# Test Coverage

Since release 1.6 a code coverage report for QuantLib’s unit test suite is provided. While for 1.6 and 1.7 the Intel compiler suite was used to generate it, the report for the current release 1.8 was generated using lcov / gcov for the first time, following these steps:

• configure with the coverage flag, switching optimizations off, debug symbols on and disabled inlining: ./configure CXXFLAGS="--coverage -O0 -g -fno-inline -fno-inline-small-functions -fno-default-inline" LDFLAGS="--coverage"

• build the library and test-suite, first by cleaning up (make clean) followed by something like make -k -j8 -w

• generate a baseline info file: lcov --no-external -c -i -d . -o ~/ql_base.info

• run the test-suite (this takes a bit due to the missing optimizations): test-suite/quantlib-test-suite

• generate the test run’s info file: lcov --no-external -c -d . -o ~/ql_test.info

• merge the baseline and test run’s info files: lcov -a ~/ql_base.info -a ~/ql_test.info -o ~/ql_total.info

• remove the test suite code itself from the coverage report: lcov --remove ~/ql_total.info "/test-suite/*" -o ~/ql_total_purged.info

• generate the html site: genhtml ~/ql_total_purged.info --demangle-cpp --output-directory ~/ql_coverage_report

# QuantLib User Meeting 2015

I gave two talks at this year’s QuantLib User Meeting. They were on

• Negative rates in QuantLib.

You can read Luigi’s full report on the conference here.

# Laplace Interpolation

Consider a (two dimensional) data matrix with some values missing and you want to fill the holes by interpolated values. One particularly simple but at the same time powerful method is called Laplace interpolation.

The easy explanation goes as follows: for each missing value take the average over the 4 surrounding values, i.e. of the entries above, below, left and right. This holds for the inner points of the matrix, special rules apply to the borders, see below. Of course, maybe some or even all of the surrounding values are missing, too. In general this leads to a system of linear equations one has to solve.

The more involved explanation: search for a harmonic function interpolating the non-missing data. A harmonic function $f(x,y)$ satisfies

$\left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}\right) f = 0$

from which the mean value property follows. This says that each function value is the average of the function values on or in a surrounding sphere $S$

$f(x,y) = \frac{1}{2\pi r} \int_{\partial S} f(x,y) ds = \frac{1}{\pi r^2} \int_{S} f(x,y) d(x,y)$

where $S = \{ (x',y') | (x'-x)^2+(y'-y)^2 < r \}$. That's deeper math of course. To find the harmonic function we have to solve its defining partial differential equation, which we discretize on the grid given by the data matrix itself assuming equidistant nodes (with distance say $h$). For an inner point we have

$\frac{f(x+h,y)-2f(x,y)+f(x-h,y)}{h^2} + \frac{f(x,y+h)-2f(x,y)+f(x,y-h)}{h^2} = 0$

which is the same as

$f(x,y) - \frac{f(x+h,y)+f(x-h,y)+f(x,y-h)+f(x,y+h)}{4} = 0$

matching the easy explanation from the beginning. For each inner point of the matrix we either get this averaging equation, or if the data point is known to be $z=z(x,y)$ simply

$f(x,y) = z(x,y)$

This way we collect $nm$ equations, if the original data matrix has $m$ rows and $n$ columns. The borders have modified equations, which I do not list here explicitly, but which are based on the conditions $f_{xx} = 0$ for the upper and lower border, $f_{yy}=0$ for the left and right border and $f_x+f_y=0$ for the four corners.

Here is a tentative implementation in QuantLib using the BiCGStab solver at its heart.

Let’s do some examples (the code can be found here, together with a gnuplot script generating the pictures.

The original data is generated by filling a $50\times 50$ matrix sample like this

    for (Size i = 0; i < N; ++i) {
for (Size j = 0; j < N; ++j) {
sample(i, j) =
sin((double)(i + 1) / N * 4.0) * cos((double)(j + 1) / N * 4.0);
}
}


By the way, the function generating the test data is not harmonic. Plotted as a heatmap it looks like this:

Now we delete $20\%$ of the points, randomly.

