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What is the go-to method to price exotic options in exotic models?

If we are in Black Scholes, then this is hard to answer, since we can both do various sorts of Monte Carlo or solve various sorts of simple PDEs.

However, in more exotic models, the PDE approach becomes harder since we typically require solving a PIDE.

So my question is, if both the model and the option is exotic, is Monte Carlo then the go-to method? Or do solving these PIDE's still remain competitive enough compared to MC?

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Most of the time the answer will depend on the dimensionality of your problem. If the payoff is simple enough, for example, to have no volatility convexity so that the Black-Scholes model is sufficient, the PDE approach will be enough: you solve by finite differences.

However, if you introduce Stochastic volatility, interest rates or for a multi-asset option, the PDE will become too complex and you will have to use Monte Carlo methods. Those do not suffer from the curse of dimensionality.

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  • $\begingroup$ What if we don't have stochastic volatility, but the model is still more complex than BS? For example, some sort of infinite-activity Levy process? Does the PDE approach still remain valid? $\endgroup$
    – fool
    Feb 22, 2020 at 16:09
  • $\begingroup$ @siou0107 what is volatility convexity? What would be a good book/paper to learn about this concept? $\endgroup$
    – fwd_T
    Apr 19, 2020 at 11:29
  • $\begingroup$ I've heard that term from a practitioner quant, though I personally prefer the term 'volatility monotonicity'. The bottom line is: is your option monotone wrt to volatility (and generally we mean realised volatility, i.e. gamma). Options that are not include barrier options or digital options. Pricing them using Black-Scholes litterally hides volatility risk. Good explanations on this phenomenon are Wilmott's books (either its 3-tome PWOQF or its FAQ book). $\endgroup$
    – siou0107
    Apr 19, 2020 at 17:08

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