Interest Rate Models

Term structure models attempt to capture the statistical properties of interest rate movements and provide us with quantitatively precise descriptions of how interest rates will change.

Equilibrium Term Structure Models

Equilibrium term structure models attempt to describe changes in the term structure through the use of fundamental economic variables that drive interest rates. While equilibrium term structure models can rely on multiple factors, the two famous models discussed in the curriculum, the Cox Ingersoll Ross (CIR) model and the Vasicek model, are both single factor models. The single factor in the CIR and Vasicek models is the short term interest rate.

The Cox Ingersoll Ross Model

The Cox Ingersoll Ross (CIR) model is based on the idea that interest rate movements are driven by individuals choosing between consumption today versus investing and consuming at a later time.

Mathematically, the CIR model consists of two components: the first part of this expression is a drift term, while the second part is the random component:

The a(b − r)dt term forces the interest rate to mean revert toward its long run value (b) at a speed determined by the mean reversion parameter (a).

Under the CIR model, volatility increases with the interest rate, as can be seen in the

The Vasicek Model

Like the CIR model, the Vasicek model suggests that interest rates are mean reverting to some long run value.

The Vasicek model is expressed as:

The difference from the CIR model that you will notice is that no interest rate (r) term appears in the second term σdz, meaning that volatility in this model does not increase as the level of interest rates increases (i.e., volatility is constant).

The main disadvantage of the Vasicek model is that the model does not force interest rates to be nonnegative.

Arbitrage Free Models

Arbitrage free models of the term structure of interest rates begin with the assumption that bonds trading in the market are correctly priced, and the model is calibrated to value such bonds consistent with their market price (hence the “arbitrage free” label). These models do not try to justify the current yield curve; rather, they take this curve as given.

The ability to calibrate arbitrage free models to match current market prices is one advantage of arbitrage free models over the equilibrium models.

The Ho Lee Model

The Ho Lee model takes the following form:

Derived using the relative pricing concepts of the Black Scholes model, this model assumes that changes in the yield curve are consistent with a no arbitrage condition.

The Ho Lee model is calibrated by using market prices to find the time dependent drift term ${\theta }_t$ that generates the current term structure. The Ho Lee model can then be used to price zero coupon bonds and to determine the spot curve. The model assumes constant volatility and produces a symmetrical (normal) distribution of future rates.

The Kalotay Williams Fabozzi (KWF) Model

The Kalotay Williams Fabozzi (KWF) model does not assume mean reversion and, like the Ho Lee model, assumes a constant volatility. However, the KWF model assumes that the short term rate is lognormally distributed. The model takes the following form:

Note that the right hand side of the equation is the same as that of the Ho Lee model. The only difference is that the KWF model assumes lognormal distribution of rates while the Ho Lee model assumed normal distribution.

Other Models

Gauss+ is a multifactor model that incorporates short , medium , and long term rates, where the long term rate is designed to be mean reverting and depends on macroeconomic variables. Medium term rates revert to the long term rate, while the short term rate is devoid of a random component, consistent with the role of the central bank controlling the short term rate. This results in a hump shaped volatility curve across tenors, with the volatility of the medium term rates being the highest.

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