We reconstruct the missing values by applying Laplace interpolation, which is done as easy as this

    laplaceInterpolation(sample2);


where sample2 is the destructed matrix with missing values marked as Null. The function laplaceInterpolation replaces these values by interpolated ones and writes them back to the input matrix sample2.

The reconstructed matrix looks is this:

As beautiful as the original, isn’t it. Let’s delete more points and look and the reconstruction.

For $50\%$ we get

for $80\%$ it is

for $90\%$

and for $95\%$

Quite amazing. Let’s look at how well the method works if we do not delete points randomly, but erase a whole stripe at the center of the matrix.

If the stripe is $10\%$ of the height

for $20\%$

and for $50\%$

What about extrapolation? So let’s delete a stripe at the top of the matrix.

For $10\%$ we get

for $20\%$

and for $50\%$

Of course the method can not see what is no longer there. Still it is working surprisingly well if you ask me.

# Adjoint Greeks VI – Source Code Transformation Pt. II

I will be talking about Automatic differentiation beyond typedef and operator overloading at this year’s QuantLib User Conference. Registration is free, but seats are limited. The blog post today is sort of a teaser, so do not miss to be in Düsseldorf on November 30th / December 1st.

To recall what has been done to make QuantLib AD-ready: We have two approaches. One is by CompatibL which in essence changes the typedef for Real in the library from double to some active type of an operator overloading tool, like CppAD::AD. Of course this doesn’t work just by relinking the typedef, but in effect it describes well what is going on.

The other approach is to transform the library’s code and introduce template parameters whenever we might potentially be interested in calculating derivatives. You can read more about this by selecting the tag Automatic Differentiation in the sidebar that can be expanded by clicking on the three horizontal lines in the upper right corner when you scroll to the top of this page.

Backward compatibility in the template approach is achieved by typedef’ing the original class names to template instantiations with T = Real, so existing code compiles without any changes. Well let’s say 99% of it, some few modifications have to be applied, but really not many (from my experience with the original examples and the test-suite).

The transformation is not applied to the full library (yet), but the vanilla IRD stuff works, so coupons, yield term structures, volatility termstructures, the GSR model. You can get it (together with some working examples) from my adjoint branch at https://github.com/pcaspers/quantlib/tree/adjoint.

Both approaches have clear advantages and disadvantages. One potential problem with the typedef approach is, that this might kill your performance in parts of the code where you do not even want to use AD. For example,

    Matrix_t<T> A(1024, 1024);
Matrix_t<T> B(1024, 1024);
...
Matrix_t<T> C = A * B;


will take 764 ms on my machine with T=double, but 8960 ms with T=CppAD::AD. This seems to be due to compiler optimizations (like using SIMD instructions) that are no longer applicable when switching from native double to the custom AD type. This is a penalty of 11.7x and you do not get anything for that. Note that I do not record any tape or something, I just use the double functionality wrapped in the AD type.

Of course there might be other operator overloading tools that are more efficient. To get a lower bound for the penalty factor I replaced T by a minimal wrapper, just storing a double and an additional native pointer and forwarding the operators +,*,+= in question here to the respective double operations. With this wrapper the penalty is 2.0x. This is much better. The perfect operator overloading tool will probably lie somewhere between 2.0 and 11.7. But even 2.0 is not good, if you can avoid it, is it?

In any case the template approach allows for selective activation of variables where AD is needed and use native instances of the library classes elsewhere. You can also mix both. For example you could do a model calibration with the double-instance of a class, then extract the calibrated parameters and plug them into the AD – version of the same model, which then maybe generates Monte Carlo paths for some CVA calculations in which you want to calculate an adjoint gradient.

Still there are use cases left where even the template approach will not help and this is when you want to apply AD to code sections like the one above, where a lot of dense double calculations take place, enabling the compiler to apply optimizations that are not possible with custom types.

