Second Cumulant of Heston

I recently stumbled upon an error in the various papers related to the Heston Cos method regarding the second cumulant. It is used to define the boundaries of the Cos method. Letting phi be Heston characteristic function, the cumulant generating function is: $$g(u) = \log(\phi(-iu))$$

And the second cumulant is defined a: $$c_2 = g’’(0)$$Compared to a numerical implementation, the c_2 from the paper is really off in many use cases.

This is where Maxima comes useful, even if I had to simplify the results by hand. It leads to the following analytical formula: $$c_2 = \frac{v_0}{4\kappa^3}{ 4 \kappa^2 \left(1+(\rho\sigma t -1)e^{-\kappa t}\right) + \kappa \left(4\rho\sigma(e^{-\kappa t}-1)-2\sigma^2 t e^{-\kappa t}\right)+\sigma^2(1-e^{-2\kappa t}) }\\+ \frac{\theta}{8\kappa^3} { 8 \kappa^3 t - 8 \kappa^2 \left(1+ \rho\sigma t + (\rho\sigma t-1)e^{-\kappa t}\right) + 2\kappa \left( (1+2e^{-\kappa t})\sigma^2 t+8(1-e^{-\kappa t})\rho\sigma \right) \\+ \sigma^2(e^{-2\kappa t} + 4e^{-\kappa t}-5) }$$

In contrast, the paper formula was:

I saw this while trying to calibrate Heston on a bumped surface: the results were very different with the Cos method than with the other methods. The short maturities were mispriced, except if one pushed the truncation level L to 24 (instead of the usual 12), and as a result one would also need to significantly raise the number of points used in the Cos method. With the corrected formula, it works well with L=12.

Here is an example of failure on a call option of strike, spot 1.0 and maturity 1.0 and not-so-realistic Heston parameters $$\kappa=0.1, \theta=1.12, \sigma=1.0, v_0=0.2, \rho=-0377836$$ using 200 points:

Update March 2014 - this is now described in my paper Fourier Integration and Stochastic Volatility Calibration.

Maxima for Symbolic Calculus

A few years ago, I found an interesting open source symbolic calculus software called Xcas. It can however be quickly limited, for example, it does not seem to work well to compute Taylor expansions with several embedded functions. Google pointed me to another popular open source package, Maxima. It looks a bit rudimentary (command like interface), but formulas can actually be very easily exported to latex with the tex command. Here is a simple example:

(%i14) D(x):=sqrt((lambda-rho*eta*x)^2+(-x^2+x)*eta^2);
(%i15) G(x) := (lambda - rho*eta*x - D(x))/(lambda - rho*eta*x +D(x));
(%i16) tex(taylor((1-exp(-t*D(x)))/(1-G(x)*exp(-t*D(x)))*(lambda - rho*eta*x - D(x)),x,0,3));

$$-{{\left(e^{t,\lambda}-1\right),\eta^2,x}\over{2,e^{t,\lambda},\lambda}}+{{\left(\left(4,e^{t,\lambda},t,\eta^3,\rho+\left(4,\left(e^{t,\lambda}\right)^2-4,e^{t,\lambda}\right),\eta^2\right),\lambda^2+\left(\left(-4,\left(e^{t,\lambda}\right)^2+4,e^{t,\lambda}\right),\eta^3,\rho-2,e^{t,\lambda},t,\eta^4\right),\lambda+\left(\left(e^{t,\lambda}\right)^2-1\right),\eta^4\right),x^2}\over{8,\left(e^{t,\lambda}\right)^2,\lambda^3}}+{{\left(\left(8,\left(e^{t,\lambda}\right)^2,t^2,\eta^4,\rho^2-16,\left(e^{t,\lambda}\right)^2,t,\eta^3,\rho\right),\lambda^4+\left(16,\left(e^{t,\lambda}\right)^2,t,\eta^4,\rho^2+\left(-8,\left(e^{t,\lambda}\right)^2,t^2,\eta^5+\left(16,\left(e^{t,\lambda}\right)^3-16,\left(e^{t,\lambda}\right)^2\right),\eta^3\right),\rho+16,\left(e^{t,\lambda}\right)^2,t,\eta^4\right),\lambda^3+\left(\left(-16,\left(e^{t,\lambda}\right)^3+16,\left(e^{t,\lambda}\right)^2\right),\eta^4,\rho^2+\left(-16,\left(e^{t,\lambda}\right)^2-8,e^{t,\lambda}\right),t,\eta^5,\rho+2,\left(e^{t,\lambda}\right)^2,t^2,\eta^6+\left(-8,\left(e^{t,\lambda}\right)^3+8,e^{t,\lambda}\right),\eta^4\right),\lambda^2+\left(\left(12,\left(e^{t,\lambda}\right)^3-12,e^{t,\lambda}\right),\eta^5,\rho+\left(2,\left(e^{t,\lambda}\right)^2+4,e^{t,\lambda}\right),t,\eta^6\right),\lambda+\left(-2,\left(e^{t,\lambda}\right)^3-\left(e^{t,\lambda}\right)^2+2,e^{t,\lambda}+1\right),\eta^6\right),x^3}\over{32,\left(e^{t,\lambda}\right)^3,\lambda^5}}+\cdots $$ Regarding Taylor expansion, there seems to be quite a few options possible, but I found that the default expansion was already relatively easy to read. XCas produced less readable expansions, or just failed.

