Longstaff Schwartz

A project based on the article "Valuing American Options by Simulation - A Simple Least-Squares Approach". We reproduce the results from the paper.

Advancement

For now, the project implements a simple version of the Longstaff& Schwartz Pricer and its finite differences counter part: only non-path dependent markovian options over 1D underlying following a Local Volatility diffusion (i.e $\mathrm{d}X_t = rX_t\mathrm{d}t + \sigma(t, X_t) X_t\mathrm{d}W_t$) can be priced (for instance the American Put).

To come

For the next version, I plan to enrich the LocalVolatilityModel by adding the possibility for the user to calibrate it to Market data. For now, the User has to instantiate the LocalVolatilityModel and right away pass it the volatility function $\sigma(t, X_t)$ through its parameter sigma (Examples: sigma = 0.2 for constant volatility or sigma = lambda t, x: 1/(1 + np.exp(x/40)) for local volatility). It would be great to also add the calibration possibily into BlackScholesModel and HestonModel.

Another 'simple' improvement would be to allow deterministic but non-constant interest rate $r(t)$.

Previous versions

v0, late april 2024: Only non-path dependent markovian options over 1D underlying following a Black Scholes diffusion (i.e $\mathrm{d}X_t = rX_t\mathrm{d}t + \sigma X_t\mathrm{d}W_t$) could be priced (for instance the American Put).

v1, mid may 2024: Current version. See Advancement section above.