Add an introduction to Monte Carlo -- with option pricing

[jstac]

This lecture will introduce Monte Carlo in an option pricing setting.

First we price European options where the distribution of the asset is known in closed form.

• integrate by hand if possible
• integrate numerically otherwise
• show that Monte Carlo is an option (but maybe not the best one)

Now switch to a setting where we only know the dynamics, and the distribution of the asset has no closed form solution.

Monte Carlo becomes the natural method -- implement invarious ways.

Consider barrier options: "For example, a European call option may be written on an underlying with spot price of $100 and a knockout barrier of $120. This option behaves in every way like a vanilla European call, except if the spot price ever moves above $120, the option "knocks out" and the contract is null and void. Note that the option does not reactivate if the spot price falls below $120 again."

Ex: pricing a put option

[jstac]

Ex: an Asian option?

[jstac]

See https://github.com/QuantEcon/rse_comp_econ_2023/blob/main/notebooks/european_option.ipynb

(cut numba and jax?)