Ho-Lee Model; Please explain

Question

I'm having trouble with the Ho-Lee model for short rates and differentiating between how to find the values for the free parameter λ versus using the model to predict future rates.

The Ho-Lee model for each step in a binomial tree: $$ \lambda_tdt + \sigma \sqrt dt $$

I've read that to set the free parameter at each step in a recombining binomial tree, you set the rate at state 0 to the current spot rate (ie: 1 month spot rate) and find a value for lambda that when plugged into the model will result in the current spot rate for the next time step (eg: starting with 1 month spot rate at state 0 and using a 1 month time step, the correct value for lambda when plugged into the model will produce the current 2 month spot rate etc).

This confuses me. Once I've determined the value of lambda for each step in my tree, what inputs do I change to use the model with my binomial tree to predict futures rates .. ie: one month rate in one month, in two months etc?

In case my description isn't clear, here is an except from Bruce Tuckman's book on the subject.

... find λ1 such that the model produces a two-month spot rate equal to that in the market. Then find λ2 such that the model produces a three-month spot rate equal to that in the market. Continue in this fashion until the tree ends.

Answer 1

You know that the Ho-Lee model is represented by the stochastic differential equations \begin{align} dr_t=\lambda_t\,dt+\sigma\,dW_t \end{align} In order to Implementation our binomial tree, we use the Euler discretization. \begin{align} r_t=r_{t-\Delta t}+\lambda_{t-\Delta t}\,\Delta t+\sigma\,\sqrt {\Delta t} \,Z \end{align} where $Z$ is a standard normal random variable.Let $t_0=0<t_1<...<t$ and expand equation, in discrete time \begin{align} r_t=r_0+\Delta t\sum_{t_0\leq t_i\leq t-\Delta t}\lambda_{t_i}+\sigma\Delta t\sum_{t_0\leq t_i\leq t-\Delta t}\ \,Z \end{align} This relation shows that the short rate is the sum of a set of non-stochastic drift terms and a set of random terms.The no-arbitrage zero coupon bond price $P(t,t+\Delta t)$ will thus be stated as

\begin{align} P(0,t_n)=E^Q\left[exp\left(-\Delta t\,\sum_{i=0}^{n-1}r(t_i) \right)\right] \end{align} For instance calculating the bond price at time $n=2$, gives us: \begin{align} P(0,t_2)=E^Q[\Delta t\,exp(-r_{t_0}-r_{t_1})]=e^{-\Delta t\,r_{t_0}}E^Q[e^{-\Delta t\,r_{t_1}}] \end{align} in other words \begin{align} P(0,t_2)=e^{-\Delta t\,r_{t_0}}\,exp\left(-\Delta t\,E^Q[r_{t_1}]+\frac{1}{2}\Delta t\,Var^Q[r_{t_1}]\right) \end{align} In this case, $r_t$ has a normal distribution,thus \begin{align} \ln P(0,t_2)=-\Delta t\,r_{t_0}-\Delta t\,r_{t_0}-\Delta t\lambda_0\,+\frac{1}{2}\sigma^2(\Delta t)^2=-2\Delta t\,r_{t_0}-\lambda_0\,\Delta t+\frac{1}{2}\sigma^2(\Delta t)^2\ \end{align} But \begin{align} \ln P(0,t_2)=\Delta t\,[-f(0,0)-f(0,t_1)] \end{align} It can be rewritten as: \begin{align} -r_{t_0}-f(0,t_1)=-2r_{t_0}-\lambda_0\ t+\frac{1}{2}\sigma^2\Delta t\ \end{align} then \begin{align} \lambda_{t_0}=f(0,t_1)-r_{t_0}+\frac{1}{2}\sigma^2\Delta t\ \end{align} This relaton give the necessary recursive relations to evolve the Ho-Lee no arbitrage model of short rates. We take a set of bond prices and structure of volatilities as an input for the short rates. Therefore we get the evolutionary equation to depict the binomial tree of the model.