Longstaff-Schwartz Algorithm

A Python implementation of the Longstaff-Schwartz linear regression algorithm for the evaluation of call rights and American options.

Seminal paper: Francis A. Longstaff, Eduardo S. Schwartz, Valuing American Options by Simulation: A Simple Least-Squares Approach (The Review of Financial Studies) (2001) Vol 14, No 1, pp. 113-147
Documentation: https://longstaff-schwartz.readthedocs.io
Free software: MIT license

Talks

PyConDE 2019-10-10: Slides, Jupyter Notebook
PyData Meetup 2019-09-18: Slides, Jupyter Notebook

Usage

from longstaff_schwartz.algorithm import longstaff_schwartz
from longstaff_schwartz.stochastic_process import GeometricBrownianMotion
import numpy as np

# Model parameters
t = np.linspace(0, 5, 100)  # timegrid for simulation
r = 0.01  # riskless rate
sigma = 0.15  # annual volatility of underlying
n = 100  # number of simulated paths

# Simulate the underlying
gbm = GeometricBrownianMotion(mu=r, sigma=sigma)
rnd = np.random.RandomState(1234)
x = gbm.simulate(t, n, rnd)  # x.shape == (t.size, n)

# Payoff (exercise) function
strike = 0.95

def put_payoff(spot):
    return np.maximum(strike - spot, 0.0)

# Discount factor function
def constant_rate_df(t_from, t_to):
    return np.exp(-r * (t_to - t_from))

# Approximation of continuation value
def fit_quadratic(x, y):
    return np.polynomial.Polynomial.fit(x, y, 2, rcond=None)

# Selection of paths to consider for exercise
# (and continuation value approxmation)
def itm(payoff, spot):
    return payoff > 0

# Run valuation of American put option
npv_american = longstaff_schwartz(x, t, constant_rate_df,
                                  fit_quadratic, put_payoff, itm)

# European put option for comparison
npv_european = constant_rate_df(t[0], t[-1]) * put_payoff(x[-1]).mean()

# Check results
assert np.round(npv_american, 4) == 0.0734
assert np.round(npv_european, 4) == 0.0626
assert npv_american > npv_european

Plots

For details see PyData Meetup Jupyter Notebook.

Approximation of continuation value

Favourable exercise

Credits

Main developer is luphord.

Primary source for the algorithm is Francis A. Longstaff, Eduardo S. Schwartz, Valuing American Options by Simulation: A Simple Least-Squares Approach (The Review of Financial Studies) (2001) Vol 14, No 1, pp. 113-147. There is no affiliation between the authors of the paper and this code.

This package was prepared with Cookiecutter and the audreyr/cookiecutter-pypackage project template.