Calibrating the Ho-Lee Short-Rate Tree to the Yield Curve

# Ho-Lee Model; Please explain


# Ho-Lee Model; Please explain












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 by user16651 (score 6)

https://quant.stackexchange.com/a/18544

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.