Hull-White: $$d r = [\theta(t) - ar]d t + \sigma d W_t.$$ There is a statement in John Hull's book:

The advantage of making $a$ or $\sigma$, or both, functions of time is that the models can be fitted more precisely to the prices of instruments that trade actively in the market.
The disadvantage is that the volatility structure becomes nonstationary. The volatility term structure given by the model in the future is liable to be quite different from that existing in the market today.

$\theta(t)$ can already match the initial curve in the market, what's the meaning of fitted more precisely to the prices?

$\sigma(t)$ can match today's implied vol of all maturities, it should be better than $\sigma.$ How to understand nonstationary?


2 Answers 2


On your first question, the fact that you fit to the yield curve $-$ which is what the standard Hull-White model with time-dependent $\theta(t)$ allows to do $-$ does not mean that you are fitting the prices of more complex products such as options. For that you need to make $a$ and/or $\sigma$ also time-dependent.

On your second question, it simply means that you no longer have a flat, constant value for $\sigma$ but a much complex structure which depends on time $-$ I don't think there is anything more to that statement.

  • $\begingroup$ Your second answer is incorrect. Stationarity has a precise meaning for stochastic processes: essentially that the law (or at least some moments) of the process are invariant by time translation. You can check wikipedia for definitions. Regards. $\endgroup$
    – AFK
    Apr 11, 2018 at 22:03
  • 1
    $\begingroup$ @AFK I know, but therefore if volatility is constant it is stationary, as do $r$ increments "w.r.t. to volatility" $-$ although increments are not stationary due to time-dependence of $\theta$. What could be the meaning otherwise? $\endgroup$ Apr 11, 2018 at 22:23

Question 1: I would say these are just assumptions they made based on the observation of real market dynamics, that is, interest rate does show inconstant volatilities and mean reversion on different term structure. Having these variables time-dependent will make the simulated tree better match the real term structure. That's basicly how interest rate trees were gradually changed from Ho-LEE to Hull-white etc.

Question 2: It is indeed non-stationary. In financial markets, any products with term structure(or different expiries) usually show non-stationary volatilities throughout time, such as interest rate, energy, grain or financial futures etc. And most cases, short-term underlying show much higher volatility than longer than longer term. (I know gold is an exception). Interest rate products also show this type of dynamics that the volatility curve normally downward sloped as tenor goes up.


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