CONTENTS

Exercise: When does antithetic fail?

Tasks

  1. Theoretical analysis. For an OTM call with S0=100,K=200S_0 = 100, K = 200, argue heuristically why f(Z)f(Z) and f(−Z)f(-Z) become positively correlated for sufficiently OTM strikes. Hint: consider the support of paths giving non-zero payoff.
  2. Numerical verification. With σ=0.2,T=1,r=0.05\sigma = 0.2, T = 1, r = 0.05, compute ρ\rho for K∈{150,175,200,225,250}K \in \{150, 175, 200, 225, 250\}. Tabulate.
  3. Straddle: f(Z)=(ST−K)++(K−ST)+=∣ST−K∣f(Z) = (S_T - K)^+ + (K - S_T)^+ = |S_T - K|. Estimate ρ\rho for the ATM straddle. Why is it ≈+1\approx +1? What's the lesson?
  4. A function ff for which ρ=0\rho = 0. Construct a simple example payoff that's neither monotone nor symmetric — for which antithetic gives no benefit. Verify numerically.
Sign in to track progress