I think there are two sorts of the effect of noise on the model prediction. One is the case of self-averaging, as Paul described. Another one is the noise amplification, which can happen in recursive schemes such as implied trees. Now, in the case of implied trees there is a kind of self-averaging as well - it is when we calculate the option price by taking the convolution of the implied distribution with the payoff function. However, if the integral is dominated by the region where the effect of noise on the distribution is particularly strong (from my experience, it is the vicinity of the maximum of implied distribution), the resulting pricing error will be large.

Scholar,your points are good ones.Re. the max_ent approach: we didn't force the tree to match the option values exactly but used a goodness of fit criterion (which weighted at-the-money and somewhat out-of-the-money options more heavily) at the optimization stepRe. the difference between discrete (nodal) transition probabilities: I think your intuition is spot on, and to be honest I don't really know why the Derman tree was so damn stable ...Hari

I never actually implemented the Derman-Kani tree but my colleagues who have, told me that (stability issues aside) the resulting deltas are off ... is that true?

PatCompletely agreed with you that a local surface extracted from vanilla prices misprices dynamic hedging of gamma products, particularly touches. If I believed in MC'ing the local surface to price exotics I'd be the biggest seller of double knockout options around.What I find interesting is that you appear to have calibrated a stochastic vol model to the market. It has been my experience that a Heston-type model calibrated to the vanilla market still mis-prices exotics in the market in the presence of high vanilla skew (really not to be expected when it seems people only favour the model due to its analytic tractability).I find myself now in a more 'heuristic' pricing world, combining a local surface solution with first passage time distributions extracted from market prices for touches. Has anyone out there had real success modelling the underlying process and getting it to match vanilla and extoics markets? ThanksFindus

Has anyone out there had real success modelling the underlying process and getting it to match vanilla and extoics markets? The JP Morgan FX research crew published a short article in Risk a year or two ago - they were using a local vols * stochastic variable model (stoch variable follows Heston).They calibrated their model to the vanillas, then compared the pricing for one touches to market one touch prices (this is in FX, where one touches are really quite liquid). They got pretty close.That said, it's a hideously complex model that takes a long time to fit and has a gazillion parameters (like all local vol variants). Also, there isn't a unique parameter fit to the vanillas - you need to choose the Heston process parameters, then back out the (current) local vols to match the vanillas. So it's not too surprising that they were able to fit the one touches as well. What's not clear, as this thread has mentioned, is how good a risk management tool it is. I've always been skeptical, but you never know.I'm not sure if they're still using it after the merger/purge - anyone else know?

Dupire's local vol model, Derman-Kani implied tree model: dS = (drift)dt + A(t,S)dWwhere A(t,S) is calibrated so that one predicts the correct implied volatility smile. Our experience is that these models are easy to fit and fitted parameters are stable, at least if one does not try to "overfit" the data ... basically at any given expiry we only extract an ATM vol, a skew, and a smile.BUT the hedges turn out to be unstable (worse than stright Black-Scholes!).To illustrate the problem, suppose we have an asset with a current price of 100. Suppose the implied volatility smile is a parabola with its minimum at 100. (Case chosen so I can work out details by hand). We calibrate A(t,S) so that our predicted option prices reproduce the parabolic implied volatility. Now that we have A(t,S), let us turn time on. Suppose due to normal market movements the asset price moves up to, say, 105. What does our model predict for the new implied vol? Using our calibrated A(t,S), we find that it predicts that the smile is a parabola with its minimum at 95!!! And if the asset price moved to 95, the parabola would move so that its minimum is at 105. IMPLIED TREE MODELS and other local volatility models predict that the Smile moves in the Opposite direction as the price of the underlying asset.(Don't take my word for it; try it out!)This is nonsense. In real markets, when the price and volatility smile shift up and down together.Now consider what this means for hedges. Our price for a European option with strike K is V = BS(S, sigma(K,S))where BS is Black-Scholes formula and sigma is the implied volatility ... as indicated, under our model it depends not only on the strike (the smile) but also on the current price S. If we compute our delta hedge, we get delta = BS delta + BS vega * dsigma/dSSo the "improvement" in the hedges come from the vega * dsigma/dS term. But the local volatility (implied tree) model predicts change-in-sigma-due-to-S changing that is exactly the opposite to the changes that actually occur, so this "improvement" has teh right magnitude but the wrong sign!I haven't tried local vol models on high gamma exotics, but it could be very bad indeed.

We use the SABR model, which is a bit different from Hestons:If R is the swap rate, we use dR = a * R^beta *dW1 da = nu * a * dW2with dW1 * dW2 = rho dtThis model is dead easy to use for plain vanilla options, because there exists a closed form formula for the implied vol sigma as a function of a, nu, beta, and rho.This formula is not exact, but it is very, very, accurate.The beta parameter and rho control the skew (vanna risk), while nu (the volvol) controls the smile (volga risk). Of course, today's value of a controls the ATM vol. Since both beta and rho control the skew (and we want to avoid overfitting) we usually set beta (to 0 or 1/2 or 1 depending on our theory du jour), and then fit a, rho, and nu to the smile/skew.In fixed income derivatives, we seldom sell barriers and other high-gamma options.

