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

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Philippe Hatstadt <phi...@ex...>
2:09 PM (0 minutes ago)
to Peter, philippe
One problem I am uncovering is that apparently, a ql.DiscountCurve()
appears to have no concept of evaluation date, as I am finding out as
described below.
This seems confirmed by this thread
https://stackoverflow.com/questions/45683889/discountcurve-is-not-aware-of-evaluation-date-in-quantlib-python

Unfortunately, the way I wrote my code is that I compound the daily short
rate from the HW engine to build a path-wise ql.DiscountCurve. In turn,
what I was doing is to compute swap.NPV() (where swap is the forward
starting swap underlying the swaption) with evalutionDate set at the
swaption expiry, then I was deflating such NPV by the value of the
numeraire at option expiry. The problem is that swap.NPV() literally
doesn't change whether I set the evaluation date to the trade_date versus
the option expiry date. So this is a problem.
However, I am still perplexed by the following: with the bank account as
numeraire, the value at option maturity of such account is
effectively 1/DiscountFactor(path curve, option maturity date). So if for a
given path the swap.NPV() at option expiry is NPV(Tj), the numeraire
deflation would effectively make such number NPV(Tj) * DiscountFactor(path
curve, option maturity date). Well, that product should be exactly equal to
swap.NPV() where I set the evaluation date to trade date instead of option
expiry date, or is that untrue?
Assuming so, and given the issue mentioned above related to a
DiscountCurve's inability to have an evaluation date other than trade_date,
I think I should just take E[swap.NPV(evaluation date = trade date)]
multiplied by N(t=0) which equals 1.

Philippe Hatstadt

On Fri, Apr 9, 2021 at 3:45 AM Peter Caspers <pca...@gm...> wrote:

> Hey Philippe,
>
> so you simulate HW paths under a specific measure (standard HullWhite
> = Bank Account, there is a class for T-Forward and the GSR which is
> basically = Hull White under T-Forward as well) and compute swap.NPV()
> on a simulation node (t, x) where t = simulation time, x = HW state.
> You'll want to compute the NPV w.r.t. t as the valuation time, i.e.
> discount the cashflows with pay date > t back to t using the
> conditional curve at (t,x). Similarly you project floating coupons
> with the conditional projection curve at (t,x). Then you'll have to
> divide the result by the numeraire, i.e. the bank account B(t) or the
> zero bond P(t,T) for the T-forward measure. The average over such
> deflated NPVs will give you the NPV at t=0 over the Numeraire at t=0,
> i.e. you have to multiply by B(0)=1 (no effect in this case of course)
> resp. P(0,T) for T-forward measure. In formulas
>
> NPV(t=0) = N(0) E ( NPV(t,x) / N(t) )
>
> where N = Bank Account or zero bond with maturity T. And yes, a change
> in measure corresponds to a change in the drift of the simulated
> process (Girsanov Theorem).
>
> Thanks
> Peter
>
>
> On Sun, 4 Apr 2021 at 17:57, Philippe Hatstadt
> <phi...@ex...> wrote:
> >
> > Hi Peter.
> > I am going back to this semi-dated thread. My question is what
> measure/numeraire is the drift adjustment in the HW engine compatible with,
> if any? In other words, if I price say a swaption via Monte-Carlo
> simulation, and say I evaluation my cash flows with the discount factor as
> numeraire, i.e. I use swap.NPV() to value the forward swap on each path, I
> overprice the option versus closed-form Jamshidian. I think swap.NPV() is
> consistent with a money market account as numeraire, so my question is
> whether I am still supposed to "deflate" the swap.NPV() to satisfy no
> arbitrage? I thought such deflation typically takes place as an additional
> drift adjustment that needs to be calibrated, but my question is whether
> this is already done or not in the HW engine for DF as numeraire, or if
> there is none and it needs to be done?
> >
> > Philippe Hatstadt
> >
> >
> > On Sun, Oct 25, 2020 at 1:47 PM Peter Caspers <pca...@gm...>
> wrote:
> >>
> >> 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
> >> >> >
> >> >> >
> >> >> >
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> >> >>
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