QuoteOriginally posted by: alluMy idea is that mixing a Wiener and Poisson component does lead to two degrees of uncertainty. One we can deal with by a riskfree rate while the other remains an open question. Can you actually eliminate the Poisson uncertainty (jump risk) completely by a perfect hedge, and leave the Brownian uncertainty (diffusion risk) completely unhedged? A more general question: Both jump diffusion models and stoch vol models improve on BS by accounting for the vol smile (as far as I'm told), do we actually need both jump diffusion and stoch vol in the same model (as Alan does in his option calculator?). Isn't possible to parametrise the smile with one extra factor?

AlluI'm glad you find this thread interesting. It's the first of the Wilmott FAQ threads to discuss incomplete markets, so it's a good place to get some agreement on some of the issues that arise with incomplete markets.With regards to adding an extra security, I absolutely do not mean to use this to calibrate your model. I really am talking about a general equilibrium solution under the very specific assumptions of Cox and Ross, as described by Alan. If jumps of any size are allowed, I'm convinced that there is no number of extra securities that can be added that enables the market to be completed. Naturally, this case (jumps of any size) corresponds to the real-world.

Johnny,Indeed, as long as we do not specify a distribution for the jumps it will not be possible to complete the market with a finite number of securities. I completely agree with you. Thus, the question remains incomplete if we do not structure jumps. The simple point jump model allows only one jump size. If we extend this to Merton's lognormal jumps or Kou's double exponential jumps, we have structured the jumps as well and just a few parameters are needed to create jumps of random size. Then completing the market with some options would do.But when one chooses not to use options for such a model, the utility approach seems the only option left. Getting towards a preference-free pricing seems then infeasible. Right? So all I can do is to make some preference-based approach.Now, in Gerber and Shiu (1994, downloadable from www.soa.org in the Actuarial Library) it is conjectured the power utility function is the only risk-averse utility function which has a certain time consistency and thus a logical choice to make risk adjustments with. Does anybody know more literature concerning this conjecture or option pricing with the power utility function in general?- Allu

In fact, the market can be complete with jumps, (Azema martingale) see " http://www.orie.cornell.edu/~protter/finance.html" paper #7.For the utility based pricing, here is one source (others may be found probably in the references of his paper)" http://www.stochastik.uni-freiburg.de/h ... s/kallsen/".

One might argue that Poisson risk can only hedged by buying insurance. (If it could be hedged why would we need insurance companies?)This perhaps explains why relatively large margin requirements are required for those selling OTM puts.

Martingale, can you summarize those papers for the FAQs?!P

I will take a shot at here. "complete market with discontinous security prices" gives an example of pure jump martingales(with an set of parameters beta) named Azema martingales which has the predictable representation theorem( every contigent claim is repicable(redundant). The main difference of Azema martingales to Levy processes are that the jumps are intrinsic to the filtration, while Levy processes the jumps are extrinsic. The model includes Brownian driven martingales as special case(beta=0). Implementation is harder than BM case. Another nice thing is Azema martingales have something to do with asymmetric information and application to credit can be found in the same page in Philip's homepage. Jan Kallsen's work has things to do with hedging in incomplete market, since the market is incomplete, we cannot perfectly hedge the option, what can we do?we can find the "best possible hedge" in some way, it depends on how to define the best. Foelmer has the so called risk-minimizing hedging, and his desendent Schweiser and so on have done quite a lof work down the road; there are also the mean-variance hedging(better than risk minimizing but analytically harder); there are also the entropy based hedging(which through utility based hedging through exponential utility), several authors have done this.(find the reference in Jan's paper, I have trouble to spell it right). Jan's work is on very general semi-martingale driven security prices, and he derived very nice results on hedging strategy using the charercteristics of the semimartingale.

