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Implied volatility expansion at 0 for BNS model
Dear willmott forum,In e.g. "Options on realized variance by transform methods: a non-affine stochastic volatility model" by Gabriel Drimus, he considers an implied volatility expansion for Heston and the 3/2 model (by the way, does that model have another name? It is impossible to google). Is there done any similarly in the BNS-model or related superposition models? Is it non-sense to consider?Thanks
Implied volatility expansion at 0 for BNS model
Generally, models with jumps do not have a t->0 limit for the implied vol. Re googling 3/2 model.Often my book gets referenced. So you could google "3/2 model, Lewis, stochastic volatility" to find stuff. 
Last edited by Alan on March 3rd, 2015, 11:00 pm, edited 1 time in total.
Implied volatility expansion at 0 for BNS model
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,
Implied volatility expansion at 0 for BNS model
Hi,Sorry to wake up this old question. How come this is so? Is it a problem of a model to not have this limit defined?
Implied volatility expansion at 0 for BNS model
No, it is a feature, not a bug. It makes it *easier* to fit short-dated smiles.The reason it happens is that, if the particle can jump from, say [$]S_0[$] to the interval [$](K,\infty)[$] with [$]K > S_0[$], then one finds(*) [$]P(S_t > K |S_0) \sim C_1 t[$] as [$]t \downarrow 0[$]. On the other hand, for a diffusion you find(**) [$]P(S_t > K |S_0) \sim e^{-C_2/t}[$] as [$]t \downarrow 0[$]. Trying to force the behavior (*) into the straightjacket of (**) essentially doesn't work. Trying to do so, you find [$]V^{imp}(S_0,K,t) \rightarrow \infty[$] for [$]K > S_0[$] as t approaches 0. Ditto for any other strike [$]K \not= S_0[$] if the particle can jump to a neighborhood of K.
Last edited by Alan on March 31st, 2015, 10:00 pm, edited 1 time in total.
Implied volatility expansion at 0 for BNS model
Cool. Thanks for the explanation Alan
Implied volatility expansion at 0 for BNS model
I've run into this topic once again and new questions has arisen. What happens ATM? - is all this written down somewhere?
Implied volatility expansion at 0 for BNS model
For diffusion models of the form [$]dS_t = \omega S_t dt + \sigma_t S_t dW_t[$], where [$]\sigma_t[$] is stochastic and independent of S,the asymptotic T->0 smile theory is well-developed. It is known that the ATM [$]\sigma_{imp} \rightarrow \sigma_0[$]. Page 131 in "Option valuation under stochastic volatility" has some discussion. The issue is more rigorouslydiscussed in my upcoming Vol II. There is literature, but not at my fingertips. For a time-homogeneous jump-diffusion, heuristically, the process never "jumps in place", so the answer would be the same.Again, I don't have a cite. If you have a particular numerical solution, you could try to confirm the jump-diffusion guess numerically to see if it looks plausible. It can be explicitly checked in, say, Merton's jump-diffusion. I can remember doing this once, but I haven't re-done it to answer yourquestion, so there is always the possibility I am mis-remembering the result.
Last edited by Alan on September 2nd, 2015, 10:00 pm, edited 1 time in total.
Implied volatility expansion at 0 for BNS model
Your conclusion is the same as what I have recently heard waking my interest. This part of the mathematical finance world which only in the clever guys' heads is pretty interesting to get a view into I am looking forward to the book!
I am looking forward to the book! Implied volatility expansion at 0 for BNS model
Thanks --it is very close. I just have to do the keyword index, send it out for comments, and then get it printed.
Implied volatility expansion at 0 for BNS model
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: http://arxiv.org/abs/1006.2294In light of recent results, you may want, for precision, to specify "Ito diffusion": in the case of stochastic volatility models, say, where the volatility process is driven by a fractional Brownian motion with Hurst index < 1/2 (the so-called "Short-memory"), you do see small-maturity explosion of the smile as well. Again, except at the money, exactly for the same reason.Alan, looking indeed very much forward to Volume II!!!Best,
Implied volatility expansion at 0 for BNS model
Antonio, thank you for the kind book mention. p.s. While cleaning out a room, I just found some old notes of relevance to this thread. After posting it, I am reminded of a more careful approach in Figueroa-Lopez & Forde. In fact, it looks like my hand-written scribbles at the bottom were attempting to check my result with theirs.(Frankly, I don't remember).Anyway, I will leave my notes posted as maybe they are of use regarding zukimaten's orig. question.
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Last edited by Alan on September 6th, 2015, 10:00 pm, edited 1 time in total.
Implied volatility expansion at 0 for BNS model
Hi Alan and Antonio. Thank you very much for your references, that seems like very interesting reads!