Re: [Quantlib-users] Drift adjustment in Jamshidian Engine and HW Prcoess

[Peter C. <pca...@gm...>]

I think this is expected, according to Brigo Mercurio Formula (3.37)
for the mean of the short rate at t (conditional on F_s with s = 0,
i.e. the unconditional mean) is

r(0) exp(-at) + alpha(t) - alpha(0) exp(-at) = alpha(t) = f(0,t) +
sigma^2 / (2 a^2) ( 1 - exp(- at) ) ^2

because alpha(0) = r(0) and so the first and last summand cancel out.
Plugging in sigma = 0.1 and a = 0.1 that means that E( r( 30 ) ) =
0.05 + 0.4514... = 0.5014.... This is all including the term
responsible for the fit to the initial flat curve at 5%.  And by the
way, I don't think you can disable this part of the drift in
HullWhiteProcess.

Also, I think sigma = 0.1 is an extremely high value, maybe on purpose
for the sake of the example?


On Sun, 25 Oct 2020 at 17:44, Philippe Hatstadt
<phi...@ex...> wrote:
>
> Actually, I have a follow-up question on the drift issue.
> I was experimenting with the Hull-White implementation from the QuantLib Cookbook, and interestingly, in Chapter 15, they build a flat curve with a forward rate of 5%, via:
> spot_curve = ql.FlatForward(todays_date, ql.QuoteHandle(ql.SimpleQuote(forward_rate)), day_count)
>
> The HW parameters are sigma=10% and a=10%, and the HW engine is built via:
>
> hw_process = ql.HullWhiteProcess(spot_curve_handle, a, sigma)
>
> They then proceed to generate paths of the short term rate, and they demonstrate that the simulated rate converges to the theoretical expected forward rate f(0,t<T) but the graph of such expected forward is not flat at 5%, instead it is monotonously increasing from 5% to as high as 50% after 30 years. So it very much looks like the paths of rates that they show are not drift adjusted otherwise the expected path should be flat at 5%.
> So I'm unsure what is going on? Is there an optional argument in the hw_process call to do the drift adjustment or not?
>
> Philippe Hatstadt
>
>
> On Sun, Oct 25, 2020 at 9:09 AM Peter Caspers <pca...@gm...> wrote:
>>
>> Hi Philippe,
>>
>> the Jamshidian engine uses a) discount bond prices conditional on the
>> state of the model (i.e. the short rate in the case of the Hull-White
>> model) and b) zero bond option prices in the model to come up with a
>> model swaption price. It retrieves this information via the
>> discountBond() and discountBondOption() methods in the
>> OneFactorAffineModel interface. The methods account for the adjustment
>> term theta(t) in the Hull-White model SDE already, there is nothing
>> that the engine needs to do in addition to that. I don't know if that
>> answers your question?
>>
>> The HullWhiteProcess also takes into account the adjustment to the
>> initial curve already. To see that you can look into the
>> implementation of HullWhiteProcess::drift()
>>
>> https://github.com/lballabio/QuantLib/blob/master/ql/processes/hullwhiteprocess.cpp#L38
>>
>> which coincides with e.g. Brigo Mercurio, Interest Rate Models, Theory
>> and Practice, Formulas (3.33) and (3.34) observing that in this
>> context
>>
>> https://github.com/lballabio/QuantLib/blob/master/ql/processes/ornsteinuhlenbeckprocess.hpp#L90
>>
>> level_ is zero, speed_ is the Hull-White mean reversion parameter and
>> x stands for the short rate at time t.
>>
>> Does that make sense?
>>
>> Thanks,
>> Peter
>>
>>
>> On Sun, 25 Oct 2020 at 12:47, philippe hatstadt via QuantLib-users
>> <qua...@li...> wrote:
>> >
>> > Is there any information about how the Jamshidian engine does the drift adjustment to match the initial curve for the Hull White model?
>> > If I want to use the QL HW Process in a MC model, I assume II have to do the drift adjustment myself, or is there existing QL functionality to do that?
>> >
>> > Regards
>> >
>> > Philippe Hatstadt
>> > +1-203-252-0408
>> > https://www.linkedin.com/in/philippe-hatstadt
>> >
>> >
>> >
>> > _______________________________________________
>> > QuantLib-users mailing list
>> > Qua...@li...
>> > https://lists.sourceforge.net/lists/listinfo/quantlib-users
>>
>>
>> _______________________________________________
>> QuantLib-users mailing list
>> Qua...@li...
>> https://lists.sourceforge.net/lists/listinfo/quantlib-users
>
>
>
>
> Brokerage services offered through Exos Securities LLC, member of SIPC / FINRA. For important disclosures, click here.