Numerical Differentiation and Bump-Reval

Setup

When Analytic Greeks Are Not Available

The Black-Scholes Greeks derived in the previous module are available in closed form because the pricing function $C(S, K, r, \sigma, T)$ is differentiable and explicit. In practice, most pricing functions used on a derivatives desk are not:

• Exotic options (barriers, Asians, lookbacks) have prices computed by Monte Carlo or finite difference solvers — no analytic formula.
• Stochastic vol models (Heston, SABR) have semi-analytic prices but whose derivatives with respect to model parameters involve numerical quadrature.
• Calibrated models where the volatility surface itself depends on the market inputs in a non-trivial way.

For all such cases, bump-reval (finite difference approximation of derivatives) is the standard method on trading desks. It is computationally expensive but model-agnostic: the same procedure works for any pricing function.

Notation

Let $f: \mathbb{R}^n \to \mathbb{R}$ be a pricing function. We want to estimate the partial derivative $\partial f / \partial x_i$ at a base point $\mathbf{x}$. The bump-reval approach: evaluate $f$ at a perturbed point $\mathbf{x} + h\,\mathbf{e}_i$ (where $\mathbf{e}_i$ is the $i$-th unit vector and $h$ is the bump size), and form a finite difference quotient.

Forward Differences

$\frac{\partial f}{\partial x_i} \approx \frac{f(\mathbf{x} + h\,\mathbf{e}_i) - f(\mathbf{x})}{h}.$

Error analysis. By Taylor expansion:

$f(\mathbf{x} + h\,\mathbf{e}_i) = f(\mathbf{x}) + h\frac{\partial f}{\partial x_i} + \frac{h^2}{2}\frac{\partial^2 f}{\partial x_i^2} + O(h^3).$

Subtracting and dividing by $h$:

$\frac{f(\mathbf{x} + h\,\mathbf{e}_i) - f(\mathbf{x})}{h} = \frac{\partial f}{\partial x_i} + \frac{h}{2}\frac{\partial^2 f}{\partial x_i^2} + O(h^2).$

The truncation error is $O(h)$: first-order accurate. The error decreases linearly as the bump $h$ shrinks.

Cost: 1 repricing per Greek (plus the base price). For $n$ inputs: $n + 1$ total repricings.

Backward Differences

$\frac{\partial f}{\partial x_i} \approx \frac{f(\mathbf{x}) - f(\mathbf{x} - h\,\mathbf{e}_i)}{h}.$

Truncation error is also $O(h)$, with the same leading coefficient but opposite sign:

$\frac{f(\mathbf{x}) - f(\mathbf{x} - h\,\mathbf{e}_i)}{h} = \frac{\partial f}{\partial x_i} - \frac{h}{2}\frac{\partial^2 f}{\partial x_i^2} + O(h^2).$

Forward and backward differences have the same $O(h)$ accuracy. Backward difference is useful when a positive bump is not feasible (e.g., interest rates near zero cannot be bumped upward without changing the sign regime).

Central Differences

$\frac{\partial f}{\partial x_i} \approx \frac{f(\mathbf{x} + h\,\mathbf{e}_i) - f(\mathbf{x} - h\,\mathbf{e}_i)}{2h}.$

Error analysis. Subtract the two Taylor expansions:

$f(\mathbf{x} + h) = f + h f' + \frac{h^2}{2}f'' + \frac{h^3}{6}f''' + O(h^4),$ $f(\mathbf{x} - h) = f - h f' + \frac{h^2}{2}f'' - \frac{h^3}{6}f''' + O(h^4).$

Subtracting: $f(\mathbf{x}+h) - f(\mathbf{x}-h) = 2hf' + \frac{h^3}{3}f''' + O(h^5)$. Dividing by $2h$:

$\frac{f(\mathbf{x}+h) - f(\mathbf{x}-h)}{2h} = f' + \frac{h^2}{6}f''' + O(h^4).$

The truncation error is $O(h^2)$: second-order accurate. Central differences are far more accurate than one-sided differences at the same bump size.

Cost: 2 repricings per Greek (up and down bump). No base-price reuse possible. For $n$ inputs: $2n$ repricings.

Second-Order Derivative (for Gamma)

$\frac{\partial^2 f}{\partial x_i^2} \approx \frac{f(\mathbf{x} + h\,\mathbf{e}_i) - 2f(\mathbf{x}) + f(\mathbf{x} - h\,\mathbf{e}_i)}{h^2}.$

Error: Adding the two Taylor expansions: $f(\mathbf{x}+h) + f(\mathbf{x}-h) = 2f + h^2 f'' + \frac{h^4}{12}f^{(4)} + O(h^6)$. Subtracting $2f$ and dividing by $h^2$: truncation error $O(h^2)$.

Gamma via bump-reval uses this formula with $x_i = S$, bump $h = \epsilon \cdot S$ (a relative bump of typically $\epsilon = 1\%$):

$\hat{\Gamma} = \frac{C(S+h) - 2C(S) + C(S-h)}{h^2}.$

Cost: 2 additional repricings (the up and down bumps also give delta via central difference, so gamma is free once delta is computed by central differences).

Step Size Selection: The Accuracy-Stability Tradeoff

There are two sources of error in numerical differentiation:

1. Truncation error: from the Taylor approximation. For central differences: $\sim \frac{h^2}{6}|f'''|/|f'|$ (relative).
2. Round-off error: from finite-precision arithmetic. The computed value of $f(\mathbf{x}+h)$ carries a rounding error of order $\varepsilon_{\mathrm{mach}} \cdot |f|$, where $\varepsilon_{\mathrm{mach}} \approx 10^{-16}$ (double precision). The finite difference quotient amplifies this by $1/h$: round-off error $\sim \varepsilon_{\mathrm{mach}} |f| / h$.