This topic is mainly about modeling of short rate \(r(t)\) which plays a central role in the theory and practice because zero coupon bonds can be priced as

\(P(t,T)=E_t[\exp(-\int^T_tr(s)ds)]\)

and so could discount factor.

Assume there is only one source of uncertainty represented by a Brownian motion \(W(t)\) under the risk-neutral measure \(Q\). We want the stochastic process to be markovian which is referred to as path independence and hence allows for the use of recombining lattice. We don't want to use non-recombining lattice cause the number of nodes growing exponentially w.r.t. the number of time steps and thus is not practical. There are several important things we need to think about when choosing a model:

-- Does the dynamics guarantee positive rates?

-- What distribution does the dynamics imply for the short rate?

-- Can the model be calibrated to fit today's structure of zero coupon bond prices, how easy, and does the calibration stable?

-- Are the zero bond prices computable by means of analytic formulas?

-- Are there analytic pricing formulas available for call/put bond option?

-- Is the model mean-reverting?

-- How do the volatility structures implied by the model look like?

-- How suited is the model for building recombining lattices?

If the parameters of a model are constant like \(dr=a(b-r(t))dt+\sigma dW\) (Vasicek) then it is called equilibrium model. It's simple but could not be calibrated to fit the current term structure of rates. If we make the parameter of the drift term time dependent then we can overcome this problem and make the model no-arbitrage.

Based on empirical observation, people find that interest rates (unlike stock prices) tend to be pulled back to some mean level. So we want our model to be mean reverting; we also want to have the analytical formulas for zero coupon bond, call/put bond options, etc., and hence we require the model be affine which means the drift term is linear to short rate and square of the volatility is also linear to short rate. Regarding the volatility, typically we have two kinds of model for volatility: normal and lognormal. It is very hard to say which one is better, normal assumption results in analytic tractability and lognormal assumption leads to positive rates. Also, normal dynamics suit low rate environments and for lognormal, the other way around.

Several important models:

Ho-Lee: \(dr=\theta(t)dt+\sigma dW\)

Hull-White: \(dr=a(\theta(t)-r)dt+\sigma dW\)

CIR model: \(dr=a(\theta(t)-r)dt+\sigma\sqrt{r}dW\)

BDT model: \(d\ln(r)=(\theta(t)+(\sigma'(t)/\sigma(t))\ln(r))dt+\sigma(t)dW(t)\)

Among these models, Hull-White has a positive probability to generate negative rates but we can cut that and it does not affect pricing very much. CIR model is the only model with full analytic tractability which also has positive rates. While doing calibration, \(\theta(t)\) term in CIR must be solved for numerically. Under lognormal assumption, the total volatility is proportional to the rate level and thus it grows/declines with the rates, which produces rate path distribution skewed upward. Theoretically, we could deduce such equation under lognormal model:

\(E_0[B(\Delta t)]=\infty\)

therefore we can make infinite money in an arbitrary small time from $1 initial investment, however in numerical implementations of a short rate model on a tree this problem does not appear as the number of states is finite.

In order to make our models better, we could add a deterministic shift:

\(r(t)=r_0(t)+\varphi(t)\)

\(dr_0(t)=a(\theta(t)-r_0)dt+\sigma\sqrt{r_0}dW(t)\).

This kind of extension could preserve the tractability but not the positiveness of rates. Shifted CIR model could perfectly fit the market price of zero coupon bond and according to this:

when rates are low: \(\sigma(t)(r(t)+\alpha)\approx\sigma(t)\alpha\)

when rates are high: \(\sigma(t)(r(t)+\alpha)\approx\sigma(t)r(t)\)

we know that shifted lognormal models capture the phenomenon that rates follow close-to-normal dynamics in low rate environments and close-to-lognormal one in high rate environments.