What I Built
Benchmarked 7 Monte Carlo variance reduction techniques for pricing European call options against the Black-Scholes analytical solution. Progressed from standard Monte Carlo through antithetic variates, control variates, and Greeks-based hedging to achieve a 125x variance reduction — standard error dropping from 0.1394 to 0.0011 at 10,000 paths.
What I Learned
Variance reduction compounds. Each technique (antithetic, delta CV, gamma CV) attacks a different source of simulation noise. Stacking all three — Delta + Gamma CV + Antithetic — hits the analytical price of 9.2270 with just 10 simulations (0.3% error). The best method is ~18x slower per run but needs far fewer paths, so total compute cost drops dramatically. At 100 paths it already beats standard MC at 10,000.
Key Results
| Method | Price | Std Error | Variance Reduction |
|---|---|---|---|
| Standard Monte Carlo | 9.0674 | 0.1394 | — |
| Antithetic Variates | 9.2005 | 0.0714 | 2x |
| Delta-based CV | 9.2230 | 0.0133 | 10x |
| Delta + Gamma CV | 9.2287 | 0.0118 | 12x |
| Delta + Gamma CV + Antithetic | 9.2209 | 0.0011 | 125x |
Black-Scholes analytical price: 9.2270
Project
Tech Stack: Python, NumPy, SciPy, Matplotlib | Course: Monte Carlo Simulation Seminar, TU Dortmund
Citation
@online{prasanna_koppolu,
author = {Prasanna Koppolu, Bhanu},
title = {Monte {Carlo} {Variance} {Reduction}},
url = {https://bhanuprasanna2001.github.io/projects/monte_carlo_vrt.html},
langid = {en}
}