Re: [Quantlib-dev] Markov Functional Model

[Luigi B. <lui...@gm...>]

Thanks Peter, I'll try and have a look.

Luigi

On Sun, Dec 2, 2012 at 9:03 PM, Peter Caspers <pca...@gm...> wrote:
> Hi,
>
> I implemented a Markov functional model which I would like to contribute
> to the lib. I would consider the development to be something like an
> alpha version.
>
> The model is an one factor interest rate model with a non parametric
> numeraire that can be fitted to a set of (arbitrage free) smiles on an
> option expiry grid. In addition a volatility function can be used to
> calibrate to a second instrument set (but only for one strike per
> expiry). An example would be a numeraire calibration to constant
> maturity swaption smiles which allows to recover market cms coupon
> prices. As a second instrument set atm coterminal swaptions can be
> chosen. The resulting model is then a good candidate for pricing
> callable cms swaps. The mean reversion parameter in the model is fixed,
> i.e. it can not be calibrated, but must be given externally. It can be
> used to control intertemporal correlations. References can be found in
> the documentation mentioned below.
>
> The package comes with the model, an implementation of Kahale's method
> for arbitrage free smile extrapolation (because arbitrage free smile
> input is essential for the numeraire to calibrate) and vanilla pricing
> engines for caps and swaptions. The swaption engine can also handle
> bermudan exercise rights. At the moment all engines are based on
> numerical integration. PDE engines and more product support will follow
> soon.
>
> You can find the code on github. I updated only the project files for
> msvc10, but it should be easy to include the relevant files in other
> enviroments. They are located in ql/experimental/models/markovfunctional.
>
> https://github.com/pcaspers/quantlib/commit/879c0fdd4f4149facdc8e771d838acafa7d6aae9
>
> A good starting point is probably running the test-suite to which I
> added a suite for the mf model which can be found in
> test-suite/markovfunctional.xpp.
>
> There are some changes in the ql I made, some essential for the model to
> work, others which I found worthwhile to do in the context and which
> support the test cases to run smoothly. I posted some of these changes
> on this list in the past, but I summarize all of them at the end of this
> mail.
>
> Any feedback is highly appreciated. There is also some documentation
> available here
>
> http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2183721
>
> Thank you and kind regards
> Peter
>
>
> Changes to the lib I made:
>
> 1 ql/math/cubicinterpolation.hpp
>
> essential for mf model, I added the Lagrange boundary condition. I
> tested against the MatLab implementation. Works fine.
>
>
> 2 ql/math/sabrinterpolation.hpp
>
> not essential for mf model, but necessary for the test suite to run.
>
> I use a derivative of swaptionvolcube1 for daily swaption smile
> calibration for quite a time now. I noticed that two problems occur from
> time to time: First, calibration is not satisfactory because of unlucky
> start values. The solution for this I implemented is not very smart, but
> it is working well and is still fast enough. Secondly (very rarely)
> calibration stops with error values. I located the problem to be the
> transformation of valid parameters to numeric error values during
> optimization. Fix is very easy.
>
> More specifically I added a try and error method for calibration. Up to
> maxGuesses start values for the calibration are generated. If the error
> is below errorAccept the current calibration is returned immedeately
> without further tries. Otherwise the best calibration seen is returned.
> The error here may be specified to be the average error instead of the
> maximum error. The parameter transformation is adjusted to ensure that
> valid parameter values are not
> transformed into error values during calibration.
>
>
> 3 ql/models/calibrationhelper.xpp
>
> not essential for mf model, but necessary for test suite to run.
>
> I made the calibration helpers lazy objects. Seems reasonable because
> otherwise market data structures loose their lazy behaviour.
>
>
> 4 ql/models/model.xpp
>
> essential for mf model. I replaced the update() call during calibration
> by generateArguments() and notifyObservers(). This allows to separate
> the smile update triggered by market data change from the numeraire
> update triggered by model volatility changes.
> The calibration to a second instrument set is faster then. Behaviour of
> existing models is not affected by this change.
>
>
> 5 ql/pricingengines/blackformula.xpp
>
> not essential for mf model, but model trace output may contain false
> vega for zero strike, which I corrected.
>
>
> 6 ql/termstructures/volatility/optionlet/strippedoptionletadapter.xpp
>
> not essential for mf model, but if you want to calibrate the numeraire
> to caplet smiles, you need a volatility structure returning a smile section.
>
> I added a smile section section implementation returning a smooth spline
> interpolated smile curve. The markov functional
> model requires the volatility structure to provide a smile section. In
> addition this should be smooth in prices, at least C^1,
> to ensure smooth digital prices. I did not put much energy in this
> enhancement, just wanted to make the model work with cap smiles.
>
>
> 7 ql/termstructures/volatility/sabrsmilesection.xpp
>
> not essential for mf model, but needed if you want to feed
> sabrsmilesections for numeraire calibration. I floored the strike at
> 0.1bp for volatility computation to avoid problems for strikes near zero.
>
>
> 8 ql/termstructures/volatility/smilesection.xpp
>
> essential for mf model. I added some supporting functions including
> arbitrage tests and option price calculation. The best place for these
> functions seemed here.
>
>
> 9 ql/termstructures/volatility/swaption/swaptionvolcube1.xpp
>
> not essential for mf model, just the same comments as under 2 apply.
>
> I added functionality to exclude negative strikes in single smiles, so
> that the -200bp quotes can be used where reasonable. Furthermore I added
> the same functionality as in sabrinterpolation above.
>
>
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