How do I calculate Hull White's Theta from the discount curve?

Question

The Question

I'm currently implementing the a finite difference method for the Hull-White model, shown below:

$$\mathrm{d}r(t)=\lambda[\theta(t) − r(t)]\mathrm{d}t + \sigma\mathrm{d}W(t)\tag{1}$$

This requires that I calculate $\theta(t)$ at each point in time, $t$.

I assume I am given a discount curve, an example of which is given below:

Time $t$ in years	Discount Factor
0	1
0.003	0.9998843
0.083	0.9968031
0.167	0.9935687
...	...
0.917	0.9629143
1	0.9599047
2	0.9200919
...	...
30	0.2699292

How do I calculate $\theta$ from such a chart?

My Attempt

On page 73 of Interest Rate Models — Theory and Practice by Brigo and Mercurio, we are given a formula for $\theta(t)$, given below:

$$ \theta(t) = \frac{1}{\lambda} \frac{\partial f^{M}(0, T)}{\partial T}\bigg\rvert_{T = t} + f^{M}(0, t) + \frac{\sigma^2}{2\lambda^2}\Big(1 - e^{-2\lambda t} \Big) \text{,}\tag{2}$$

where $f^{M}(0, t)$ is the market instantaneous forward rate at time $0$ for the maturity $t$. This can be calculated by

$$f^{M}(0, T) = -\frac{\partial P^M(0, T)}{\partial T}\text{,}\tag{3}$$

where $P^M(0, T)$ is the market discount factor for the maturity $T$.

Suppose we wanted to calculate $\theta(t)$. Then we just need to calculate

$$\frac{\partial P^M(0, T)}{\partial T}\tag{4}$$ and $$\frac{\partial^2 P^M(0, T)}{\partial T^2}\tag{5}$$

at $t$ using Taylor series discretizations and plug the formulas into (2).

Choose an $\epsilon$ such as $\epsilon = .000001$. We can try to calculate (5) at $t$ using the three points, $P^M(0, t-\epsilon)$, $P^M(0, t)$, and $P^M(0, t+\epsilon)$.

Well, now our answer boils down to how we estimate $P^M(0, t \pm \epsilon)$, which we estimate using (4) and (5). If we assume that $P$ is piecewise linear, for example, then (5) will always be $0$, except at times $t$ appearing in the chart, where it will blow up.

A different question might be "How do we interpolate $P^M$ for the purpose of calculating $\theta$?"

Answer 1

In practical situations you will never know $P^M(0,t\pm\epsilon)$ for a continuum of $t$ and $\epsilon\,.$ In other words, $\theta$ will practically always depend on an interpolation method between the $t$-points in your chart. There is no model that will tell you what the right interpolation method should be. Many different ones can be used in practice. Interpolation methods of order higher than linear typically use not only adjacent points $t_{i-1}$ and $t_i$ to interpolate inbetween but also $t_{i-2}$ and $t_{i+1}$ and perhaps more. This will have the effect that changing, say, the input $P^M(0,t_{i+1})$ affects your $\theta$ in a 'remote' interval $[t_{i-2},t_{i-1}]\,.$ It is a matter of judgment if such spillover effects of interest rate risk are desired.