# Why should future short rates tend towards the current term structure of interest rates?

I'm currently looking at the Hull-White model reproduced below:

$$\mathrm{d}r = \lambda(\theta(t)-r)\mathrm{d}t + \sigma\mathrm{d}W(t)\text{.}\tag{1}$$

I have a simplistic understanding of the model. My understanding is that $$\theta(t)$$ is a long term "mean-interest rate function" that $$r$$ tends towards. My thinking is that $$\theta(t)$$ can be anything. I see two motivations that could help you choose $$\theta$$.

a. You may choose $$\theta$$ based on what you think the market is going to do. I could, for example, base my beliefs about $$\theta(t)$$ on what I think the U.S. Federal Reserve is going to do. If I think the Fed will target a lower short rate in the near future, but will increase this target in several years, then I may choose a $$\theta$$ based on that. My beliefs may differ from the current term structure. Is this pure speculation, or is this kind of reasoning relevant for hedging?

b. You may choose $$\theta$$ with the goal of "hedging." In this case, you will calculate $$\theta$$ from the current term structure of interest rates.

That leaves me with several question, the most important of which is (3):

1. Is (a) above ever used in practice? If yes, is (a) above relevant for hedging?
2. How is (b) above relevant for hedging?
3. Do I expect the current term structure of interest rates to causally modulate future short-rates? Not just that it matches expectations or market prices, but will, at time $$t_i$$, there be market forces that push $$r$$ towards $$\theta(t_i)$$? To put it another way, why should future short rates tend towards the current term structure of interest rates?

Edit: $$\theta$$ represents current expectations of future short rates based on the market. You will adjust your hedge at time $$t$$ based on the price of the underlying (and $$v$$ and $$r$$) at time $$t$$. The short rate at time $$t$$ may be different from $$\theta(t)$$. So it seems to me that what is more important for hedging is what the hedger's expectation of what the short rate will be rather than what the market thinks what the short rate will be.

It really depends for what purpose you are using the model. Let’s say you are using it for valuation of some instrument. If you want the fair market value, then a) is irrelevant and you would instead calibrate to the current term structure. For hedging , one usually means hedging the market value so again b) is appropriate. The only reason to use a) is to determine your own view of the valuation , but this will be inconsistent with the values of other traded instruments in the market.

As for 3, there is a lot of literature concerning whether the current term structure is or is not a good estimator of future interest rates. It appears that there is a slight bias to overestimate future interest rates (ie the term structure is slightly more upward sloping than would be justified by pure expectation of interest rates). Please google term premium. To answer your question, the short rate does tend to mean revert (ie when rates are very high, they are expected to fall, and vice versa). But this effect is not particularly precise ( the exact level of $$\theta$$ is not known. Hope that gives some ideas.

• Your first paragraph does not make it clear how this hedging is done. WRT your second paragraph, I'm not asking if the term structure is accurate. I'm wondering if there are reasons future short-rates will converge to interest rates implied by the term structure, e.g., by the market being "populated" with bonds following the old term structure and some peculiarity about how bond contracts/markets work. Jul 17 at 7:11
• Hi how hedging is done? Literally, by buying or selling bonds or swaps ( whatever is being used to calibrate the model).
– dm63
Jul 17 at 11:20
• Second part: there are definitely no reasons why rates will converge to a level implied by an earlier term structure.
– dm63
Jul 17 at 11:22
• no, I mean that $\theta$ represents current expectations of future short rates based on the market. You will adjust your hedge at time $t$ based on the price of the underlying (and $v$ and $r$) at time $t$. The short rate at time $t$ may be different from $\theta(t)$. So it seems to me that what is more important for hedging is what the hedger's expectation of what the short rate will be rather than what the market thinks what the short rate will be. Jul 30 at 16:22
• No, practitioners usually hedge derivatives based on a model that is calibrated to the market. To the extent that a trader disagrees with the expectations inherent in the market , a separate decision may be made to position the book accordingly. Point is the 2 things are usually considered separately.
– dm63
Jul 30 at 22:34