Brownian Bridge™

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\times

100\times

(if

h = 0.01

)

10\times

Identical accuracy — the leading error coefficient differs but the order is the same

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:

10^{-16}

(machine epsilon)

10^{-5}

10^{-8}

10^{-3}

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?

Because complex arithmetic avoids all floating-point errorsBecause no subtraction of nearly equal quantities occurs — the derivative information is in the imaginary part, which is computed independently of the real partBecause the Taylor series for analytic functions converges for any

h

Because

h = 10^{-100}

makes the truncation error exactly zero

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

truefalse

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?

O(n)

times the function evaluation (same as bump-reval)Constant: approximately

3

–

5

times the function evaluation, regardless of

n

O(\sqrt{n})

times the function evaluation

O(\log n)

times the 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?

35 (one per pillar)70 (two per pillar: up and down bump)71 (70 bumped plus 1 base)140 (for up and down bumps, with a separate base price)

Quiz: Numerical Differentiation and Bump-Reval

Quick Quiz