A couple of things on the bi-modal distribution to clarify my understanding. Assuming the underlying is not, say log-normal, but follows a n-modal distribution, is there any case where the returns distribution will not have n peaks. Also if the real world distribution is n-modal, is there any case where the risk-neutral transformation may not result in n-modal as well? Thx vm.

In my view, the "real world" distribution is never really n-modal, but when you transform it using the Radon-Nikodym derivative (i.e. multiply the distribution by [$]e^{-\lambda x}[$]) then the resulting "risk neutral" distribution is highly skewed and may be multi-modal. HTH

in fact I see most of the stock indices price level distribution in recent time (say last 1 year daily price) is bi-modal, sometime strongly so. The returns are also bi-modal, but less pronounced. But besides the facts, let's just assume the price levels are bi-modal.

There's a couple of related issues here that make the question somewhat ill-defined.Let's use 'P' as an abbreviation for 'real-world' and 'Q' for 'risk-neutral'.1. Under the 'usual theoretical setup' (P and Q stochastic processes exist), is there any reason for the modal type of a Q-distributionat some horizon to be related to the modal type of the P-distribution at the same horizon? AFAIK: no.2. Is the last year of daily returns a good proxy for a (forward-looking or backwards-looking) P distribution at a one-day horizon? My opinion: noSo, if such distribution was bi-modal, I'd say `so what?'; it is just a short-time realization. Having said that, if a broad-based index like the SPX is currently showing a forward-looking bi-modal Q-distribution at some horizon, (presumably inferred from an option chain), please post a chart -- would be interesting to see. If that kind of evidence existed, then the issue of the realizedbehavior over the last year might have some more weight.My two cents.

Last edited by Alan on March 9th, 2016, 11:00 pm, edited 1 time in total.

I agree the past distribution is no good proxy for forward. Also the implied distribution across asset classe (diversified equities, swaptions, fx) rarely is bi-modal. Not now for most I have checked. I think there is one reference of it in GBP/USD around the collapse of FRM mechanisms in early 90s I think (sorry I do not have the reference right now).The chain of thoughts comes from a separate source - more to do with distributional arbitrage. In times around big event that will be one big jump either way (i.e. it will not set in motions a series of moves or jumps) it is reasonable to expect the realized forward distribution will be bi-modal. Given the implied smiles never match that, it means in such rare cases (like ECB today, may not be very rare for single stocks around earnings) there are some over-pricing of the tails. And probably the fair tail will be different for speculators and market makers. See here for a bit more on thisThe question is does this still holds for even market makers. If a real world bi-modal outcome corresponds to a risk neutral one too, then probably it does, as the pricing error is in the wrong assumption of risk neutral distribution shape, irrespective of distribution parameters (i.e. vol parameters, whatever the model may be). The FX case above was such an event with a bi-modal expectaion. A thought!

The issue of jumps puts a different spin on the question. If there is a scheduled event with a jump distribution,then the P and Q distributions have to have the same support. Taking your example, suppose theP distribution will jump up or down but only to two possibilities, say up 5% or down 8%.Then, the Q distribution must also and *only* either jump up 5% or down 8%. The relative probabilities can be different under Q (as long as both are positive).If the Q-distribution allows *anything else*, there will be an arbitrage opportunity.That's because bets under the Q-distribution on *anything else* occurring will command a positive price under Q, but are impossible under P.In reality, there will be a continuum of outcomes. However, suppose we knew that the absolute value of the jump-return would be at least 2% under P.Then, the same property would have to hold under Q and so perhaps this is the way to, more realistically, make a bi-modal connection between the two distributions.

Last edited by Alan on March 9th, 2016, 11:00 pm, edited 1 time in total.

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