Is there a modern theory for the probability distribution of stock returns?<p>It is relatively easy to deduce that under idealized conditions stock returns follow a log normal distribution. One arrives at this by considering the product of ratios of prices ("stock returns"), applying a natural logarithm to convert the product into a sum and then applying the Central Limit Theorem under the condition that the ratios are iid (independent and identically distributed) and have finite mean and variance.<p>The problem is of course that we cannot just assume that returns are iid or that they have finite variance. So I am seeking alternative theories that try to address these shortcomings.<p>I am aware of Mandelbrot's Multifractal Model of Asset Returns. Is this considered SOA in the field? Is there something else that is considered a better model or easier to work with?
For synthetic distributions Mandelbrot's Multifractal is good enough.<p>You may consider using real historical distributions derived from real data. You can record different distributions from different time periods and different economic conditions.