Why Finance Models Use Normal Returns and How They Depart from Them

# Why is it so rare for finance theory to depart from the normal distribution?


# Why is it so rare for finance theory to depart from the normal distribution?












I understand almost all of the theory that has been built upon in quantitative finance is based on the normal distribution, and obviously you wouldn't want to throw all of it out the window on a whim but since stock returns are clearly not as normal as we like to believe why is it so uncommon to treat them as not being normal. For example, could you not as easily simulate a price process based on, say, a student T random distribution as well as a normal one? Looking at stock returns we often see that a student T distribution fits it better than a normal one so how come something like this is never done (at least I have never seen it done)?

Would is the biggest limitation that prevents us from going further with these ideas?

## Answer by Stéphane (score 8, accepted)

https://quant.stackexchange.com/a/51827

The Geometric Random Walk: The Starting Point

Let me begin by being a little more specific. The simplest, yet relatively sound model of asset prices that we have is this one: \begin{equation} ln S(t+1) = \mu - \Psi_{t+1}(-1) + ln S(t) + \epsilon(t+1), \; \epsilon(t+1) | F_t \sim N(0,\sigma^2). \end{equation} where $\Psi_{t+1}(u) := ln E_t \left( \exp( -u \epsilon_t) \right)$ is the log conditional MFG of the normal distribution and $F_t$ is the natural filtration of the process. You normally do not see the term $\Psi_{t+1}(-1)$ in a geometric random walk, but I appended this term because it ensures: \begin{equation} E_t\left( \frac{S(t+1)}{S(t)} \right) := E_t(R(t+1)) = \mu. \end{equation} In essence, the log MFG is a convexity correction. Unsurprisingly, when $\epsilon(t+1) \sim N(0,\sigma^2)$, we have $\Psi_{t+1}(-1) = \sigma^2/2$. In continuous time, we have the equivalent to a geometric Brownian motion: \begin{equation} \frac{dS(t)}{S(t^-)} = \mu dt + \sigma dW(t) \end{equation} where $(W(t))_{t \geq 0}$ is a standard Brownian motion under the physical measure. If you apply Ito's Lemma to move toward $dlnS(t)$ you will see that convexity correction term appear as well. This way, everything is treated the same way.

The nice properties:

- Over any future interval of time, expected returns are just compounds of $\mu$. One thing we do know about financial markets is that first conditional moments are hard to estimate, so it's not stupid to just rule it out;

- This ensures stock prices never become negative;

- Prices, conditional on today, are log-normally distributed, so you have a bit of an heavy tail thing going on. However, it does say that returns are conditionally normally distributed.

- The first two moments summarizes the normal distribution, so you can more or less equate risk with variance and, since it is subadditive, you immediately have an obvious advice: diversify.

That's probably why the first tight framework for the valuation of European options was constructed under a simple geometric Brownian motion (Black and Scholes, 1973). But people were quick to work on departures.

A Few Common Examples of Departures from Conditionally Normal Arithmic Returns

Heston (1993) proposed to model the dynamic of stock prices using a stochastic volatility model where the volatility followed an Ornstein-Uhlenbeck process where both Brownian motions were correlated. This model takes into account the fact that volatility estimates seem to exhibit temporal dependance and that they tend to be negatively correlated to returns (at least for stock market indexes). Note that, now, returns no longer are conditionally normal because they are build from a mixture of two normal distributions. Going forward, there is a window where the model will build sknewness and kurtosis.

Cool addition: As for the Black-Scholes-Merton world of geometric Brownian motion, Heston's model allows for a quasi-closed form European option pricing formula. In fact, all models which allows for an exponentially linear conditional MFG of log prices allow this as well.

Duan (1995) proposed a GARCH model for option prices. The model uses normal innovations and since for a GARCH model the conditional variance is known one step ahead, returns are conditionally normal only for one period ahead. Like the Heston (1993) model, this builds up conditional non-normality over time as past shocks enter the variance dynamic in a non-linear manner.

Bakshi, Cao and Chen (1997) have a nice (and famous) study where they compare the pricing and hedging performance of many option pricing models. They have time-varying conditional volatility like Heston, but they also look into adding jumps. When you add jumps in continuous time (or heavy tailed innovations as in the IG-GARCH model of Christoffersen, Heston and Jacobs (2006)), you have an immediate departure from conditional normality that can be quite severe, even in the very short run unlike in GARCH and SV models where it sorts of builds up over time.

As a sidenote with regards to option pricing, conditional nonnormality has consequences on risk-neutralization. Christoffersen, Elkamhi, Fenou and Jacobs (2010) showed that it is one way to force a variance risk premium at all horizons in a GARCH model (otherwise, the Q and P conditional expectations of variances only diverge over time) and a negative VRP is a pervasive empirical feature (it's one way to resolve a variance forecasting puzzle).

Conclusion

I barely scratched the surface for one very narrow part of financial literature and we have many examples of people departing from models that impose the conditional nonnormality of arithmetic returns. It's not absurdly complicated to talk about these things. If you know a bit of econometrics and a bare minimum of stochastic calculus, you can learn these things just by reading the relevant papers.

## Answer by demully (score 2)

https://quant.stackexchange.com/a/51849

The benefits of normal/lognormal have been well-described already. The problems with normal/lognormal, especially in the tails, are equally almost-universally known and appreciated.

They persist because they make a variety of applied derivative problems easily solvable (and easily adaptable to similar but slightly different problems).

Imagine you could derive a model that perfectly described the distribution of asset returns. It says you should expect this much skew and that much kurtosis next week. Which would give you a X% probability of a Y% drawdown.

So what then is the probability of a (Y+a)% drawdown next week, (Y+a)% next month, (Y+b) next week and (Y+b) next month? How would you price (why anyone would want to is moot, but they do) an exotic autocallable one-touch double reverse gobbledegook option using his model? And then manage the risk in terms of time decay, price sensitivity and appropriate equivalents to the Greeks in your model?

The point here is that normal/lognormal is almost right or "close enough most of the time" (at least for stocks and commodities, less-so bonds, and not credit). So it gives fast, scalable and almost-right answers to the kind of questions above, and myriad others.

The alternatives might be slightly more accurate historically. But: (1) this doesn't actually reduce the inherent uncertainty about the true nature of the distribution of returns. (2) they struggle to give answers to practical questions. Or at least, it's so darn computationally expensive, the marginal questionable precision just doesn't seem worthwhile. (3) if the application of any model/distribution requires any subjective human finger-in-air estimate, the technical advantages versus shortcomings of the various models you might use won't represent the significant source of error in the first place! This alone puts some premium on model simplicity, which favours (log)normality.

## Answer by Martin Vesely (score 1)

https://quant.stackexchange.com/a/51843

You are right that many models are based on normal distribution or log-normal distribution which are connected. It seems that there is three reasons (of course, maybe more) for using normal distribution:

- During "good times" price differences and other variables really behave according to a normal distribution

- You can easy calculate with normally distributed variables

- More philosophical reason: any manager know what is Gauss bell-shaped curve

Mainly in times after crises, fat-tail distribution and Black swan events start to be disccused. However, to use such distribution is not as straighforward as in case of normal distribution. Moreover, fat-tailed distribution have some counterintuitive featues, for example they do not have standard deviation or even average. See family of so-called stable distributions.

In past there were some attempts to switch from normally distributed variables to some observed on markets in reality. I would recommend an article New methods in statistical economy writen by Benoit Mandelbrot in 1963 for more information on this.