Suggested by nsande

once again just as a starting point:considering the implied distribution (cf Dupire, Derman, Kani) from the options market, it is well known that the distribution tails are fatter than a Normal one (leptokurticity) and there is an assymetry as wellthere are many different ways of dealing with this:- Jarrow and Rudd (and later Corrado and Su) use a Gram-Charlier series where there is a non-Normal correction for the moments of the distribution- Merton (and later Kou) add jumps to the Normal Diffusion. - Carr and Madan (as well as Geman and Yor) use pure jumps (so they reject Normal component altogether) in VG or CGMY- Stochastic Volatility is another way of dealing with the non Normalityothers?

when I said Stochastic Volatility, GARCH would fall under the same categoryalso all this is not just true for implied distributions from optionsit could be applied to time series as well

The normal distribution assumption seems to give a very poor discreption of financial markets. Some work done by Prof. Eberlein and his co-workers shows that time series of financial returns follow a pure jumps process, similar to those chosen by Carr et al. Have a look at Prof. Eberlein regards, Anton

Perhaps a good question to ask is: does it matter if real-world financial returns are not normally distributed?

wee it does for derivatives pricing formulaefor instance Black Scholes assumes Normality of log returns etc etc

QuoteOriginally posted by: JohnnyPerhaps a good question to ask is: does it matter if real-world financial returns are not normally distributed?Johnny, I would like to re-phrase your question, perhaps more theoretically, as follows:Suppose that financial returns are not normally distributed, but that the rest of the BS assumptions are still valid: continuous trading, no transaction costs, infinite short selling, constant volatility, etc, etc. Now suppose we incorrectly assume that the returns are normally distributed, use the BS equation to price options, and to dynamically hedge them. What will go wrong?My point is that, we are hedging on a trade by trade basis. If we see a big move (which is really more probable than we should expect, because the distribution is really Levy rather than Gaussian), we will attribute it to a rare event in the Gaussian distribution, never question its occurence, and go ahead and rebalance our delta in accordance with Gaussian BS. I know that something must go wrong, but I don't know what it is.

Omar - absolutely. I think this is a good way to get a clear theoretical view of what's going on. In addition, I also intended the question as a practical question. For example, when a market maker sells a call on, say, the S&P500, he will often buy an adjacent call or buy the equivalent put and sell stock on a 100%, etc etc. So in practice, the difference it makes (of assuming normal log returns) depends on what other derivatives exist to allow a market-maker to get the risk profile he wants.The original question of the thread "how good is the assumption of normal distribution" surely needs to be viewed in the context (a) of what alternatives there are and (b) how much difference it makes theoretically and practically speaking to assume the "wrong" distribution.

i.e. in a market with many derivatives it is possible for a market-maker to reduce substantially his exposure to his choice of asset returns distribution; in a market with few or no other derivatives, this is not possible. In a developed market it may not particular matter what distribution is assumed; in an emerging market - say the market for derivatives on hedge funds - it may not be possible to say much at all about the distribution so there is no better alternative.I wish the editing function worked on this forum.

My private concern is completely theoretical. I have another related question: Suppose we have two theoretically valid models of stock return movements: 1. Gaussian.2. Azema.All the rest of the BS assumptions are valid. Trader A chooses model (1). Trader B chooses model (2). Both A and B are capable of, and will hedge dynamically. Who will break even? who will not? Since models (1) and (2) are different, they must give different prices, so I don't see how both traders can both break even.

It cannot always be one or the other, otherwise I would sell myself options all the time and hedge my short position differently than my long position! So ... it must depend on circumstances. Unfortunately until I can see into the mists of the Azema process I can't say more. Martingale?

There is one more thing to add: by the central-limit theorem the P&L of traders A and B must tend to zero as the number of trades becomes large. So presumably the consequences of mis-specifying the distribution (a) are circumstantial and (b) only matter if the number of trades is small.

"How good is the assumption of normal distributions for financial returns?"The assumption of normal returns is descriptively unrealistic for many asset classes. For example, empirical studies provide strong evidence for kurtosis and weak evidence for skewness in share price returns.However, it is interesting to analyse the circumstances in which an options market-maker will systematically lose money by (falsely) assuming that log returns are normally distributed:1. If there exist other derivatives similar to the one being priced, it will be possible for the market-maker substantially to reduce his exposure to the (false) assumption of normally distributed log returns. "Hedging model risk".2. If the market maker is unable to "hedge" model risk in this way, but executes many trades in derivatives on assets with similar (but non-normal) returns distributions then the law of large numbers will diversify away the model risk. "Diversification of model risk".3. If the market maker is unable either to hedge or diversify away model risk, he may widen his bid-ask spreads to such a degree that no other counterparty with the same information will be interested in trading.Therefore the only circumstances in which a market maker should regularly lose money through (falsely) assuming a normal distribution for log returns is when he has less information than his counterparties but doesn't realise it. A rational trader that started losing money systematically might not understand in what respect he was wrong about the distribution of returns, but would nonetheless have an incentive to widen bid-ask spreads to the point where he was no longer losing money on new trades.It is also interesting to consider alternative assumptions:1. Models incorporating jumps or untraded quantities (stochastic volatility) require knowledge of risk preferences, which themselves are badly defined. There is therefore no a priori reason to prefer these models.2. Normal distributions are considered "unrealistic" at short horizons; non-normal stable Levy distributions are equally "unrealistic" at distant horizons. Once again, there is no a priori reason to prefer these models. In summary, the assumption of normal log returns allows the construction of models in which preferences can be ignored, is no less counter-factual than some other competing assumptions and doesn't lead to systematic losses unless the trader doesn't learn to adjust his bid-ask spreads when losing money!Any takers?

Not beeing a taker ... but an additional question: May you relate it to the volatility smile? Returns are ATM, options are not.

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