# Stochastic Processes (Applying Ito's Lemma on Ho-Lee Model )

I seek a basic form (SDE) to understand the Ho-Lee model.

I already understand the models from Vasicek, Merton and Cox-Ingereoll-Ross, etc.. For example,

\begin{align*} dX_t &= -1/2 \alpha X_t dt + \sigma dWt, \\ r_t &=f(t,X_t)=(X_t)^2. \end{align*} Then, $$f_t(t,x)=0$$, $$f_x(t,x)=2x$$ and $$f_{xx}(t,x)=2$$. By Itô's Lemma,

\begin{align*} dr_t&= \left(-1/2 \alpha X_t \cdot 2X_t+ 1/2\sigma^2 \cdot 2\right) dt+2\sigma X_t dW_t \\ &= \left(\sigma^2 -\alpha r_t\right)dt + 2\sigma \sqrt{r_t} dW_t. \end{align*}

So, what about the Ho-Lee model?

I know that the instantaneous forward is given by $$f(t, T) = f(0,T) + \sigma^2 (Tt - 1/2t^2) + \sigma W_t.$$

The short rate is optained using $$r_t=f(t,t)$$, that is:

$$r_t= r_{0} + 1/2 \sigma^2 t^2 + \sigma W_t.$$

But is there a way of defining the model via an SDE?

## 1 Answer

Could you please verify that I edited your question correctly, i.e. that this is indeed your question.

In this case, the Ho-Lee (1986) model reads as $$dr=\theta_t dt +\sigma dW_t$$. Do you can use $$f(t,x)=x$$ such that $$X_t=r_t$$. In this Sense, the Ho-Lee model is a Vasicek model with time dependence. This is further generalised in the model from Hull and White (1990). In all of these models, the short rate is normally distributed. They allow for closed-form solutions of zero-coupon bond (option) prices.

• The post has been edited correctly. So, If I understood well, the final solution would be Theta t dt as Ft(t,x)=0 ; Fx(t,x)= 1 ; Fxx(t,x)= 0 – Mark Sep 23 '19 at 14:52
• Yeah, these derivatives are correct. Thus, $X_t=r_t$. – Kevin Sep 23 '19 at 14:53