I was looking at literature and found that for hybrid models, most of the literature only gives hybrid models where the volatility of the interest rate process(e.g Hull White) is constant.

Is there a way to generalize this to a deterministic time dependent vol for the IR process in the hybrid model? Could i just replace it with a term-structure dependent model vol and it works fine?

Best, Ben


1 Answer 1


You can simply introduce a piece-wise constant vol parameter. The extension is rather trivial as it will involve simple sums. Also, the derivations of the ChF will not be affected by this extension.

  • 1
    $\begingroup$ Thanks Lech! Would you be able to walk through briefly how to extend with simple sums? $\endgroup$
    – Benedict
    Commented Aug 19, 2021 at 11:31
  • $\begingroup$ Yes, of course. As you may remember, in the Heston model we are able to introduce piece-wise constant parameters. The derivation of the corresponding characteristic function involves the iterative solution of the Riccati type of equation. If you extend the Heston model with HW the same would apply. On the other hand, if you seek the solution of the HW model for time-dependent parameters (piecewise constant would be a natural choice) then instead of integrals you would end up with summations. You can search for the thesis of Dionl van Lange at TUDelft for more details. Good luck! $\endgroup$
    – Lech
    Commented Aug 20, 2021 at 11:12

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