Hi! As we know that to implement the Heston Model, we need the complex logarithm log(z)=log|z|+i(arg(z)+2*pi*n). But Matlab, Excel and C++ usually restrict the logarithm to its principle branch by restricting arg(z) in [-pi, pi] and setting n=0. This makes the log-function discontinuous at the cut along the negative real axis. But we need the log-function to maintain continuous in this model. Does any one know how to deal with this problem and the way of numerical implementation of complex logarithm? Many Thanks!

See "Not-so-complex logarithms in the Heston model", by Christian Kahl and Peter Jackel

Another paper dealing with this topic is "Why the rotation count algorithm works" , where the equivalence between the rotation count algorithm and another (more numerically stable) formulation of Heston's characteristic function is shown.

The solution and the proof of it can be found in our paper : The Little Heston trap. The paper is available on my webpage -http://www.schoutens.be/HestonTrap.pdf -and will be published shortly.The solution to the accuracy problem is extremely simple ...

The representation as a shifted contour integral by the aid of the Carr-Madan formula (essentially using the Lewis-Gatheral-etc way of expressing the characteristic function), using an optimal shift alpha (as in Lord-Kahl), plus the conversion to a finite integral using the asymptotic behaviour (as in Kahl-Jaeckel), plus an adaptive integration technique such as adaptive Gauss-Lobatto, is possibly the most efficient way of solving for the vanilla option price.Merry Christmas,Peter

The best way is to use the Lewis-Lipton formula instead of the original Heston representation.With the Lewis-Lipton formula we do not have the problem with branch-cuts of the complex logarithm.The proof of this is actually quite simple. I wrote a paper on this issue which will be published shortly.If anybody is interested I can post a preprint (which I do not have at hand at the moment).Moreover the Lewis-Lipton formula has nice convergence properties, so that it is totally sufficient to use standard numerical integration.

I'd be very interested to read that if you have a preprint. The advantage of that representation is that it uses an alpha/damping factor in the region [-1,0], which means that the corresponding moment - the (alpha+1)st moment has to be finite - is guaranteed to be finite, so it's an excellent default value. However, you will find out that for options that are out-of-the-money and or have short maturity, the integrand will become very oscillatory. For these cases choosing the optimal contour is very welcome indeed...As for standard vs. adaptive numerical integration, in principle it's fine to use a fixed no. of abscissae with standard Gaussian quadratures once you've chosen the optimal contour, but if you want a robust routine, you should use adaptive integration techniques.

I just posted a preprint on ssrn: "Modern Logarithms for the Heston Model".

GoGoFa, I have not checked, whether it is the same formula as in Gatheral.For that it is true as well (I think: Schoutens et al, "The little HestonTrap"). It may be worth to note, that one can use Maple to work it out inway that complex arithmetics is almost eliminated. That's what I did fora Excel implementation, see this thread.But the problem of possible ugly oscillations remain and besides uselessshort times a nice example is given by enginkuru in that thread.

I was just reading Gatheral book, and was wondering about the conjecture he talks about (which is also mentioned in the paper by Kahl and Jäckel) about the argument of the complex log. Nayone has an idea on how to prove (or maybe disprove) this ?I also used "The Little Heston trap" trick, which does work very fine.

GoGoFa:If I read your paper correctly, your proof is just for the Lewis formula (i.e. alpha = -1/2 in Carr-Madan terms), and also under the restriction that rho <= 2*kappa/omega. The proof in our paper also covers this case.Also, see my recent post in this thread (applied to the VG model) and this paper (for examples from Heston) for a demonstration that alpha = -1/2 can certainly be improved upon.Antonio:the papers mentioned in this thread deal with the conjecture that Gatheral mentions.

Roger, if you read my paper even more correctly, you will note that I emphasize that i) the proof is simple and intuitiveii) it covers all relevant cases in practiceLook at the second figure and you will see that putting additional effort to the proof it can probably extended to the whole parameter space, however I don't care, sinceI have not yet met any application for those parameter ranges.

GoFaFa, LordG: It seems your papers don't cover the case of Heston with time-dependent parameters. Would appreciate if you could share some idea on this model.

GoFaFa, LordG: It seems your papers don't cover the case of Heston with time-dependent parameters. Would appreciate if you could share some idea on this model.

Hi all. I'm currently dealing (PhD thesis) with the consistent pricing of SPX and VIX options. As you may know, to reproduce the positive skew of VIX options the Heston model doesn't work well. So I'm working with the SVJJ of Duffie et al. (2000) which features correlated jumps in price and in volatility. Sepp (2008, Risk) showed jumps in volatility are probably the good way to get the positive skew.Now my, question, I've found both of yours papers very interesting and fruitful, but I'm wondering whether some of you tackle the (similar) problem of the complex logarithms due to jumps in the SVJJ model characteristic function.I asked this question in another post: characteristic function SVJJ model - complex logarithm before finding this one. So apologize for multiple posts.Thanks