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.
