Sorry if this equation is not phrase in precise mathematical form. I am open to suggestions to improve the explanation, and I have tried to formulate the problem as precisely as I could.
I was talking to a friend--postdoc in PDEs--today and he was asking me about obtaining some Yahoo data on stock prices for his students. He explained that he had some students working on stock price data and he was looking for software packages/libraries for time series analysis--like ARIMA, GARCH, ARMA, ETS, etc.
I told him that as far as I remember, most many models to stock prices or asset prices use stochastic differential equation like the Black-Scholes models. So I was asking if he actually needed like a stochastic differential equation solver, rather than a time series package.
He said something that surprised me. He said that many people used to use Black-Scholes models, but implied that they are not used as much any more. He basically said that many people lost money using these models.
I myself don't work in mathematical finance or in banking at all. However, I was unaware of any major critiques of stochastic differential equations and their applications. From a formal perspective, I imagine that these SDE models have a set of parameters (means, variances, etc.) that users can tune against some data using an optimization method. I am not sure what optimization would make sense, I can imagine approximate bayes or something, could work, but there are probably a million choices. So is the claim of a bad fit, mean that the error between the model predictions and the ground truth data too high, or perhaps changing over time, etc. Also, are there a different set of finance models being used as replacements for Black-Scholes?