Calibrating the Ho-Lee Model’s Drift from a Yield Curve

Summary

The document explains how the time-varying drift in the Ho-Lee short-rate model can be recovered from zero-coupon bond prices implied by an initial yield curve. It presents the bond-pricing relationship and shows that the drift function depends on the second time derivative of the logarithm of the initial bond price, together with the model’s volatility contribution. This gives a route from observed curve data to the model parameter rather than requiring a software implementation.

For a curve represented with piecewise interpolation, the derivative can be evaluated piecewise analytically, as the answer notes. This is relevant to the question’s use of quarterly Treasury curve points and its difficulty applying a continuous-time formula in a discrete setting. The response does not spell out a discrete algorithm, address noisy or irregular market data, or discuss smoothing and boundary effects. Calibration quality therefore depends on how the yield curve is constructed and differentiated.

Key ideas

• Ho-Lee bond prices link the initial zero-coupon curve to the model’s time-varying drift.
• The drift can be expressed using the second derivative of log initial bond prices and the volatility term.
• A piecewise interpolation scheme allows the required derivative to be evaluated by segment.
• The answer gives a continuous-time relationship but does not provide a full discrete calibration procedure.
• Curve construction and differentiation choices can affect the resulting drift estimate.

Tags

# How does one estimate theta in the Ho-Lee model from a yield curve?


# How does one estimate theta in the Ho-Lee model from a yield curve?












I have a yield curve constructed using linear interpolation with data points every 3-months for US treasuries.

I would like to use that calibrate a Ho-Lee model, but I can't wrap my head around how to calibrate theta.

Is there any implementation that I could use (preferably matlab, r or c++) or a detailed description of the algorithm that I could use for reference? I have found some notes on the optimal form of theta*, but it's described in continuous rather than discrete terms, so it's of limited use in my case.

## Answer by Gordon (score 4)

https://quant.stackexchange.com/a/27527

Given the Ho-Lee interest rate model of the form \begin{align*} dr_t = \theta_t dt + \sigma dW_t, \end{align*} the price at time $t>0$ of a zero-coupon bond, with maturity $T$ and unit face, has the form \begin{align*} B(t, T) &=E\Big(e^{-\int_t^T r_s ds} \mid r_t \Big)\\ &=e^{-(T-t)r_t - \int_t^T (T-u)\theta_u du + \frac{\sigma^2}{6}(T-t)^3}. \end{align*} See this question for the details. In particular, \begin{align*} B(0, t) = e^{t\, r_0 - \int_0^t(t-u)\theta_u du + \frac{\sigma^2}{6}t^3}. \end{align*} Moreover, \begin{align*} \theta_t = \sigma^2 t - \frac{\partial^2 \ln B(0, t)}{\partial t^2}, \end{align*} which can be computed piecewise-analytically based on the interpolation scheme of the yield curve.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.