Deriving the Ho-Lee Short-Rate Drift from the Forward Curve
Summary
The document derives the time-dependent drift in a Ho-Lee short-rate model, where the short rate follows a diffusion under the risk-neutral measure. The method integrates the short-rate process to obtain the distribution of the accumulated rate, which is Gaussian. It then prices a zero-coupon bond as the expected exponential of the negative integrated short rate.
Taking the logarithm of that bond-price expression and differentiating twice yields the drift as the forward curve’s slope plus a term proportional to volatility squared and time. The derivation illustrates how an initial term structure can determine the drift needed to fit a short-rate model. The stated result assumes the given diffusion specification and constant volatility; the answer also notes that a related approach applies to Hull-White models, without providing a separate derivation for them.
Key ideas
- Integrating the short-rate diffusion gives a Gaussian distribution for the accumulated short rate.
- A zero-coupon bond price follows from the risk-neutral expectation of discounted cash flows.
- Differentiating the log bond-price expression connects the model drift to the forward curve slope.
- The stated drift relationship includes a volatility-squared term that grows with time.
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# Determining bond price based on diffusion process for the short rate model
# Determining bond price based on diffusion process for the short rate model
Suppose the diffusion process for the short rate $r_t$ under the risk-neutral measure $Q$ is given by: $$ dr_t = \theta(t)dt+\sigma dZ_t $$
where $Z_t$ is a Brownian motion.
I am trying to show that the parameter $\theta(t)$ is related to the slope of forward rate curve: $$ \theta(t) = \left. \frac{\partial F}{\partial T}(0,T) \right|_{T=t} + \sigma^2 t $$ where: $$ \partial F(0,T) = -\frac{\partial \log P(0, T)}{T} $$
I am lost as to where to start. Any help would be appreciated
## Answer by byouness (score 2, accepted)
https://quant.stackexchange.com/a/64345
You can apply this approach for your model (it's the Ho-Lee model) but also to other short rate models such as Hull-White model.
(1) First, integrate twice the short rate SDE to get $\int_0^t r_u du$, you will find out that it's gaussian with this distribution: $$ \begin{aligned} -\int_0^t r_udu &\sim \mathcal{N}(m_t = -r(0)t - \int_0^t \int_0^u \theta(v)dv, v_t = \frac{\sigma^2t^3}{6}) \end{aligned} $$ This might help you get there: Integral of Brownian motion w.r.t. time
(2) compute the zero-coupon bond price given by the model: $$ P(0,t) = \mathbb{E} \left[e^{-\int_0^t r_udu}\right]=e^{m_t + \frac{v_t}{2}} $$
(2) Then, take the logarithm and differentiate twice to get the desired expression: $$ \theta(t) = \left. \frac{\partial F(0,T) }{\partial T} \right|_{T=t} + \sigma^2 t $$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.