Variance Reduction Techniques in Simulation › Summary

This chapter introduced three fundamental variance reduction techniques that dramatically improve Monte Carlo simulation efficiency:

Antithetic variates exploit the symmetry of the normal distribution by pairing simulations using $Z$ and $-Z$. The negative correlation between pairs reduces variance for monotonic payoffs. Implementation is straightforward: generate half as many random numbers and use each twice with opposite signs.

Control variates adjust Monte Carlo estimates using related quantities with known expectations. The optimal adjustment coefficient equals the regression slope of the target variable on the control. The discounted terminal stock price provides a natural control for option pricing, with delta-based controls offering even larger improvements at greater computational cost.

Stratified sampling ensures uniform coverage of the random input space by dividing it into strata and sampling from each. Latin Hypercube Sampling extends this to multiple dimensions without exponential growth in strata.

Key takeaways for practice:

• Start simple. Implement antithetic variates first since they add minimal overhead and work for most options.
• Exploit known expectations. Look for control variates with high correlation to your target.
• Combine techniques. Layer multiple methods for multiplicative benefits.
• Benchmark empirically. Theoretical variance reduction may not match reality.
• Consider computational cost. The best method balances variance reduction against time per simulation.

The variance reduction factor tells you how many times fewer simulations you need. A VRF of 10 means the same accuracy in one-tenth the time, a transformation that makes real-time derivatives pricing possible.