Short-rate 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 ${\displaystyle r_{t}\,}$.

The short rate

Under a short rate model, the stochastic state variable is taken to be the instantaneous spot rate.[1] The short rate, ${\displaystyle r_{t}\,}$, then, is the (continuously compounded, annualized) interest rate at which an entity can borrow money for an infinitesimally short period of time from time ${\displaystyle t}$. 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 ${\displaystyle r_{t}\,}$ as a stochastic process under a risk-neutral measure ${\displaystyle Q}$ then the price at time ${\displaystyle t}$ of a zero-coupon bond maturing at time ${\displaystyle T}$ with a payoff of 1 is given by

${\displaystyle P(t,T)=\mathbb {E} ^{Q}\left[\left.\exp {\left(-\int _{t}^{T}r_{s}\,ds\right)}\right|{\mathcal {F}}_{t}\right]}$

where ${\displaystyle {\mathcal {F}}}$ is the natural filtration for the process. The interest rates implied by the zero coupon bonds form a yield curve or more precisely, a zero curve. 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

${\displaystyle f(t,T)=-{\frac {\partial }{\partial T}}\ln(P(t,T)).}$

Particular short-rate models

Throughout this section ${\displaystyle W_{t}\,}$ represents a standard Brownian motion under a risk-neutral probability measure and ${\displaystyle dW_{t}\,}$ its differentialTemplate:Disambiguation needed. Where the model is lognormal, a variable ${\displaystyle X_{t}\,}$, is assumed to follow an Ornstein–Uhlenbeck process and ${\displaystyle r_{t}\,}$ is assumed to follow ${\displaystyle r_{t}=\exp {X_{t}}\,}$.

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 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 observed market prices ("calibration"). This problem is overcome by allowing the parameters to vary deterministically with time.[2][3] In this way, 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. Here, the implementation is usually via a (binomial) short rate tree;[4] see Lattice model (finance) #Interest rate derivatives.

1. Merton's model (1973) explains the short rate as ${\displaystyle r_{t}=r_{0}+at+\sigma W_{t}^{*}}$: where ${\displaystyle W_{t}^{*}}$ is a one-dimensional Brownian motion under the spot martingale measure.[5]
2. The Vasicek model (1977) models the short rate as ${\displaystyle dr_{t}=(\theta -\alpha r_{t})\,dt+\sigma \,dW_{t}}$; it is often written ${\displaystyle dr_{t}=a(b-r_{t})\,dt+\sigma \,dW_{t}}$.[6]
3. The Rendleman–Bartter model (1980) explains the short rate as ${\displaystyle dr_{t}=\theta r_{t}\,dt+\sigma r_{t}\,dW_{t}}$.[7]
4. The Cox–Ingersoll–Ross model (1985) supposes ${\displaystyle dr_{t}=(\theta -\alpha r_{t})\,dt+{\sqrt {r_{t}}}\,\sigma \,dW_{t}}$, it is often written ${\displaystyle dr_{t}=a(b-r_{t})\,dt+{\sqrt {r_{t}}}\,\sigma \,dW_{t}}$. The ${\displaystyle \sigma {\sqrt {r_{t}}}}$ factor precludes (generally) the possibility of negative interest rates.[8]
5. The Ho–Lee model (1986) models the short rate as ${\displaystyle dr_{t}=\theta _{t}\,dt+\sigma \,dW_{t}}$.[9]
6. The Hull–White model (1990)—also called the extended Vasicek model—posits ${\displaystyle dr_{t}=(\theta _{t}-\alpha r_{t})\,dt+\sigma _{t}\,dW_{t}}$. In many presentations one or more of the parameters ${\displaystyle \theta ,\alpha }$ and ${\displaystyle \sigma }$ are not time-dependent. The model may also be applied as lognormal. Lattice-based implementation is usually trinomial.[10][11]
7. The Black–Derman–Toy model (1990) has ${\displaystyle d\ln(r)=[\theta _{t}+{\frac {\sigma '_{t}}{\sigma _{t}}}\ln(r)]dt+\sigma _{t}\,dW_{t}}$ for time-dependent short rate volatility and ${\displaystyle d\ln(r)=\theta _{t}\,dt+\sigma \,dW_{t}}$ otherwise; the model is lognormal.[12]
8. The Black–Karasinski model (1991), which is lognormal, has ${\displaystyle d\ln(r)=[\theta _{t}-\phi _{t}\ln(r)]\,dt+\sigma _{t}\,dW_{t}}$.[13] The model may be seen as the lognormal application of Hull–White;[14] its lattice-based implementation is similarly trinomial (binomial requiring varying time-steps).[4]
9. The Kalotay–Williams–Fabozzi model (1993) has the short rate as ${\displaystyle d\ln(r_{t})=\theta _{t}\,dt+\sigma \,dW_{t}}$, a lognormal analogue to the Ho–Lee model, and a special case of the Black–Derman–Toy model.[15]

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".[16]

1. The Longstaff–Schwartz model (1992) supposes the short rate dynamics are given by: ${\displaystyle dX_{t}=(a_{t}-bX_{t})\,dt+{\sqrt {X_{t}}}\,c_{t}\,dW_{1t}}$, ${\displaystyle dY_{t}=(d_{t}-eY_{t})\,dt+{\sqrt {Y_{t}}}\,f_{t}\,dW_{2t}}$, where the short rate is defined as ${\displaystyle dr_{t}=(\mu X+\theta Y)dt+\sigma _{t}{\sqrt {Y}}dW_{3t}}$.[17]
2. The Chen model (1996) which has a stochastic mean and volatility of the short rate, is given by : ${\displaystyle dr_{t}=(\theta _{t}-\alpha _{t})\,dt+{\sqrt {r_{t}}}\,\sigma _{t}\,dW_{t}}$, ${\displaystyle d\alpha _{t}=(\zeta _{t}-\alpha _{t})\,dt+{\sqrt {\alpha _{t}}}\,\sigma _{t}\,dW_{t}}$, ${\displaystyle d\sigma _{t}=(\beta _{t}-\sigma _{t})\,dt+{\sqrt {\sigma _{t}}}\,\eta _{t}\,dW_{t}}$.[18]

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.

References

1. Short rate models, Prof. Andrew Lesniewski, NYU
2. Continuous-Time Short Rate Models, Prof Martin Haugh, Columbia University
3. Binomial Term Structure Models, Mathematica in Education and Research, Vol. 7 No. 3 1998. Simon Benninga and Zvi Wiener.
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13. Short Rate Models, Professor Ser-Huang Poon, Manchester Business School
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15. Pitfalls in Asset and Liability Management: One Factor Term Structure Models, Dr. Donald R. van Deventer, Kamakura Corporation
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