Approximating the Ho–Lee Calibration Derivative from Discount Factors
Summary
The document addresses how to compute the second derivative of the logarithm of market discount factors in the Ho–Lee interest-rate model’s calibration function. Since observed discount factors are discrete values rather than a continuous formula, the derivative must be approximated computationally. For equally spaced maturities, it presents the central finite-difference approximation: take the log discount factors at adjacent points, form the second difference, and divide by the squared time spacing.
The answer favors evaluating suitable finite-difference methods over fitting a smooth interpolation and differentiating it, while the question itself raises spline interpolation as a possible approach. The displayed approximation assumes equally spaced data and uses neighboring points around the maturity being considered. The note does not compare error properties or specify how to handle irregular spacing, endpoints, or noisy market inputs. Those choices affect calibration and require an appropriate method for the available curve data.
Key ideas
- The Ho–Lee calibration function requires the second time derivative of log market discount factors.
- Discrete discount factor observations require a numerical derivative approximation.
- For evenly spaced maturities, a central second finite difference uses adjacent log discount factors.
- The choice of finite-difference method should suit the data; the document does not provide a universal method.
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Full text
# Ho & Lee yield curve fitting with zero coupon bond market prices
# 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?
## Answer by Trevor Hansen (score 1, accepted)
https://quant.stackexchange.com/a/42836
When no functional form is available in differential analysis then one should use a computational method. As Daneel comments a common computational approximation of the second order derivative can be obtained using finite differences.
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}$
I wouldn't first fit an interpolation and then differentiate as I have not seen that used in practice. I would assess which finite difference method is the most appropriate and use that.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.