hard · Quantitative Finance numerical

What is the precise reason?

1. For the discontinuous payoff, only paths landing within h of the barrier differ between bumps, so the finite-difference increment is O(1)/h on an O(h)-probability set; its variance scales like 1/h and diverges as hto 0 at fixed N.
2. Common random numbers actually break down for digital payoffs because the up- and down-bumped simulated prices become statistically independent as hto 0 on the paths near the barrier, so differencing adds rather than cancels variance.
3. The central difference's own O(h^2) truncation error dominates at small h, and squaring that deterministic bias term inside the reported variance estimate is what produces the observed blow-up as h steadily shrinks toward zero at fixed N.
4. Round-off error from subtracting two nearly equal simulated Monte Carlo prices dominates only once h falls below machine epsilon on double-precision hardware, and this floating-point effect alone is the sole source of the observed variance increase.

Sign up free to see the explanation and track your rank →