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So, as far as I know, we have 3 main numerical methods. Monte Carlo, PDE-methods (FDM), and numerical integration methods (Fourier transforms and so on).

How do these methods generally compare to each other? What are the rules of thumb? When should you use one over the other?

I am not asking for specifics, just general ideas for the most typical situations that one might run into.

For example, for European option pricing in Levy models, which are most often used, and why? What if we switch to stochastic volatility models? What if we want to price Americans or path-dependents?

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  • $\begingroup$ "European option pricing in Levy models" ? - do you mean, how to price European options for calibrating the said model (i.e. Levy model in this case). Otherwise, for pricing of European options, we always use the Black Scholes analytical formula with implied volatility extracted (interpolated or extrapolated) from the implied vol surface. This is the standard way how market values (i.e. prices) all European options. $\endgroup$
    – bhutes
    Commented Jul 10, 2019 at 2:08
  • $\begingroup$ On a second thought, the greeks generated by non-Black Scholes model may be better for vanilla European options too. E.g a delta-hedge portfolio of Vanilla options using a stochastic vol model may have a lower variance than another delta-hedged portfolio which uses BS greeks. $\endgroup$
    – bhutes
    Commented Jul 10, 2019 at 8:17

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1> Analytical - Black Scholes formula for Vanilla European options, Digitals. These valuations are just an "interpolation" of traded options. We interpolate the implied volatility from the traded points on the implied volatility surface.

There is no modeling assumption involved here. Market uses this formula for implementing the "interpolation". Scope for going wrong using this method (compared to market participants) is very limited. Most issues arise only when volatility surface needs to be extrapolated.

Quanto Europeans also use this method frequently, although this is more than just an interpolation. Modeling assumptions are implicit - (i) Lognormal terminal distributions of the underlying and the FX rate, (ii) correlation input to the analytical formula is often not an "interpolated" value from traded options; hence needs further modeling assumptions.

2> Numerical integration - all payoffs which depend only on the terminal distribution of the underlying, e.g. self-quanto options

3> PDE Method (Finite difference or Trees) - for path dependent payoffs with single underliers (low dimensionality).

4> Monte Carlo - for path dependent payoffs with higher dimensionality e.g. basket options or rates products with multiple Forward Libor rates as the underliers.

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