[Quantlib-dev] Markov Functional Model

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

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.