Finite Difference Greeks And Bump Sizes

If a pricing model is too complicated to differentiate by hand — an exotic payoff under Monte Carlo, say — you can still estimate a Greek numerically: price the option, nudge one input (spot, volatility, rate) up by a small amount $h$, re-price, and divide the change in price by $h$. This is called "bump and reprice," and the hard part is not the formula, it's choosing $h$.

Make $h$ too large and you introduce truncation error: the finite difference approximates a curve with a straight line over an interval, and if that interval is wide, the line misses the curve's actual curvature. Make $h$ too small and you introduce rounding error: the two re-priced values become nearly identical, so subtracting them and dividing by a tiny $h$ amplifies whatever floating-point noise or Monte Carlo sampling noise is already in each price. The two errors move in opposite directions as $h$ shrinks, so there is a sweet spot, not a "smaller is always better" rule.

A central difference — pricing at spot $+h$ and spot $-h$ and dividing the difference by $2h$ — roughly squares the truncation error compared to a one-sided bump, at the cost of one extra re-pricing, which is why it's the default choice whenever an extra valuation is affordable. For a Monte Carlo price with simulation noise on the order of 0.01, a bump of $h = 1\%$ of spot is a typical starting point; too much smaller and the signal drowns in the simulation's own randomness.