In the following, I will say what the smile problem is, and I am not sure everyone will follow me. This is how deeply and how long I’ve been thinking about the smile problem. I’ve pursued thinking about it in directions exactly opposite to the quantitative one, as you will see. You wouldn’t be asking, at this stage, 30 years later, what the smile problem is, if the answer were ordinary. The tools that it took me so long to develop, in order to understand the smile problem, are not quantitative tools. They are philosophical tools, which question and dig into the categories of thought: writing, trading, time, formalism, etcetera.
The smile problem isn’t something that happens to the Black–Scholes–Merton (BSM) model from outside. It is not a falsification of the BSM model. The smile problem isn’t that BSM assumes the underlying price process to be lognormal and that it suddenly happens in reality, externally to BSM, that the process is different, i.e., admitting of stochastic volatility and jumps. The smile problem is produced from inside.
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