# Ho & Lee yield curve fitting with zero coupon bond market prices

The Ho & Lee model for interest rates is given by the SDE: $$\mathrm d r = \eta(t) \mathrm d t + c\,\mathrm d X$$ The calibration function for $$\eta(t)$$ is given by $$\eta^*(t)=c^2(t-t^*)-\frac{\partial^2}{\partial t^2}\operatorname{log}(Z_M(t^*;t))$$ where $$Z_M(t^*, t)$$ are the discount factors in the market from today $$= t^*$$ to maturity $$t$$ (Source: Paul Wilmott on Quantitative Finance, p. 526).

The term $$\frac{\partial^2}{\partial t^2}\operatorname{log}(Z_M(t^*;t))$$ confuses me.

I have a set of discount factors $$Z_M$$, which are numbers (e.g. $$Z_M(0;\,0.5)=0.99750, Z_M(0;\,1)=0.989060)$$.

So, the $$\operatorname{log}Z_M$$ is also a number.

How can I compute the partial derivative $$\frac{\partial^2}{\partial t^2}\operatorname{log}(Z_M(t^*;t))$$ of a number?

EDIT: My current understanding is that I have to use some interpolation method which is twice differentiable (for example spline interpolation) using the discount factors as support points. Would this be correct?

• You would need to estimate it numerically from your discount factors $Z_M(0,t)$ using finite differences. – Daneel Olivaw Nov 28 '18 at 0:50

For example if we assume the points you have available for your discount factors $$Z_M$$ are equally spaced with gap $$\Delta t$$ then you get the following approximation via the second order central finite difference method:
$$\eta^*(t) = c^2(t) - \frac{\partial^2}{\partial t^2}\log(Z_M(0;t))$$
$$\approx c^2(t) - \frac{\log(Z_M(0;t+\Delta t))-2\log(Z_M(0;t))+\log(Z_M(0;t-\Delta t))}{(\Delta t)^2}$$