We saw an example in https://quantlib.wordpress.com/2015/04/14/adjoint-greeks-iv-exotics. The time equivalent of around 80 NPV calculations (holy cow!) is consumed for one adjoint gradient calculation of a Bermudan swaption in the Hull White model. From theory we would expect a complexity of 4x for the adjoint sweep, in practice we are probably happy with any factor up to 10x. But with 80 NPV calculations you can do a lot of things and even if your gradient vector has 100 – 200 entries, the application of AD would merely result in a speed up of around 2x compared to a naive bump and revalue approach. This would not really justify the whole AD-thing.

One possible way out of this is to write the adjoint code by hand.

Another approach is to use source code transformation. I started with a toy example in one of my last posts.

In the meantime I implemented a Bermudan swaption engine that produces an AD gradient w.r.t. the $H$ and $\zeta$ vectors describing an LGM model and the relevant discount factors of an interest rate curve. For this I extracted the computational core of the Gaussian1dSwaptionEngine (which relies on numerical integration) and rewrote it in Fortran. Then I applied OpenAD/F to the code to generate adjoint code and compiled everything into a library. You can have a look at the project here: https://github.com/pcaspers/quantlib/tree/master/QuantLibOAD/lgm.

On the QuantLib side I wrote a pricing engine acting as a wrapper for the Fortran core: It can be used just as any other pricing engine, but provides the adjoint gradient vector as an additional result. The gradient is computed w.r.t. the model parameters, but it is easy to tranform this into the usual market variables’ sensitivities. This is done exactly the same way as in the post I mentioned earlier.

Testing the engine on a 10y Bermudan swaption with yearly calls (using 49 grid points per expiry for the numerical integration) yields the following results (you can find the example code here)

• single pricing on a Intel(R) Core(TM) i7-2760QM CPU @ 2.40GHz, single threaded, non-transformed code: 4.2 ms
• pricing + gradient (which is 105 dimensional in this example): 25.6 ms
• transformation of gradient w.r.t. model parameters: 6.2 ms

Looks good, no? In particular, because no extra optimization of the input or transformed code was necessary to achieve the good multiple of 6.1, it just came out of OpenAD/F this way.

As a side note when implementing it as described here you can use the power of AD even from the original / “classic” QuantLib just through specialized pricing engines (and by linking against the Fortran support library of course). Or you can mix operator overloading in the template – QuantLib with source code transformation applied to critical sections of the code which slow down too much with operator overloading. Maybe this hybrid solution is the way to go.

More on this and other topics in Düsseldorf.

# Dynamics II

This is a sequel to the recent post on smile dynamics, generalizing things a bit. It’s just a stream of thoughts, nothing that is actually backed by own experience or anything. Hopefully it makes sense nevertheless.

The PL of a interest rate derivative portfolio reacts on movements in market data like broker quotes for FRAs, Euribor futures, swaps, implied volatilities for caps / floors and swaptions. These quotes are processed into rate curves and volatility surfaces, which are then fed into pricing models which value the portfolio and thereby produce its PL.

A trader is interested in how his portfolio reacts on market movements, i.e. in its greeks, which are by definition partial derivatives of the portfolio value by the single market data inputs. So a portfolio may for example be particularly sensitive on the 10y and 15y Euribor 6M swap quotes and on the 5y/15y and 10y/10y swaption volatilities. The trader would see a large interest rate delta on these two buckets of the 6M curve and a large vega in these two cells of the vega matrix for swaptions.

The input variables are by far not independent. For example if the 10y swap quote rises by 10 basis points, it is likely that the 15y swap quote rises too, and often even roughly by the same amount. Similarly, movements of the cells of implied volatility surfaces are correlated, in particular if their distance is not too big. Even more, an interest rate move will possibly affect the implied volatility, too, in a systematic way: If rates rise, typically lognormal volatilities will decrease, so to roughly keep the product of the forward level and the lognormal volatility (which is by definition the so called (approximate) basispoint volatility) constant. The dependency is obviously dependent on the nature of the volatility, so it will be quite different for normal implied volatilities.