could somebody explain the practicalities of hedging this thing to a guy who doesn't work in the eqderiv business any longer?yes, I could wade through all the papers, but what's the idea? It's easy to set up a discrete tree with extra branches to accomodate the jumps, and then take the appropriate limits as dt -> 0. Then, you'd take a close look at the linear algebra that's needed to make the extra derivatives jointly non-redundant. I.e. if you have a jump up and a jump down, then I guess you'll need a tree that branches four times at each node, and you'll need two options to complete the market. But what do you specify - i.e. which market do you choose as the complete one? You could take sigma to be your diffusion vol, and then chose +/- 1 sigma strikes - might be especially smart when the jump component is taken to be a multiple of sigma greater than one. I'm thinking from the really practical arbitrage POV. Let's back up a moment and say that we model our underlying as Black-Scholes. If we observe a smile, then you might assume that the wings options are redundant securities because their vols are high relative to ATM. If this were really true, then you'd sell the wings and hedge like a maniac until you finally... found the underlying process to be inadequate. Is this what we're supposed to do if we find a JD underlying model that we like? I.e. we take spot and our +/- 1 sigma options, and we see that there are 'mispricings'. So we use a portfolio of the three instruments + a rates hedge, and then replicate like mad? That is, replicate until we're mad because either we decide that the underlying params were incorrectly calibrated, or that the model itself is wrong?

kr,Suppose near OTM Put options are cheap (oversold) and far OTM Put options are very expensive. Suggest a way to take advantage of this situation without excessive Backspread risk?-newton

QuoteOriginally posted by: newtonkr,Suppose near OTM Put options are cheap (oversold) and far OTM Put options are very expensive. Suggest a way to take advantage of this situation without excessive Backspread risk?-newtonOf course, I meant ratio (vertical) spread risk (not backspread). (I don't seem to be able to edit anything posted to this forum.)

I think people should be aware of the works of Svetlana Boyarchenko and collaborators on option pricing under Levy processes (brought to my attention by Alan)http://www.eco.utexas.edu/facstaff/Boyarchenko/She has also co-authored a recent book on the subject published by World Scientific. I haven't seen it, but I think Alan has it. Maybe he can tell us something about it.

QuoteOriginally posted by: Omar<blockquote>QuoteA more general question: Both jump diffusion models and stoch vol models improve on BS by accounting for the vol smile (as far as I'm told), do we actually need <u>both</u> jump diffusion <u>and</u> stoch vol in the same model (as Alan does in his option calculator?). Isn't possible to parametrise the smile with one extra factor?I think the jury is still out on this. Each class of model has its distinct properties that are hard for the other to reproduce.For example, stoch. vol. models are good at capturing vol. mean reversion, so that when you get in a very volatile period, theshort-dated options can be priced at a high vol. but not the same 3 years out. Or the so-called "leverage effect", that forequities like the SPX, lowers the implied vols. for the higher strikes, among other things. (see the smile charts in my article).However the continuous sample path models (stoch. vol.) have a hard time with very short dated options. Mathematically, that whole class has OTM option values decaying something like e^(-c/T) as expiration approaches (don't quote me).But for jump-diffusions you get option value decaying like c T (think about Merton's formula, or the prob. of a jump).Well, when you try to match a c T decay to Black-Scholes, which has the exponential decay, then the only way it works isfor the implied sigma to go through the roof! And that's the way the market seems to price it. Having said all that, remember I just said "hard", not impossible. You can coerce s.v. models into pretty steep smiles,for example. Or make lambda mean revert (although I would call this a jump-diffusion with a stochastic parameter, similar to s.v.)

Alan,Which model would give you a vol smile that flattens as expiry date gets further away in time? Is there a simple/natural way to get that?

Another source of very interesting works on Levy processes in finance is Wim Schoutens sitehttp://www.wis.kuleuven.ac.be/applied/wim.htmHe mentions that a book on the subject will appear in 2003.

QuoteOriginally posted by: OmarAlan,Which model would give you a vol smile that flattens as expiry date gets further away in time? Is there a simple/natural way to get that?all the models do that (s.v. & jump-diffusions). In fact, the market smile curves flatten rather slowly,so you want the models to do that.