# Is it always better to use the entire distribution of a financial returns series, not just $\mu$ and $\sigma$?

In finance models that use historical returns for inputs, including option pricing models, forecasting and portfolio optimization, only the statistical moments of the returns distribution, $$\mu$$ and $$\sigma$$ (expected value, or mean, and standard deviation), are used as inputs because the moments summarize a return series' probability distribution (pdf). How strong is the argument that the user would be better off, and would get more accurate results, in using the data's entire pdf, instead of only $$\mu$$ and $$\sigma$$?

And would using the entire pdf also be better than models that try to extend to the third and fourth moments (skewness, kurtosis)? given that you could even create a distribution of the rolling skewness and rolling kurtosis of a return series, i.e. the distribution of each moment

• At least if you have a decent prior view about the right distribution and a short series of observations, it can be better to just estimate the few parameters of the theoretical distribution. Here the empirical distribution can be noisy.
– fes
Jul 21, 2020 at 6:31
• in finance, noisy observations are valuable though, otherwise we would be trimming away indicators of excess outperformance and financial crises. Models built on the signal alone (if any), without noise, would not be able to cope with the empirical world Nov 16, 2020 at 16:39
• For option pricing, you don't care about (you don't need) the actual historical volatility and the historical returns. You care about the cost of replicating the option via delta-hedging, and for that, you need the cost of borrowing money and the cost of borrowing the stock: these parameters would be captured by the (forward-looking) repo-rate, and money-market rate. You take the implied vol from quoted options, and interpolate the vol surface to price non-quoted strikes & maturities. In other words, you are completely oblivious to the past returns and volatility of the underlying. Nov 17, 2020 at 7:57

It depends.

For example, if you're doing option pricing in the log normal world returns are completely described by the mean and standard deviation. If you add jumps, you would also need to parametrize the underlying Poisson process which is fully described by one parameter and the jump size. In other words, if you have a (log)normal distribution and the mean and standard deviation you're using the complete distribution. Of course, there is no need to keep the parameters fixed over the whole time horizon. This is what local volatility and stochastic volatility models are all about.

If you're doing portfolio optimization taking into account more than mean and variance, you should definitely use higher moments and do so almost by definition.

For risk management taking other moments into account is common place. In the Gaussian world, the number of extreme events is heavily underestimated after all. Performing scenario analysis can be seen as specifying the return distribution and you're not really considering the statistical mean or standard deviation in any case.

If you want to use the empirical distribution of past returns, you can use filtered historical simulation.

• in portfolio optimization, how much more beneficial is it to use the entire pdf rather than the first four moments? and how can the entire pdf be captured in that application Jul 21, 2020 at 2:41
• If you use a known distribution, you already use the entire pdf. I think working with multivariate empirical distributions would be quite cumbersome in any case. If the number of dimensions goes up, observations will be far apart from each other giving a jumpy pdf. Maybe you can try to bootstrapping? Jul 21, 2020 at 10:26
• i think distributions of common portfolios are never known, so averaging bootstraps is often the case. the question then becomes whether the averaged bootstrapped pdf is better to use than moments alone, nevermind the fact that there's no way to target the averaged boostrapped pdf during optimization because it just forms on its own Jul 31, 2020 at 4:52
• I don't follow. Jul 31, 2020 at 6:45