Hello all!I'm currently working on implementations of the Heston and Schöbel/Zhu (Stein & Stein with non-zero vol-asset correlation) stochastic volatility models. As we all know, there is a problem with the complex logarithm in these models, but we also know the algorithm for log branch correction and can thus keep the logarithm continuous.Well, the specific problem I unfortunately see myself confronted with is, that I'm using time-dependent parameters, which are chosen piecewise constant. Thus, a solution to the system of coupled ODEs needed to recover the moment generating function can not be obtained in one single step. Instead I have to solve it iteratively over the different time buckets. But how do you treat the arising complex logs in this setting? My first intuitive try was to adopt the branch correction algo for each time bucket, so that over each fixed interval the logs remain continuous. This does not seem to work properly, though. I must be missing something....Does anyone have experience with time-dependent stochastic vol models, or, better yet, has anyone implemented them or the like and can help me out?The slightest hint or clue will indeed be very much appreciated!Thx

I have solved tme-dependent volatility for one-factor mdoels (barriers). Use exponentail FDM fitting.Have you thought of solving the Heston PDE using ADI and/or spliting methods?Personanly I find the ODE way of doing things in prcing a bit unfounded (it's usualy an ANSATZ or assumption). You should confront the PDE head-on by using FDM, FEM or similar.Are you using the 1993 Heston article?

Time-dependent coefficients in SV models are best handled using homogenization methods. See for example here (you can disregard most of the stuff about forward Libor models and just look at Section 6, and in particular Sec 6.3 for the time-dependent volatility). as for the relative merits of ODEs vs PDEs, I think one should go for the most numerically efficient algorithm possible. And when ODE methods are applicable they are typically much superior to a brute-force ADI scheme-V

Don't agree.We use splitting methods and ADI methods that have proved themselves both in USA and Soviet Union since the late 1950's, both mathematically and numerically, people like Douglas, Peaceman, Rachford, Mitchell, Gourlay, Janenko, Marchcuk D'yakonov all has worked on these problems and have come up with efficient schemes. These guys worked in oil and defense.ADI and spliitting are very easy to program, a sequence of 1 d problen with tridiagonal matrices. Your conclusions are too emotive, for example the remark about ADI. Personally, if you want to tallk about brute force, try the binomial method. All statements need to be backed up by facts, for example:Is scheme based on rigorous maths or an ANSATZ?Stability of the schemeAccuracy of schemeFinally, transforming to another equation is a bit like cheating IMO, you still have to tranform back. Another assumption (which may be OK) is that the model is affine, then ...I go from the worst case, i.e. nothing can be transformed. So head on tackle.

My conclusions are based on comparing the numerical complexity of the two methods. To price a European option in an SV model using Lewis's method requires nuemrically computing a Fourier integral (complexity O(Nz)) where the integrand at each point is computed by solving an ODE in the time domain (complexity O(Nt)). Thus total required operations is O(Nz Nt). To solve a PDE would require O(Nt Nz Nx) operations, or roughly Nx times slower. Here Nx is number of discretization points for the underlying, Nz for the stoch vol process, and Nt for the time. Typically Nx should be of order 50 to 100 to get a reasonable space resolution so a direct nuemrical solution of the PDE is 50 times slower. -V

I'll have a look at the Lewis formula, where can I find it.When you say O(NX, NS, NT) what do you mean exactly? Linear increase in execution time, product etc.? Are you basing the estimate on what source?So, if Lewis takes 1 second to calc, then ADI takes 50 seconds?I'll get back on this but I that your estimates are a bit 'tongue in cheek'. Have you taken into account?the number of operations needed to calculate the special functions?Truncation and roundoff?

Well, yes, the ODE way usually does involve an ansatz, or the somewhat anticipated "guessing" of the solution's form in advance.Working with piecewise constant parameters allows for an, though iterative, but analytical solution of the arising Riccati-type ODE and thus the whole system over each time bucket. So, to get the MGF, I merely have to evaluate functions. Then I'm left with the numerical integration of the Heston integrals. To this end I have implemented an FFT algorithm also using several "smoothing" methods as proposed by Carr & Madan. Also tried "regular" Gauss-integration of the integrals, because I thought the complex log correction might screw up the FFT. Still, the results seem to be wrong.Admittedly I have not yet tried to tackle the problem via FD or similar. But given the analytical solutions and the FFT algo, I can not imagine FD being faster or more efficient. I do not really dare to make a judgement, though, because I just haven't tried FD in this case.Thx a lot, piterbarg, for the link, although I have already been well familiar with the paper beforehand (and it certainly is on my 'TO DO' list for things to try out). Yet, using DD, it is supporting a somewhat different "philosophy" of smile/skew generation. For now I'm stuck with stochastic vol à la Heston and Schöbel/Zhu, where the skew evolves due to asset-vol correlation.It is actually only the latter model that seems to give me problems with the complex log. For a reason I do not know yet, the Heston case seems to work fine..... (???)

> Still, the results seem to be wrong.Are these accuracy problems?

I am working on a FDM Spltting method for the Heston PDE. Does anyone have any good input for this problem?thx

Bit of an old thread - but in "Why the rotation count algorithm works" we show that you can rewrite the characteristic function for the Schöbel-Zhu model in "Heston" form, and therefore either use the rotation count algorithm or the more numerically stable form of the characteristic function (e.g. the one that Duffie/Pan/Singleton and Gatheral use), which automatically avoids any problems with the complex log.

The problem is in fully discussed and solved 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 ...

Having spent more time on "Why the rotation count algorithm works" rather than "The Little Heston Trap" it would seem the proofs provided apply only to the time-homogeneous case and so don't specifically accommodate for the subject of this thread - correct me if i'm wrong please