Dupire's local vol model, Derman-Kani implied tree model:
...
so that its minimum is at 105. IMPLIED TREE MODELS and other local volatility models predict that the Smile moves in the Opposite direction as the price of the underlying asset.
(Don't take my word for it; try it out!)
>>

Sorry to resurrect this thread so much after the fact, but I'm looking at building this sort of model (based on Chriss / Haug). While I'm not doubting your results, I'm slightly confused as to how you generated the skew at underlying prices different than that at which you calibrated the surface for and why this is going on at all.

I'm guessing that one would build the tree and generate a new set of option prices for a different underlying price by applying the deltas and using the new set as inputs for the calibration of the tree at a different underlying price (Is this a reasonable way to do things, BTW? The only other approach I could think of was an FD to give you multiple underlying prices at time 0).

My naive thought was that this approach would give a vol surface that roughly follows its own tangent, though I haven't really thought this out as much as I need to. Given that, the results you're getting seem counterintuitive. Anyway, wanted to get some feedback before I plunge in to implementing something that will wind up broken anyway. Thanks in advance.

--Pete

pfein@pobox.com

Local vol models: For simplicity, let F be the forward price of the asset (in the forward measure). Then F is a martingale. Thus
dF = something dW
Local vol model assumes that "something" is Markovian. So
dF = A(t,F)FdW , F(0) = f
where f is today's value and A(t,F) is the "local" vol.

For simplicity, assume A = A(F). Then
(*) dF = A(F)F dW.

Suppose that somehow we knew the correct A(F) (we will calibrate it below). Consider a European option with strike K. If we use MC or finite differences (or singular perturbation techniques!) we can find the correct dollar price of the option, and from this dollar price we could find the implied Black volatility that would give the same price. One would find that
sigma(K) = A(0.5[f+K]) + ...
Ie, to leading order the implied vol is the local vol halfway between today's forward and the strike.
(WARNING: this formula is not very good near a minimum of A(F); one needs to go to the next order!!) Now suppose that when the forward F is f, we observe the implied volatility curve
sigma(K). then picking A(F) to match the observed volatility curve yields A(F) = sigma(2F-f):

dF = sigma(2F-f)FdW

Now that we have our model calibrated, suppose we turn time on. Then the underlying is going to move up or down to some new value f'. The new implied volatility smile curve for the European options is then

sigma(K) = A(0.5[f'+K]) = sigma(K + f'-f) + ...

This means that if f increases to f', the smile curve moves to the left; if f decreases to f', then the smile curve moves to the right. Ie, the smile moves in the opposite direction as the forward.

One can clean this argument up by going to higher order, or carrying out the calculations exactly using finite difference methods, but the conclusion is the same. One can also develop much better perturbation schemes based on the average of A(F) between f and K instead of A at the average point (f+K)/2.

One also finds out that, under local vol models, that the volatilitiy increases (regardless of whether f increases or decreases) when one is near a local minimum

I would like to ask the following very general (and probably very naive, sorry for my ignorance) question. A general tree can be viewed in two alternative ways: either as a discrete approximation to the stochastic process, or simply as a grid of points (which maybe be discrete in time, but space continious) defining a set of conditional transition densities of interest. Given a specific method to solve the linear inverse problem (i.e. calibration) such as minimum entropy method, transition densities can be uniquelly fixed, but the step from conditional densities to an implied stochastic process is ambiguous: only you we assume the validity of local volatility assumption, the correspondence is one to one. Going over to stochastic volatility models, there is much freedom out there as many different models can match the same data and the same conditional transition densities. My question is: what are the reasons that force us to go to the language of volatility (local, stochastic) instead of staying with implied transition probabilities? I understand that one reason is purely historical (reference to BS), and also there are instruments such as volatility swaps where the former formulation is more natural. But are there other reasons ?

I guess it's also useful to see the connection between the local vol (implied from the options) and instantaneous vol from for instance a stochastic vol model ...

Is that your answer ? I was asking a different question

what you actually think I read everything you write???
Dude, I've got work to do !

But your response was pretty fast...
The next time I post on this forum, I start it like this: "Guys, I've really got lots of work to do and no time to read what you wrote here, but I think that ..."
Just kidding, just kidding

but seriously you asked the valid question:what are the reasons that force us to go to the language of volatility (local, stochastic) instead of staying with implied transition probabilities? >>and I thought one reason was to be able to compare local vol to stochastic volatility models. how would you do this in probability language?
you can't compare implied distribution (Dupire) to a two-factor model ... or are you saying we can?