Re: [Quantlib-users] Option-market-implied probabilities for interest rate moves

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

Hey Michael,

just a few pointers that might be helpful: The QuantLib SmileSection has
methods to get the smile implied cdf (via digitalOptionPrice() which is
basically the cdf) and pdf (density())

https://github.com/lballabio/QuantLib/blob/master/ql/termstructures/volatility/smilesection.hpp#L71

with a simple default implementation using finite differences. In this
context it's good to use an arbitrage free smile parametrization which in
general is not provided by the Hagan 2002 SABR solution. One popular and
modern variant of the SABR model is the normal free boundary SABR proposed
by Antonov which has a semianalytic arbitrage free solution. An
experimental (!) implementation can be found here

https://github.com/OpenSourceRisk/Engine/blob/master/QuantExt/qle/models/normalsabr.hpp#L37

The inversion of the option price is done using an exact implied vol
formula due to Peter Jaeckel

https://github.com/OpenSourceRisk/Engine/blob/master/QuantExt/qle/models/exactbachelierimpliedvolatility.hpp#L33

Finally there is another approach in a discrete setting due to Carr and
Madan that can be used to derive a smile implied density

https://github.com/OpenSourceRisk/Engine/blob/master/QuantExt/qle/models/carrmadanarbitragecheck.hpp#L49

This implementation is restricted to the Equity / FX setting, but we'll add
a variant handling IR normal and (shifted) lognormal vols in the next
release of that library.

Thanks
Peter

On Tue, 31 Aug 2021 at 16:10, Michael (DataDriven portal) <
mi...@da...> wrote:
>
> Hi Giuseppe:
>
> Thanks a lot for your very insightful reply!
>
> I will do B-S (Breeden-Litzenberger) implied probabilities first to see
if the results make sense before going to more sophisticated modelling.
>
> This is great, thank you!
>
> Michael
>
> On Tue, Aug 31, 2021, 3:27 AM Giuseppe Trapani <tr...@gm...> wrote:
>>
>> Hi Michael,
>>
>> apologies but I'm not getting your point: the result is valid for
whatever underlying (be it interest rates / stocks / commodities) and so
on, it only relies on option prices. You can compute those option prices
with a model or you can get them from the market and THEN you obtain the
risk-neutral density for the underlying.
>>
>> As for the SABR being "specific for rates" I think it's a slight
misconception: the SABR model describes the dynamics of the underlying
under the forward measure so you can use it with whatever underlying. It
became sort of "standard" in IR markets since it's a very simple way to
improve the computation of hedging ratios: when pricing vanilla derivatives
in the Black-76 (or Bachelier) framework you can use it to parametrize the
volatility curve and compute smile-coherent greeks. Such a framework is the
most intuitive since it allows you to price the most common and liquid
derivatives (Swaptions and Caps/Floors) without the need to model
dependencies between the knots of the yield curve.
>>
>> If instead you are interested more specifically in the dynamics of the
ENTIRE yield curve, you have to rely on term-structure models (for example
short-rate models or maybe more involved quasi-gaussian models accounting
for the skew or more modern markov-functional models). You work in a
Monte-Carlo fashion like this:
>>
>> 1) calibrate the parameters of the model to some "information carrying
derivatives"
>> 2) simulate the yield curve at the horizon you need
>> 3) compute whatever you are interested in (for example the 10y swap fair
rate) on that curve
>> 4) aggregate all the simulation results
>>
>> Now as for 1) term-structure models are typically low-dimensional so you
cannot "calibrate" to all the IR market derivatives. A common choice is a
set of coterminal swaptions or ATM caps/floors spanning the analysis
horizon.  Also depending on the model, 2) and 3) can have some analytic
formulas / approximations so maybe you can spare yourself the whole
simulation and compute directly what you need.
>>
>>
>>
>>
>> Il giorno mar 31 ago 2021 alle ore 03:18 Michael (DataDriven portal) <
mi...@da...> ha scritto:
>>>
>>> Thanks a lot!
>>>
>>> I see links like below Breeden-Litzenberger
>>>
>>>
https://quant.stackexchange.com/questions/29524/breeden-litzenberger-formula-for-risk-neutral-densities
>>>
>>> This is very useful for B-S distributional assumptions and will work
well for stocks. I am not sure if it will produce good results for interest
rates which have different distributions (e.g. rates now are so low that
are less likely to decrease than increase). But I will definitely give it a
try. Another way to do this is to use SABR volatility model (which is
specific for rates) but I am not sure if a simple solution exists to derive
probabilities there.
>>>
>>> Thanks,
>>>
>>> Michael
>>>
>>>
>>>
>>>
>>>
>>> On Mon, Aug 30, 2021 at 11:36 AM Giuseppe Trapani <tr...@gm...>
wrote:
>>>>
>>>> Hi Michael,
>>>>
>>>> to add on the previous answer, the derivative is taken with respect to
the strike price of the option.
>>>>
>>>> It's pretty easy to derive by yourself starting from the general
payoff of an option (yielding a "model free result") or from the Black-76
formula.
>>>>
>>>> For extra directions check online "Breeden-Litzenberger result".
>>>>
>>>> Giuseppe Trapani
>>>>
>>>> Il lun 30 ago 2021, 15:28 Michael (DataDriven portal) <
mi...@da...> ha scritto:
>>>>>
>>>>> Yes. Thanks! If you could point me in the right direction on where I
can get code for this that would be great.
>>>>>
>>>>> Thanks
>>>>>
>>>>> On Mon, Aug 30, 2021, 8:47 AM Gorazd Brumen <gor...@gm...>
wrote:
>>>>>>
>>>>>> There is a well known formula that relates call/put prices to implied
>>>>>> pricing probabilities, related to the second derivative of the
>>>>>> call/put prices. You might need an implied option value
>>>>>> parametrization for that.
>>>>>> Regards,
>>>>>> G
>>>>>>
>>>>>> On Sun, Aug 29, 2021 at 5:56 PM Michael (DataDriven portal)
>>>>>> <mi...@da...> wrote:
>>>>>> >
>>>>>> > Hi All,
>>>>>> >
>>>>>> > I am looking for an algo to calculate option-market-implied
probabilities of interest rates moves derived from the premiums of interest
rate swaptions.
>>>>>> >
>>>>>> > E.g. market-implied probabilities from the prices of swaptions on
10-year-swap rates. What is the market-implied probability that 10Y swap
rate will increase 25, 50, 75 bps, etc?
>>>>>> >
>>>>>> > Thanks,
>>>>>> >
>>>>>> > Michael
>>>>>> >
>>>>>> > _______________________________________________
>>>>>> > QuantLib-users mailing list
>>>>>> > Qua...@li...
>>>>>> > https://lists.sourceforge.net/lists/listinfo/quantlib-users
>>>>>
>>>>> _______________________________________________
>>>>> QuantLib-users mailing list
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>>
>>
>> --
>> Giuseppe Trapani
>
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