Okay, maybe the title is somewhat misleading. My university calls this a BSc Project, but it is limited to between 4000 and 6000 words, so it isn't particularly long. Anyhow, one of the projects offered is on the subject of econophysics, which seems quite interesting to me. However, I'm a little bit stuck with regard to what kind of direction this project could take, and whether there is sufficient physics to write a good report on. Based on some reading, beginning with the Black-Scholes (B-S) model of options pricing would be a good place to start (it also allows for the introduction of a random walk -> Brownian motion -> geometric Brownian motion, and stochastic calculus). Additionally, there are the parallels between (B-S) and the heat PDE.
One assumption in B-S is that returns are assumed to be distributed as per a normal distribution. Fitting to empirical data generally shows this not to be the case, because in reality, "crashes" and other significant events are more likely at the tails of the distribution, and thus "fat-tail" distributions are better suited (at least empirically) to model these phenomena. I can apply goodness of fit tests to determine which distribution works best, but from what I can see, based purely on empirics, it seems to be lévy-stable distributions which are favoured (though their variance is undefined, and for sufficiently long periods of time, stock returns converge to a normal distribution anyway).
Additionally, BS also assumes the volatility, $\sigma$, to be constant. I believe this is a result of the efficient market hypothesis which essentially states that stock prices have the Markov property, and thus future stock prices depend only on the current price, and not historical prices (or more generally, new information entering the system is assumed to instantaneously be incorporated into the stock price, therefore not allowing investors to gain an advantage). In reality, there are information asymmetries, which means that the volatility is variable and in fact when implied volatilities (determined by inverting B-S) with the same expiration dates are plotted, a "volatility" smile is observed.
Beyond this, I'm struggling to find new material to research. It seems not much has progressed in the last 20-30 years. I would appreciate it if people could give me some hints as to what I could look into.
