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 $r_t \,$.

The short rate

Under a short rate model, the stochastic state variable is taken to be the instantaneous forward rate. The short rate, $r_t \,$, 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. 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

$P(t,T) = \mathbb{E}\left[\left. \exp{\left(-\int_t^T r_s\, ds\right) } \right| \mathcal{F}_t \right]$

where $\mathcal{F}$ 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

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

Particular short-rate models

Throughout this section $W_t\,$ represents a standard Brownian motion under a risk-neutral probability measure and $dW_t\,$ its differential. Where the model is lognormal, a variable $X_t \,$, is assumed to follow an Ornstein–Uhlenbeck process and $r_t \,$ is assumed to follow $r_t = \exp{X_t}\,$.

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. 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 tree (lattice).

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

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".

1. The Longstaff–Schwartz model (1992) supposes the short rate dynamics are given by: $dX_t = (a_t-b X_t)\,dt + \sqrt{X_t}\,c_t\, dW_{1t}$, $d Y_t = (d_t-e Y_t)\,dt + \sqrt{Y_t}\,f_t\, dW_{2t}$, where the short rate is defined as $dr_t = (\mu X + \theta Y)dt + \sigma_t \sqrt{Y} dW_{3t}$.
2. The Chen model (1996) which has a stochastic mean and volatility of the short rate, is given by : $dr_t = (\theta_t-\alpha_t)\,dt + \sqrt{r_t}\,\sigma_t\, dW_t$, $d \alpha_t = (\zeta_t-\alpha_t)\,dt + \sqrt{\alpha_t}\,\sigma_t\, dW_t$, $d \sigma_t = (\beta_t-\sigma_t)\,dt + \sqrt{\sigma_t}\,\eta_t\, dW_t$.

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.

References

Notes

Primary references

• Black, F.; Derman, E. and Toy, W. (1990). "A One-Factor Model of Interest Rates and Its Application to Treasury Bond Options". Financial Analysts Journal: 24–32. 
• Black, F.; Karasinski, P. (1991). "Bond and Option pricing when Short rates are Lognormal". Financial Analysts Journal: 52–59. 
• Lin Chen (1996). "Stochastic Mean and Stochastic Volatility — A Three-Factor Model of the Term Structure of Interest Rates and Its Application to the Pricing of Interest Rate Derivatives". Financial Markets, Institutions, and Instruments 5: 1–88. 
• Cox, J.C., J.E. Ingersoll and S.A. Ross (1985). "A Theory of the Term Structure of Interest Rates". Econometrica 53: 385–407. doi:10.2307/1911242. 
• T.S.Y. Ho and S.B. Lee (1986). "Term structure movements and pricing interest rate contingent claims". Journal of Finance 41. doi:10.2307/2328161. 
• John Hull and Alan White (1990). "Pricing interest-rate derivative securities". Review of Financial Studies 3 (4): 573–592. 
• Kalotay, Andrew J.; Williams, George O.; Fabozzi, Frank J. (1993). "A Model for Valuing Bonds and Embedded Options". Financial Analysts Journal (CFA Institute Publications) 49 (3): 35–46. doi:10.2469/faj.v49.n3.35. 
• Longstaff, F.A. and Schwartz, E.S. (1992). "Interest Rate Volatility and the Term Structure: A Two-Factor General Equilibrium Model". Journal of Finance 47 (4): 1259–82. 
• Merton, Robert C. (1973). "Theory of Rational Option Pricing". Bell Journal of Economics and Management Science 4 (1): 141–183. doi:10.2307/3003143. 
• Rendleman, R. and B. Bartter (1980). "The Pricing of Options on Debt Securities". Journal of Financial and Quantitative Analysis 15: 11–24. doi:10.2307/2979016. 
• Vasicek, Oldrich (1977). "An Equilibrium Characterisation of the Term Structure". Journal of Financial Economics 5 (2): 177–188. doi:10.1016/0304-405X(77)90016-2. 

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