Pricing European Options: Martingale Derivation, Lattice Convergence and Discrete Hedging Error

We derive the Black–Scholes–Merton (BSM) price twice, from a self-financing replication argument and from the martingale representation under an equivalent measure, and quantify how fast implementable schemes approach it. For an at-the-money call (S₀ = K = 100, T = 1, r = 5%, σ = 20%, C = 10.45058), the Cox–Ross–Rubinstein lattice converges at O(N⁻¹) with a parity-dependent oscillation. Richardson extrapolation and the Leisen–Reimer lattice both raise the rate to O(N⁻²). A control variate cuts Monte Carlo variance by a factor of 6.8, matching 1/(1 − ρ²). The dispersion of a discretely rebalanced delta hedge agrees within 2% with the asymptotic law σₚₙₗ ≈ √(π/4) · σV/√n.