Disclaimer: I have some knowledge of statistics, machine learning and probability theory, but next to zero knowledge of finance (I had to look up Wikipedia to refresh my knowledge of the difference between a bond and a stock), so please don't shoot the newbie :)
I was reading this question on Cross Validated, and I noted that some users (included the original poster) noted that the task of predicting the stock market was impossible. To be pedantic, I think they were actually referring to the simpler (?) task of predicting the future values of a specific stock price based on the stock price past values, not predicting the overall trend of the whole stock market, but you get my point. This is something which I've seen often in pop-finance books, and which is usually explained intuitively this way: if there was a model which could be used to predict reliably the future price of one or more stock, everyone would use it, this would affect the future stock exchanges and prices, changing the data generating process (the stochastic process corresponding to the stock price time series), and thus the model wouldn't work anymore. Another argument which is sometimes given is that market crises are never predicted (the subprime mortgage crisis, dot-com bubble, etc.).
However, there are companies, university degrees, research centers, etc. which work in the field of quantitative finance, so, even if the endeavor of predicting the stock and bond markets is understandably hard, there must be some degree of success. What is currently predictable with some margin of accuracy, and what is not? Concerning the level of the answer, I can follow you if you talk about expectation, stochastic processes, martingales, Monte Carlo and Markov Chain Monte Carlo, neural networks, etc.. However I've only heard about stuff such as the Black-Scholes equation, without knowing the actual details. I think I could understand if you explained me the concept, but I cannot say for sure.