Martin Odersky teaches Scala to the Masses

I tried today the Scala courses on coursera by the Scala creator, Martin Odersky. I was quite impressed by the quality: I somehow believed Scala to be only a better Java, now I think otherwise. Throughout the course, even though it all sounds very basic, you understand the key concepts of Scala and why functional programming + OO concepts are a natural idea. What’s nice about Scala is that it avoids the functional vs OO or even the functional vs procedural debate by allowing both, because both can be important, at different scales. Small details can be (and probably should be) procedural for efficiency, because a processor is a processor, but higher level should probably be more functional (immutable) to be clearer, easier to evolve and more easily parallelized.

I recently saw a very good example at work recently of how mutability could be very problematic, with no gain in this case because it was high level (and likely just the result of being too used to OO concepts).

I believe it will make my code more functional programming oriented in the future, especially at the high level.

Setting Values in Java Enum - A Bad Idea

Making Classic Heston Integration Faster than the Cos Method

A coworker pointed to me that Andersen and Piterbarg book “Interest Rate Modeling” had a chapter on Fourier integration applied to Heston. The authors rely on the Lewis formula to price vanilla call options under Heston.

More importantly, they strongly advise the use of a Black-Scholes control variate. I had read about that idea before, and actually tried it in the Cos method, but it did not improve anything for the Cos method. So I was a bit sceptical. I decided to add the control variate to my Attari code. The results were very encouraging. So I pursued on implementing the Lewis formula and their basic integration scheme (no change of variable).

My impression is that the Lewis formula is not so different from the Attari formula in practice: both have a quadratic denominator, and are of similar complexity. The Lewis formula makes the Black-Scholes control variate real (the imaginary part of the characteristic function is null). The Cos formula looks quite different, but it actually is not that much different as the Vk are quadratic in the denominator as well. I still have this idea of showing how close it is to Attari in spirit.

My initial implementation of Attari relied on the log transform described by Kahl-Jaeckel to move from an infinite integration domain to a finite domain. As a result adaptive quadratures (for example based on Simpson) provide better performance/accuracy ratio than a very basic trapezoidal rule as used by Andersen and Piterbarg. If I remove the log transform and truncate the integration according by Andersen and Piterbarg criteria, pricing is faster by a factor of x2 to x3.

This is one of the slightly surprising aspect of Andersen-Piterbarg method: using a very basic integration like the Trapezoidal rule is enough. A more sophisticated integration, be it a Simpson 3/8 rule or some fancy adaptive Newton-Cotes rule does not lead to any better accuracy. The Simpson 3/8 rule won’t increase accuracy at all (although it does not cost more to compute) while the adaptive quadratures will often lead to a higher number of function evaluations or a lower overall accuracy.

Here is the accuracy on put options with a maturity of 2 years:

I had to push to 512 points for the Cos method and L=24 (truncation) in order to have a similar accuracy as Attari and Andersen-Piterbarg with 200 points and a control variate. For 1000 options here are the computation times (the difference is smaller for 10 options, around 30%):

Here is the accuracy on put options with a maturity of 2 days:

All methods used 200 points. The error is nearly the same for all. And the Cos method takes now only 0.02s. The results are similar with a maturity of 2 weeks.

Conclusion

The Cos method performs less well on longer maturities. Attari or Lewis formula with control variate and caching of the characteristic function are particularly attractive, especially with the simple Andersen-Piterbarg integration.

Better Fonts in Fedora than in Ubuntu

cd /etc/fonts/conf.d

ln -s /usr/share/fontconfig/conf.avail/10-autohint.conf

ln -s /usr/share/fontconfig/conf.avail/11-lcdfilter-default.conf

yum install freetype-freeworld

Attari, Lord-Kahl & Cos Methods Comparison on Heston

I recently wrote about the Cos method. While rereading the various papers on Heston semi-analytical pricing, especially the nice summary by Schmelzle, it struck me how close were the Attari/Bates methods and the Cos method derivations. I then started wondering if Attari was really much worse than the Cos method or not.

I noticed that Attari method accuracy is directly linked to the underlying Gaussian quadrature method accuracy. I found that the doubly adaptive Newton-Cotes quadrature by Espelid (coteda) was the most accurate/fastest on this problem (compared to Gauss-Laguerre/Legendre/Extrapolated Simpson/Lobatto). If the accuracy of the integration is 1e-6, Attari maximum accuracy will also be 1E-6, this means that very out of the money options will be completely mispriced (might even be negative). In a sense it is similar to what I observed on the Cos method.

“Lord-Kahl” uses 1e-4 integration accuracy, “Attari” uses 1E-6, and “Cos” uses 128 points. The reference is computed using Lord-Kahl with Newton-Cotes and 1E-10 integration accuracy. Well here are the results in terms of accuracy:

As expected, Lord-Kahl absolute accuracy is only 1E-5 (a bit better than 1E-4 integration accuracy), while Attari is a bit better than 1E-6, and Cos is nearly 1E-7 (higher inaccuracy in the high strikes, probably because of the truncation inherent in the Cos method).

