter & Andrew Rennie (1996). Financial Calculus. Cambridge University Press. ISBN 978-0-521-55289-9. Damiano Brigo; Fabio Mercurio (2001). Interest Rate Models – Theory and Practice with Smile, Inflation and Credit (2nd ed. 2006 ed.). Springer Verlag. ISBN 978-3-540-22149-4. Gerald Buetow & James Sochacki (2001). Term-Structure Models Using Binomial Trees. The Research Foundation of AIMR (CFA Institute). ISBN 978-0-943205-53-3. Andrew J.G. Cairns (2004). Interest Rate Models – An Introduction. Princeton University Press. ISBN 978-0-691-11894-9. Andrew J.G. Cairns (2004). Interest-Rate Models; entry in Encyclopaedia of Actuarial Science. John Wiley and Sons. 2004. ISBN 978-0-470-84676-6. K. C. Chan, G. Andrew Karolyi, Francis Longstaff, and Anthony Sanders (1992). An Empirical Comparison of Alternative Models of the Short-Term Interest Rate (PDF). The Journal of Finance, Vol. XLVII, No. 3 July 1992.{{cite book}}: CS1 maint: multiple names: authors list (link) Lin Chen (1996). Interest Rate Dynamics, Derivatives Pricing, and Risk Management. Springer. ISBN 978-3-540-60814-1. Rajna Gibson, François-Serge Lhabitant and Denis Talay (1999). Modeling the Term Structure of Interest Rates: An overview. The Journal of Risk, 1(3): 37–62, 1999. Lane Hughston (2003). The Past, Present and Future of Term Structure Modelling; entry in Peter Field (2003). Modern Risk Management. Risk Books. ISBN 978-1-906348-30-4. Jessica James & Nick Webber (2000). Interest Rate Modelling. Wiley Finance. ISBN 978-0-471-97523-6. Robert Jarrow (2002). Modelling Fixed Income Securities and Interest Rate Options (2nd ed.). Stanford Economics and Finance. ISBN 978-0-8047-4438-6. Robert Jarrow (2009). "The Term Structure of Interest Rates". Annual Review of Financial Economics. 1 (1): 69–96. doi:10.1146/annurev.financial.050808.114513. F.C. Park (2004). "Implementing Interest Rate Models: a Practical Guide" (PDF). CMPR Research Publication. Archived from the original (PDF) on 2010-08-16. Riccardo Rebonato (2002). Modern Pricing of Interest-Rate Derivatives. Princeton University Press. ISBN 978-0-691-08973-7. Riccardo Rebonato (2003). "Term-Structure Models: a Review" (PDF). Royal Bank of Scotland Quantitative Research Centre Working Paper.