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American Option Pricing via Longstaff-Schwartz Monte Carlo

This project implements and benchmarks several methods for pricing American options, centred on the Longstaff-Schwartz Least-Squares Monte Carlo (LSM) algorithm. It covers GBM and Heston stochastic-volatility path simulation, a Crank-Nicolson finite-difference benchmark, antithetic and control-variate variance reduction, and a full convergence analysis of the LSM estimator.

Methods

  • Longstaff-Schwartz LSM with weighted Laguerre polynomial basis (Longstaff & Schwartz 2001)
  • GBM paths — exact log-normal discretisation, no Euler error
  • Heston stochastic volatility — full-truncation Euler scheme (Lord, Koekkoek & Van Dijk 2010)
  • Crank-Nicolson finite difference — tridiagonal PDE solver, O(n) per step, used as benchmark
  • Antithetic variates* — mirrored normal increments; empirically ~3x variance reduction for ATM puts
  • European-put control variate — optimal BS-European adjustment; empirically ~2–3x variance reduction

LSM vs Crank-Nicolson (ATM put: S₀=100, K=100, r=5%, σ=20%, T=1yr)

Method Price Uncertainty
LSM (100k paths) 6.0429 ±0.0225 (1 SE)
Crank-Nicolson 6.0880 —

The two methods agree to within 0.05, well within two standard errors of the LSM estimate.

Variance Reduction (20k paths, 30-trial empirical comparison)

Method Empirical VRF
Antithetic variates ~3x
Control variate ~2x

VRF = (plain variance) / (reduced variance) — how many times fewer paths achieve the same precision.

Convergence

LSM price error decays as 1/√n with respect to path count, consistent with standard Monte Carlo theory. The price stabilises at ~20 time steps; higher polynomial degrees (>3) show no improvement over the paper's default.

Project Structure

american-option-pricing/
├── src/
│   ├── models/
│   │   ├── gbm.py                  # GBM path simulator (exact log-Euler)
│   │   └── heston.py               # Heston path simulator (full-truncation Euler)
│   ├── pricing/
│   │   ├── longstaff_schwartz.py   # LSM backward-induction pricer
│   │   ├── finite_difference.py    # Crank-Nicolson PDE solver
│   │   └── payoffs.py              # American put/call payoff factories
│   ├── variance_reduction/
│   │   ├── antithetic.py           # Antithetic variates wrapper for LSM
│   │   └── control_variate.py      # European-put control variate for LSM
│   └── analysis/
│       ├── convergence.py          # Convergence experiment functions
│       └── plots.py                # Figure functions
├── tests/
│   ├── test_gbm.py
│   ├── test_lsm.py
│   ├── test_benchmarks.py
│   └── test_variance_reduction.py
├── config/
│   └── params.yaml                 # Shared model parameters
└── requirements.txt

Limitations

  • LSM low bias — backward-induction regression underestimates the continuation value, producing a slightly low price estimate. See Broadie & Glasserman (2004) for the upper-bound dual.
  • No Heston closed-form benchmark — the Heston American option has no analytic price; the GBM comparison in notebook 03 is a consistency check, not a validation.
  • Uniform S grid in Crank-Nicolson — a log-S grid would give better accuracy near the strike with fewer nodes.
  • LSM exercise boundary — the boundary extracted from regression is an approximation; the true critical S* requires solving C(S*) = h(S*) from the regression coefficients.

References

  • Longstaff, F. A. & Schwartz, E. S. (2001). Valuing American Options by Simulation: A Simple Least-Squares Approach. Review of Financial Studies, 14(1), 113–147.
  • Broadie, M. & Glasserman, P. (2004). A Stochastic Mesh Method for Pricing High-Dimensional American Options. Journal of Computational Finance, 7(4), 35–72.