* martingale originates in gambling…
* Brownian motion originates in biology.
* Heat equation, Monte Carlo, … all have roots in physical science.
These models worked well in the original domains, because the simplifications and assumptions are approximately valid even though clearly imperfect. Assumptions are needed to simplify things and make them /tractable/ to mathematical analysis.
In contrast, financial mathematicians had to make blatantly invalid assumptions. You can find fatal flaws from any angle. Brian Boonstra told me all good quants appreciate the limitations of the theories. A small “sample”:
– The root of the randomness is psychology, or human behavior, not natural phenomenon. The outcome is influenced fundamentally by human psychology.
– The data shows skew and kurtosis (fat tail).
– There’s often no way to repeat an experiment
– There’s often just a single sample — past market data. Even if you sample it once a day, or once a second, you still operate on the same sample.