Dear lovenatalya, I would look at the following paper:  http://www.sciencedirect.com/science/article/pii/0304414987902018. This is not Hajek's original article ( B. Hajek,  Mean stochastic comparison of diffusions) but should provide some useful elements.  I would be more than happy to discuss about...

Hello Alan,
If you just write out the integrals, you have something of the form
$$
\int_{\mathbb{R}}\exp\left\{\alpha\int_{0}^{T}e^{z\sqrt{t}}dt\right\}\exp\left\{-\frac{z^2}{2}\right\}dz.
$$
As z tends to infinity, the integrand is positive and does not tend zero, for any $\alpha>0$.

Dear Fadai88,You could try related MSc in Business School, but in a Maths department, this sounds very unlikely. Typically, when we say "mathematical background", we mean some combination (at undergrad level) of analysis, functional analysis, ideally proba/stats, algebra.

<t>Hello lovenatalya, For fixed maturity, one way to potentially prove some results along these lines would be to use some (Hajek's) comparison theorem. The first step would be to see if the total variance increases/decreases (almost surely) with the vol of vol, and then transfer that to the stock p...

<r>In the case of models with jumps, the at-the-money smile depends on whether you have a Brownian component or not. This is treated precisely in the following paper: <URL url="http://arxiv.org/abs/1006.2294In">http://arxiv.org/abs/1006.2294In</URL> light of recent results, you may want, for precisi...

Quantuplet, thank you for your comment.Indeed, the power law seems to have nicer properties.As far as I have seen, though, if you take the SPX, for maturities not too small, the Heston function phi is also pretty good.PS: I know the remark on Page 14, as I am the other author.....

And I agree with Quantuplet's fit for the SSVI slice.Will check the other Python optimisers and let you know.One question though, why did you take this function phi and not the other function phi proposed in the paper?Best,

Just did the standard SVI fit (with a, b, rho, m, sigma) and I agree with Cuch that the parallel shift up, then down gives a perfect fit, with optimal parameters:a, b, rho, m, sigma = 0.4806313 0.08346917 -0.41060706 -0.13041738 0.25963651

At first sight, the L-BFGS-B does not seem to be able to catch the left wing. The rest of the fit seems ok, though.I will try more carefully tomorrow.Best,

I agree. I just check it afterwards.

<r>Dear Cuchulainn, Quantuplet,Jim G. does indeed use R.I personally use Python/IPython, with the L-BFGS-B constrained optimisation algorithm (<URL url="http://docs.scipy.org/doc/scipy-0.15.1/reference/generated/scipy.optimize.fmin_l_bfgs_b.html"><LINK_TEXT text="http://docs.scipy.org/doc/scipy-0.15...

Dear mbunea,Just out of curiosity, why this journal, and not a mathematical finance journal, which will be read by people working in the area?Best wishes,

<t>Dear JacobYW,I would say being in a top university is probably the most important factor. Then, the topic, your supervisor (his industry contacts) and your motivation are (not necessarily in that order) very important factors. Regards your list, it looks odd to put Imperial College Business Schoo...

Dear zukimaten,As Alan mentioned, models with jumps do not have a small-maturity smile, except at the money. This exact meaning is that, as the maturity tends to zero, the smile blows up to infinity. Or you need some rescaling in order to observe some genuine limit.Best,

<t>Hello,It may depend on what you are trying to achieve: are you calibrating a given model, for which you need to extrapolate or are you trying to calibrate some parametric function to the observed smile, and then extrapolate this function outside the observed range?Another possibility is to consid...