Furthermore we have technical dependencies. Say you work with sensitivites on zero rates instead of market quotes. Then a movement in the EONIA zero rates will affect the Euribor 6M forward curve, because this curve is bootstrapped using the EONIA curve as a discounting curve. So a sensitivity on the EONIA curve might come from a direct dependency of a deal on the EONIA curve, or also from an indirect dependency via forward curves.

So the abstract picture is that you have tons of correlated data points as an input, a very long gradient vector describing the sensitivity of the PL to the input, and maybe also a big Hessian matrix describing second order effects.

Wouldn’t it be reasonable to apply a principal component analysis first, select the most important risk drivers and only manage the sensitivities on these drivers ? Conceptually this is nothing more that a suitable rotation of coordinates such that the new coordinates are independent. Once this is done you can sort the dimensions by the ratio of the total variance of the input data they carry and focus on the most important ones. Which may be only 5 to 10 maybe instead of hundreds on input data points.

Hedging would be be more efficient, because you could hedge the same amount of PL variance with fewer hedge positions. The main risks drivers could be summarized in terms of only a few numbers for management reporting. Sensitivity limits can be put on these main drivers. PL could be explained accurately by movements and sensitivites of only a few quantities. Stress tests could be formulated directly in the new coordinates, yielding more realistic total scenarios, since natural co-movements of other variables not directly stressed are automatically accounted for (e.g. huge rate shifts would imply appropriate adjustments to volatilities and keep them realisitic).

If you think about it, the kind of interest rate delta you see for swaptions and caps / floors this way would include a historically calibrated smile dynamics. Which you could compare to the dynamics we discussed last time. For example.

Also, typically the new coordinates allow for a natural interpretation like parallel movements of curves, rotations and so on, so the view wouldn’t necessarily be more abstract than the original one.

What is needed is the coordinate transformation layer, ideally directly in the front office system, so that it can be used in real-time. Plus a data history has to be collected and maintained, so that the principal components can be updated on a regular basis.

# Adjoint Greeks V – Source Code Transformation

In my last post on automatic differentiation we saw that the operator overloading approach in some situations (quite naturally) comes with a dramatic performance breakdown.

An alternative approach that might do better is called source code transformation. Here the code is differentiated at compile time. While there are quite a few operator overloading tools that work on existing code bases more or less out of the box, this does not seem to be the case for source code transformation tools.

I looked around quite a while but did not really find a mature framework for C++. There is one under development (ADIC 2.0) that should be mentioned. There is also quite an old version of this same project for C (ADIC 1.x).

There is a Fortran framework though called OpenAD/F. Quite interestingly this framework separates the abstract ad engine (“OpenAD”) from language specific parts (“/F”), using a XML representation for the computational core to be differentiated. So in princinple you can implement a front end for C++ replacing the Fortran front end and reuse the algorithms of OpenAD. This is actually what ADIC is doing.

Still I have this faint idea of using the intermediate representation (IR) of the LLVM compiler infrastructure to interface to OpenAD. There are already backends that can translate back IR to C, so the “only” thing left to do would roughly be the transformation of IR to XAIF and back and the implementation of a run time support library. Very roughly. I certainly do not have enough experience in both AD and the IR to really overlook this and also not the time to delve into this. Anyway.

I played around with OpenAD/F some months ago. I solved a toy pde for american option pricing with Fortran code written from scratch. The results were promising in terms of performance. So here is the plan how to benefit from that in a productive setting:

The idea is that a typical core pricing algorithm is actually not too big and can effectively be excavated from the rest of the code.

If this is the case we could reimplement this computational core in Fortran, apply OpenAD/F to differentiate it and use the differentiated code via a plain interface from QuantLib pricing engines to retrieve adjoint greeks.

This post aims to present a minimal example for that procedure to prove the technical feasibility. Later I will try to apply this to the Bermudan Swaption example I mentioned above.

The libraries are organized as the usual QuantLib shared object library and a Fortran shared object library that contains the differentiated code for the numerical cores. The minimal example library is called simplelibad (simple lib ad). An application will then typcially be an usual C++ program linking against QuantLib and the new ad library.

