hard · Quantitative Finance
You compute the Delta of a digital (cash-or-nothing) option by bumping the spot and using a central finite difference on a Monte Carlo price, reusing the same random numbers (common random numbers) for the up- and down-bumps. As you shrink the bump size h at FIXED number of paths N, the variance of the estimated Delta is observed to BLOW UP rather than settle.
What is the precise reason?
- 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.
- Common random numbers break down for digital payoffs because the up- and down-bumped prices become independent as hto 0, so differencing adds rather than cancels variance.
- The central difference's O(h^2) truncation error dominates at small h, and squaring it in the variance estimate produces the observed blow-up.
- Round-off error in subtracting two nearly equal Monte Carlo prices dominates once h falls below machine epsilon, which is the sole source of the variance increase.
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