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Re: [Quantlib-users] Option-market-implied probabilities for interest rate moves
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From: Michael (D. portal) <mi...@da...> - 2021-09-01 14:34:47
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Hi Peter: Thanks a lot - extremely useful! Michael On Wed, Sep 1, 2021 at 3:37 AM Peter Caspers <pca...@gm...> wrote: > 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 > >>>>> Qua...@li... > >>>>> https://lists.sourceforge.net/lists/listinfo/quantlib-users > >> > >> > >> > >> -- > >> Giuseppe Trapani > > > > _______________________________________________ > > QuantLib-users mailing list > > Qua...@li... > > https://lists.sourceforge.net/lists/listinfo/quantlib-users > |