You can find the example code here. It comes with a make file that by default builds a differentiated version of the library functions. You can also run the target plain to build only the original functions without AD, which seems useful for testing the ported code against the original one before doing anything fancy.

Actually only one function is provided in our simplelib wich computes a discount factor from a given continuously compounded zero yield and a day count fraction. The coding looks as follows (sorry, no syntax highlighting)

subroutine discount(zeroyield, dcf, df)
implicit none
double precision:: zeroyield, dcf, df
!$openad INDEPENDENT(zeroyield) df = exp(-zeroyield * dcf) !$openad DEPENDENT(df)
end subroutine discount


To keep the interface simple no return parameters are present, the result is written to the last input parameter instead.

There are two OpenAD – specific lines starting with !\$openad which declare the active variables. The make file invokes a wrapper script openad that is shipped with OpenAD as a simple driver for the actual tools

openad -c -m f simplelib.f90


This produces the differntiated version of the above code. The option -m f sets the mode to forward. The resulting code is then compiled into an object file

gfortran -g -O3 -o simplelib.pre.xb.x2w.w2f.post.o -c simplelib.pre.xb.x2w.w2f.post.f90 -fpic


The same is done with some additional runtime support files provided by OpenAD

gfortran -g -O3 -o w2f__types.o -c w2f__types.f90 -fpic


Finally we need a driver that acts as the interface to QuantLib later on

subroutine discount_ad(zeroyield, dcf, df, ddf)
implicit none
external discount
double precision:: zeroyield, dcf, df, ddf
df = res%v
ddf = res%d


This returns the derivative together with the original result of the computation and does nothing more than invoking the differentiated code. We compile the driver as well

gfortran -g -O3 -o driver_simplelib.o -c driver_simplelib.f90 -fpic


and then everything is linked into a shared object library simplelibad

gfortran -shared -g -O3 -o libsimplelibad.so w2f__types.o OAD_active.o driver_simplelib.o simplelib.pre.xb.x2w.w2f.post.o


In the minimal example here we actually do not use any QuantLib classes, but directly talk to the ad library from our example application, like this

#include <iostream>
#include <cmath>

extern "C" void discount_ad_(double *zeroyield, double *dcf, double *df,
double *ddf);

int main() {

double zeroyield = 0.02;
double dcf = 10.0;
double df, ddf;

std::cout << "result1 = " << df
<< " should be: " << std::exp(-zeroyield * dcf) << std::endl;

std::cout << "result2 = " << ddf
<< " should be: " << -dcf * std::exp(-zeroyield * dcf)
<< std::endl;
}


The application can be compiled as any other C++ code, it just needs to be linked against simplelibad. Running the code yields the expected output

result1 = 0.818731 should be: 0.818731
result2 = -8.18731 should be: -8.18731


Now of course the real work starts … hope to be back with a meaningful example soon.

# Smile Dynamics by Densities

The term smile dynamics refers to a rule how an implied volatility smile behaves when the underlying moves. This rule can be estimated from actual historical data, implied by a pricing model or set up from scratch.

Today I am looking at some common specifications, but from a slightly different angle. I am not looking at the effect on the implied volatility smile, but rather on the underlying’s density in its natural pricing measure.

This can be done using the Breeden-Litzenberger Theorem which links the density $\phi$ of the underlying to non deflated call prices $c$ for options struck at strike $k$ by

$\phi(k) = \frac{\partial^2 c}{\partial k^2} (k)$

The idea is to have a more natural look at such dynamics – while implied volatility may be something we think we have an intuition for, a density seems to be a somewhat more grounded description of what is actually going on.

I am looking at an underlying from the interest rate world with forward level $3\%$ and options on this underlying expiring in 5 years. Think of these as european swaptions for example. I am assuming an implied volatility smile generated by the SABR model, in terms of lognormal volatilities. The observations easily translate to shifted lognormal volatility smiles, as they are currently used for some currencies due to the low rates.

