Numerical Differentiation and Bump-Reval

1. Central differences have truncation error $O(h^2)$ while forward differences have $O(h)$. For the same bump size $h = 0.01\sigma$, approximately how many times more accurate are central differences for vega computation?

2. The optimal step size for central differences balances truncation and round-off error. For a function $f$ with $|f'''|/|f| \approx 1$ and machine epsilon $\varepsilon_{\text{mach}} \approx 10^{-16}$, the optimal step size $h^*$ is approximately:

3. The complex-step method estimates $f'(x) \approx \text{Im}[f(x+ih)]/h$ for small real $h$. Why can $h$ be taken as small as $10^{-100}$ without loss of accuracy?

4. The complex-step method works for Monte Carlo pricers that use indicator functions of the form $\mathbf{1}_{S_T > K}$.

5. Adjoint Algorithmic Differentiation (AAD) computes the gradient of a pricing function with respect to $n$ inputs. What is its computational cost relative to a single function evaluation?

6. For a vega ladder over a $5 \times 7$ implied vol surface (5 maturities, 7 strikes), how many full repricings are needed using central differences?