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FRM Part I · FRM Exam Part I · Simulation and Bootstrapping

A risk analyst estimates the price of a European call by Monte Carlo simulation. For each random draw Z, she also uses -Z to generate a second price path, and averages the two payoffs for that pair. Which variance reduction technique is this, and what condition makes it effective?

This is antithetic variates. Each draw Z is paired with -Z, and the pair average has variance σ²(1+ρ)/2, so variance falls only when the two payoffs are negatively correlated. Control variates would need a variable with a known expected value.

  1. AAntithetic variates; the payoffs from Z and -Z must be negatively correlatedCorrect
  2. BControl variates; the payoff must be correlated with a variable having a known expected value
  3. CImportance sampling; the draws must be concentrated in the tail region
  4. DStratified sampling; the draws must be split into equal-probability intervals

Explanation

Pairing each draw Z with -Z is the antithetic variates method. The variance of the pair average is (1/2)σ²(1+ρ), so it falls only if the correlation ρ between the two payoffs is negative. Control variates use a correlated variable with known mean, which is not described here.

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