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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.

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    $\begingroup$ Is there sufficient "content" … ? In general, opinion-based questions are off-topic on this site. I would appreciate it if people could give me some hints as to what I could look into. In general, list-based questions tend to be off-topic as well. $\endgroup$ Commented Feb 26, 2023 at 4:45
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    $\begingroup$ I’m voting to close this question because it is about quantitative finance not physics. You could try asking on the Quantitative Finance Stack Exchange as this is exactly the sort of thing they discuss there. $\endgroup$ Commented Feb 26, 2023 at 7:01
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    $\begingroup$ There are reviews in RMP and books written on the subject (e.g., there's a book by Voit, if I am not mistaken). So the answer is yes. Particularly what concerns statistical physics and the theory of random processes. The subject is however less rich than biophysics, because economics/finance are not really about physics - the things that a physicist brings there is good background in math and research experience, but almost no domain knowledge. $\endgroup$ Commented Feb 26, 2023 at 7:04
  • $\begingroup$ @RogerVadim how are you defining "domain knowledge" here? Because as a physicist working in finance, I know plenty of physics PhDs with ample knowledge of SDEs, finance/economics, etc (and not just knowing the math, but having worked with it as part of their dissertations). $\endgroup$ Commented Mar 1, 2023 at 17:17
  • $\begingroup$ @KyleKanos what I mean is that a biophysicist is actually studying physical processes, whereas a physicist working in finance is using the math background, which is not itself physics - one could acquire similar knowledge studying mathematics, or finance or another discipline. If we define physics as studying certain kinds of natural phenomena, then finance is not physics, even if one uses the same methods. No offense, just a matter of definitions. E.g., I am a computational biologist/bioinformatician, and I wouldn't call myself a biophysicist, because what I do is not physics. $\endgroup$ Commented Mar 1, 2023 at 17:37

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I don't think this is an opinion-based question as the answer is pretty straightforward: there are papers on papers on the dynamics of markets using all manner of stochastic differential equations. So there is more than enough content out there. Even just Black Scholes and similar models would be enough to write a thesis on themselves - there are lots of odds and ends to think about including the one you've mentioned.

That being said, you're at best on the verge of being off-topic and maybe beyond that point, for a physics site. So if you have any followups I'd recommend quant stackexchange.

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