Optimal Bump Size for Greeks

Junior Quant · Numerical Implementation

Question

Compute the Black-Scholes delta numerically using a central-difference bump:

\Delta \approx [C(S+h) - C(S-h)]/(2h)

. Compare against the analytic formula. Derive the optimal bump size

h

that minimises the sum of truncation error

O(h^2)

and floating-point cancellation error

O(\varepsilon_{\mathrm{mach}} / h)

, and state the resulting

h^* \sim \varepsilon_{\mathrm{mach}}^{1/3}

. Explain why the analysis differs for a forward-difference scheme.

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