The question is now, what happens to a smile if the underlying’s forward level rises to say $3.5\%$ or $4\%$ ? Nobody really “knows” this of course. But you have to make an assumption whenever you look at effects resulting from rate shifts. This assumption can be crucial for the effectiveness of your delta hedges. It is also of paramount importance for the specification of stress tests, where bigger rate shifts are applied. This will become clear shortly.

I am looking at five different rules for the smile dynamics:

• Sticky strike dynamics, meaning that after the rate shift the volatilities remain the same for each fixed strike
• Sticky absolute moneyness assigning the original volatility at strike $k-s$ to the strike $k$ if the rate shift is $s$
• Sticky model keeping the (SABR) model parameters constant, i.e. the volatilities after the shift are computed with the same SABR parameters as before, except for the forward, which is now $f+s$ instead of $f$ before
• Sticky approximate basispoint volatility keeping the approximate basispoint volatility, defined as the product $f\cdot\sigma$ of the forward $f$ and the implied lognormal volatility $\sigma$, constant. At the same time the volatility smile is moved in absolute moneyness space, meaning that the (lognormal) volatility at strike $k$ is $f\cdot\sigma_{k-s} / (f+s)$ after the rate shift.
• Sticky basispoint volatility keeping the exact basispoint volatility constant, defined as the volatility parameter of the normal Black model, in our case matching a price produced by the lognormal Black model. Again, the price at $k-s$ is relevant for strike $k$ after a shift by $s$.

Obviously, one would need some extrapolation rule for absolute monenyness and approximate basispoint volatility dynamics, for strikes below the shift size. We just ignore this here and draw the pictures starting at $s$ instead, this is interesting enough for the moment. Also note that for the sticky basispoint volatility dynamics for strikes below $s$ the implied basispoint volatility is zero.

Let’s start with a SABR smile given by parameters $\alpha=0.2, \beta=1.0, \nu=0.0 (, \rho=0.0)$. This is simply a Black76 model, so we have a flat, lognormal smile. It shouldn’t come as a surprise that in this case three of the five dynamics produce the same result. For sticky strike, sticky moneyness and sticky model we have the following underlying densities for the original rate level, the shift to $3.50\%$ and the shift to $4.00\%$.

The densities keep their lognormal nature and since the log volatility remains the same the effective absolute variance of the underlying rates get bigger when rates rise. Under big upshifts coming from a low initial rate this can lead to much too fat, unrealistic densities.

The sticky approximate basispoint variant produces a different result

keeping the lognormal nature, but adjusting down the log volatility for higher rates such that the absolute variance of the rates stays more (but not totally) constant than in the other alternatives, which is probably more realistic under big upshifts.

I do not have a picture for the last dynamics (sticky exact basispoint), but it is clear, that this dynamics actually just translates the density by $s$, preserving the shape exactly.

The second exapmle is again without stochastic volatlility (i.e. a pure CEV example, $\nu=0$), but now with $\alpha=0.015, \beta=0.3$ producing a skew

Sticky strike,

and sticky model

lead to densities again with slightly increasing absolute variance, while sticky approximate basispoint

roughly keeps the absolute variances constant. However there are some artefacts arising for low strikes. This is because formerly rather high volatilities belonging to low strikes are now applied to higher strikes. This is an obvious problem of this dynamics.

Sticky aboslute moneyness has similar problems

at the same time leading to higher absolute variances for higher rates.

Again, sticky exact basispoint volatility will just translate the density.

Now let’s add stochastic volatility and look at $\alpha=0.015, \beta=0.30, \nu=0.50, \rho=0.2$. The smile now looks like this

Sticky strike,

sticky model,

sticky approximate basispoint

show a similar qualitative behaviour as for the CEV case, the approximate basispoint dynamics again with the artefacts for low strikes. Similarly for sticky absolute moneyness

elsewhere keeping the absolute variance of rates quite the same across the shifts.

Sticky exact basispoint will again translate the original density by $s$.

What kind of dynamics matches reality best, is in the end subject to a statistical check on historical data (with all the difficulties in that procedure). Or what traders consider to be the most appropriate dynamics. If I had to choose from the alteratives above, sticky (exact) basispoint or sticky model seem most reasonable.