How Good is the Assumption of Normal Distributions for Financial Returns?

Short answer

The answer has to be 'it depends.' It depends on the timescale over which returns are measured. For stocks over very short timescales, intraday to several days, the distributions are not normal, they have fatter tails and higher peaks than normal. Over longer periods they start to look more normal, but then over years or decades they look lognormal.

It also depends on what is meant by 'good.' They are very good in the sense that they are simple distributions to work with, and also, thanks to the Central Limit Theorem, sensible distributions to work with since there are sound reasons why they might appear. They are also good in that basic stochastic calculus and Ito's lemma assume normal distributions and those concepts are bricks and mortar to the quant.

Example

In Figure 2.14 is the probability density function for the daily returns on the S&P index since 1980, scaled to have zero mean and standard deviation of 1, and also the standardized normal distribution. The empirical peak is higher than the normal distribution and the tails are both fatter.

On 19 October 1987 the SP500 fell 20.5%. What is the probability of a 20% one-day fall in the SP500? Since we are working with over 20 years of daily data, we could argue that empirically there will be a 20% one-day fall in the SPX index every 20 years or so. To get a theoretical estimate, based on normal distributions, we must first estimate the daily standard deviation for SPX returns. Over that period it was 0.0106, equivalent to an average volatility of 16.9%. What is the probability of a 20% or more fall when the standard deviation is

Figure 2.14: The standardized probability density functions for SPX returns and the normal distribution.

0.0106? This is a staggeringly small 1.8 • 10-79. That is just once every 2 • 1076 years. Empirical answer: Once every 20 years. Theoretical answer: Once every 2 • 1076 years. That's how bad the normal-distribution assumption is in the tails.

Long answer

Asset returns are not normally distributed according to empirical evidence. Statistical studies show that there is significant kurtosis (fat tails) and some skewness (asymmetry). Whether this matters or not depends on several factors:

• Are you holding stock for speculation or are you hedging derivatives?

• Are the returns independent and identically distributed (i.i.d.), albeit non normally?

• Is the variance of the distribution finite?

• Can you hedge with other options?

Most basic theory concerning asset allocation, such as Modern Portfolio Theory, assumes that returns are normally distributed. This allows a great deal of analytical progress to be made since adding random numbers from normal distributions gives you another normal distribution. But speculating in stocks, without hedging, exposes you to asset direction; you buy the stock since you expect it to rise. Assuming that this stock isn't your only investment then your main concern is for the expected stock price in the future, and not so much its distribution. On the other hand, if you are hedging options then you largely eliminate exposure to asset direction. That's as long as you aren't hedging too infrequently.

If you are hedging derivatives then your exposure is to the range of returns, not the direction. That means you are exposed to variance, if the asset moves are small, or to the sizes and probabilities of discontinuous jumps. Asset models can be divided roughly speaking into those for which the variance of returns is finite, and those for which it is not.

If the variance is finite then it doesn't matter too much whether or not the returns are normal. No, more important is whether they are i.i.d. The 'independent' part is also not that important since if there is any relationship between returns from one period to the next it tends to be very small in practice. The real question is about variance, is it constant? If it is constant, and we are hedging frequently, then we may as well work with normal distributions and the Black-Scholes constant volatility model. However, if it is not constant then we may want to model this more accurately. Typical approaches include the deterministic or

local volatility models, in which volatility is a function of asset and time, a(S, t), and stochastic volatility models, in which we represent volatility by another stochastic process. The latter models require a knowledge or specification of risk preferences since volatility risk cannot be hedged just with the underlying asset.

If the variance of returns is infinite, or there are jumps in the asset, then normal distributions and Black-Scholes are less relevant. Models capturing these effects also require a knowledge or specification of risk preferences. It is theoretically even harder to hedge options in these worlds than in the stochastic volatility world.

To some extent the existence of other traded options with which one can statically hedge a portfolio of derivatives can reduce exposure to assumptions about distributions or parameters. This is called hedging model risk. This is particularly important for market makers. Indeed, it is instructive to consider the way market makers reduce risk.

• The market maker hedges one derivative with another one, one sufficiently similar as to have similar model exposure.

• As long as the market maker has a positive expectation for each trade, although with some model risk, having a large number of positions he will reduce exposure overall by diversification. This is more like an actuarial approach to model risk.

• If neither of the above is possible then he could widen his bid-ask spreads. He will then only trade with those people who have significantly different market views from him.

References and Further Reading

Mandelbrot, B & Hudson, R 2004 The (Mis)Behaviour of Markets: A Fractal View of Risk, Ruin and Reward. Profile Books

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