# Discount factor in Hull-White model

Consider a Hull-White model

$$dr(t)=\left(\theta(t)-a(t) r(t)\right) dt + \sigma dW(t)$$

with parameters

• $$a=0.1$$
• $$\sigma=0.3$$
• $$\theta(t)$$ was calibrated to match
• $$P(0,t)=\exp(-\mu t)$$
• with: $$\mu=0.2$$

At time $$t=0$$ the forward discount factor is:

$$P(0,t,T) = \exp (-\mu (T-t))$$

For example $$P(0,5,10)=\exp(-0.2\times (10-5))\approx 0.37$$

I can calculate this discount factor also in HW model (the discount factor will depend on the observed rate $$r(t)$$) at time $$t=5$$ with the formula:

$$P(t,T)=\exp\left(A(t,T)-B(t,T) r(t)\right)$$

If I do so (with $$t=5$$ and $$T=10$$ below), I get:

• $$\mathbb{E}\ r(5)=0.9$$
• $$A(5,10)=-2.4$$
• $$B(5,10)=3.9$$
• $$P(t,T)=\exp(-2.4 - 3.9 \times 0.9)=0.003$$

Now I ask myself: is this result correct? The discount factor between $$t=5$$ and $$T=10$$ was diminished from $$0.36$$ to $$0.003$$! Of course I do not have very realistic numbers (if $$t$$ is in years, then at $$t=5$$ the interest rate is 90% per year), but maybe I misunderstand how to use the formulas?!

A good advice when it comes to the Hull-White model is to never work with the short rate $$r(t)$$ directly. It will typically be quite unstable and depend on interpolation on the yield curve. Instead introduce the new variable $$x(t) = r(t) - f(0,t),$$ where $$f(0,t)$$ is the instantaneous forward rate at time $$t$$ seen from time $$0$$. By rewriting the Hull-White model in this variable, the only quantities you will need from the yield curve to price the bond $$P(t,T)$$ are the market discount factor to $$t$$ and $$T$$.
When you use $$E[r(t)]$$, you need to include an additional term to ensure the no-arbitrage condition.
Try to use simulated $$r(t)$$ and calculate the expectation of $$P(t,T)$$ because $$P(t,T)$$ is a random variable. ($$N$$ = number of simulation)
$$P^1 (t,T) = \exp(A(t,T) - B(t,T)r^1(t))$$ $$P^2 (t,T) = \exp(A(t,T) - B(t,T)r^2(t))$$ $$...$$ $$P^N (t,T) = \exp(A(t,T) - B(t,T)r^N (t))$$
$$E[P (t,T)] = \frac{1}{N} \sum_{i=1}^{N} P^i (t,T)$$