The relative error tells a different story, Lord-Kahl is 1E-4 accurate here, over the full range of strikes. It is the only method to be accurate for very out of the money options: the optimal alpha allows to go beyond machine epsilon without problems. The Cos method can only go to absolute accuracy of around 5E-10 and will oscillate around, while the reference prices can be as low as 1E-25. Similarly Attari method will oscillate around 5E-8.

What’s interesting is how much time it takes to price 1000 options of various strikes and same maturity. In Attari, the charateristic function is cached.

The Cos method is around 7x faster than Attari, for a higher accuracy. Lord-Kahl is almost 8x slower than Cos, which is still quite impressive given that here, the characteristic function is not cached, plus it can price very OTM options while following a more useful relative accuracy measure. When pricing 10 options only, Lord-Kahl becomes faster than Attari, but Cos is still faster by a factor of 3 to 5.

It’s also quite impressive that on my small laptop I can price nearly 100K options per second with Heston.

Giving Fedora Another Chance

I have had some stability issues with the Ubuntu 13.04 on my home computer, not on my laptop. It might be related to hard disk encryption (out of curiosity I encrypted my main hard drive in the installer option, resulting in a usable but quite slow system - it’s incredible how much the hard drive is still important for performance). I did not have any particular issue on my work laptop with it.

Anyway I gave Fedora 19 a try, with Gnome Shell, even if I am no particular fan of it. So far so good, it seems more stable than my previous Fedora 17 trial, and I get used to Gnome Shell. It’s quite different, so it takes a while to get used to, but it is as productive as any other env (maybe more so even).

It made me notice that Ubuntu One cloud storage is much less open that it seems: it’s extremely difficult to make it work under Fedora. Some people manage this, I did not. I moved to owncloud, which fits my needs.

Julia and the Cumulative Normal Distribution

I just stumbled upon Julia, a new programming language aimed at numerical computation. It’s quite new but it looks very interesting, with the promise of C like performance (thanks to LLVM compilation) with a much nicer syntax and parallelization features.Out of curiosity, I looked at their cumulative normal distribution implementation. I found that the (complimentary) error function (directly related to the cumulative normal distribution) algorithm relies on an algorithm that can be found in the Faddeeva library. I had not heard of this algorithm or this library before, but the author, Steven G. Johnson, claims it is faster and as precise as Cody & SLATEC implementations. As I previously had a look at those algorithms and was quite impressed by Cody’s implementation.The source of Faddeeva shows a big list (100) of Chebychev expansions for various ranges of a normalized error function. I slightly modified the Faddeva code to compute directly the cumulative normal distribution, avoiding some exp(-xx)exp(xx) calls on the way.Is it as accurate? I compared against a high precision implementation as in my previous test of cumulative normal distribution algorithms. And after replacing the exp(-xx) with Cody’s trick to compute it with higher accuracy, here is how it looks (referenced as “Johnson”).I also measured performance on various ranges, and found out that this Johnson algorithm is around 2x faster than Cody (in Scala) and 30% faster than my optimization of Cody (using a table of exponentials for Cody’s trick).

The COS method for Heston

Fang, in her thesis, has the idea of the COS method and applies it to Heston. There are several published papers around it to price options under various models that have a known characteristic function, as well as to price more exotic options like barriers or bermudans.

The COS method is very close to the more standard Heston quasi analytic formula (use transform of characteristic function for the density and integrates the payoff with the density, exchanging summation), except that the more simple Fourier series are used instead of the standard Fourier transform. As a consequence there are a few more approximations that are done related to the truncation of the domain of integration and the result is already discrete, so no need for a Gaussian quadrature.

In practice, the promise is to be faster. I was wondering how stable it was, especially with regards to short maturities/large strikes.

It’s quite easy to code, I made only one mistake initially: I forgot to handle the first element of the sum differently. It is however very unstable for call options prices, because the upper integration boundary is then used in an exponential, which explodes in most cases I have tried, while for put options, the lower boundary is used in an exponential, and the lower boundary is negative.

So one has to rely on the put-call parity formula to compute call prices. This means that we are limited to something around machine epsilon accuracy and can’t compute a very out-of-the-money call price, contrary to the Lord-Kahl method. However it seemed stable for the various Heston parameters I have tried and accurate as long as the resulting price is not too small as the following graph shows.

I was surprised to see that the more in-the-money put options also have inaccuracy: the price given is actually less than the final payoff. This is related to the domain of truncation. If I double it (L=24 instead of L=12), those disappear, what remains is that OTM puts can’t go beyond 1e-12 for the COS method.

In practice the COS method was effectively 2x to 3x faster than my Lord-Kahl implementation. As a side note, on this problem, Java is only 2x faster than Octave.

As long as we don’t care about very small option prices, it is an interesting alternative, especially because it is simple.

Update April 2014 - There is more information on the subject in my paper at http://papers.ssrn.com/abstract=2362968