Central first derivative
V'(x) approx [V(x+h) - V(x-h)] / (2h)Symmetric bumps cancel the leading one-sided truncation error.
Architecture & Systems
Finite-Difference Checks for Greeks
Finite differences approximate sensitivities numerically and provide an independent check on analytic Greeks.
Mental model
Small controlled input bumps provide an independent numerical check on analytic sensitivities.
Analytic Greeks are derivatives of the pricing formula; finite differences approximate them by bumping inputs and re-pricing.
Central differences, using up and down bumps, usually give better accuracy than one-sided bumps for smooth payoffs.
Explore the mechanism
Does an independently bumped price reproduce the analytic sensitivity at a stable step size?
Finite-difference checks for Greeks
Bump-and-reprice provides an independent check on analytic sensitivities.
S
100
K
100
r
0.020
vol
0.25
T
1.00
Finite difference
bump h
0.50
Monte Carlo noise (optional)
paths
2000
noise
1.0
seed
42
Call price
10.87056
Analytic delta
0.581214
FD delta
0.581197
|FD − analytic|
0.000017
Error vs bump size
Interpretation
- Central differences usually dominate forward differences for smooth payoffs.
- Bump choice is a trade-off: numerical noise versus curvature bias.
- With Monte Carlo, CRN reduces noise in the difference by correlating bumps.
UNDERSTAND
What to notice
Choosing the bump size involves a trade-off: too small leads to numerical noise, too large breaks the linear approximation.
For Monte Carlo pricers, using common random numbers across bumps reduces sampling noise in the finite-difference estimate.
The quantitative structure
Notation
V(x)- price as a function of input x
h- finite-difference bump size
Central second derivative
V''(x) approx [V(x+h) - 2V(x) + V(x-h)] / h^2The second difference estimates local curvature.
A small example
Checking delta
A pricer gives V(101) = 12.6 and V(99) = 11.4 around spot 100, using h = 1.
- Compute the central difference: (12.6 - 11.4) / (2 * 1).
- The numerical delta is 0.60.
- Compare 0.60 with the analytic delta under a stated tolerance.
ResultThe independent finite-difference delta is 0.60.
Where it appears
- checking analytic and algorithmic Greeks
- testing pricer implementations
- diagnosing sign and unit errors
Where the intuition breaks
What this view leaves out
- A bump that is too large adds truncation error, while one that is too small exposes floating-point noise and Monte Carlo variance.
Common mistakes
“Agreement at one bump size proves the Greek is correct.”
Not quite.A robust check examines stability across bump sizes and controls numerical or simulation noise.
Where this concept sits
Follow the strongest prerequisites, applications, extensions, and contrasts.
Connections for Finite-Difference Checks for Greeks
Finite-Difference Checks for GreeksYOU ARE HERE
Used byValidation Toolbox for PricersIndependent finite-difference sensitivities are a practical component of pricer validation.
ValidatesGreeks and Risk FactorsBumping inputs and repricing provides an independent numerical check of reported sensitivities.
Supported byNumerical Tolerances for Regression TestsFinite-difference checks need explicit tolerances because discretisation and floating-point effects prevent exact equality.
Used by · Validation Toolbox for Pricers
Independent finite-difference sensitivities are a practical component of pricer validation.
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