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The Greeks: Sensitivity Analysis via PDE Differentiation

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The Formal Theorem

Let V(S,t) V(S, t) denote the price of a European derivative satisfying the Black-Scholes PDE:
∂V∂t+rS∂V∂S+12σ2S2∂2V∂S2−rV=0 \frac{\partial V}{\partial t} + rS\frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} - rV = 0
Given the linearity of the differentiation operator, the sensitivity Greeks Δ=∂V∂S \Delta = \frac{\partial V}{\partial S} , Γ=∂2V∂S2 \Gamma = \frac{\partial^2 V}{\partial S^2} , and Θ=∂V∂t \Theta = \frac{\partial V}{\partial t} satisfy the derived sensitivity PDEs obtained by differentiating the primary PDE with respect to S S or t t . Specifically, Δ \Delta satisfies:
∂Δ∂t+rS∂Δ∂S+12σ2S2∂2Δ∂S2+(r+σ2)Δ−rΔ=0 \frac{\partial \Delta}{\partial t} + rS\frac{\partial \Delta}{\partial S} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 \Delta}{\partial S^2} + (r + \sigma^2) \Delta - r\Delta = 0

Analytical Intuition.

Imagine you are standing on a landscape where the altitude is the value of a derivative V(S,t) V(S, t) . The 'Greeks' are not mere abstract numbers; they are the fundamental topographic map of this financial surface. Δ \Delta tells us the slope (first derivative) as the stock price S S shifts, effectively measuring our hedge ratio. Γ \Gamma reveals the curvature of the landscape, warning us how rapidly that slope changes—like the suspension in a car hitting a bump. When we differentiate the Black-Scholes PDE itself, we are performing a 'sensitivity of a sensitivity' analysis. We move from asking how the price changes to asking how the hedge itself must evolve as time t t flows and the underlying S S fluctuates. By differentiating the PDE, we reveal that these sensitivities are not chaotic; they are governed by their own internal dynamics, effectively creating a 'shadow' PDE that dictates how the Greeks propagate through time. It is the mathematical equivalent of watching a ripple move across a pond, where the PDE describes the water's surface and the Greeks describe the shifting motion of the individual droplets.
CAUTION

Institutional Warning.

Students often assume Greeks are static snapshots. In reality, Greeks are dynamic solutions to auxiliary PDEs. A common trap is conflating 'Total Delta' with the partial derivative ∂V/∂S \partial V / \partial S , forgetting that changes in t t and σ \sigma fundamentally alter the hedge requirement in time-dependent environments.

Academic Inquiries.

01

Why differentiate the PDE rather than the closed-form solution?

Differentiating the PDE allows us to analyze Greeks in models where closed-form solutions do not exist, such as American-style options or exotic paths with complex boundary conditions.

02

What is the physical interpretation of the additional terms appearing in the differentiated PDE?

These terms represent the 'coupling' between the Greeks; for example, the Gamma term emerges because the Delta sensitivity is itself sensitive to the curvature of the option price surface.

Standardized References.

  • Definitive Institutional SourceWilmott, P., Derivatives: The Theory and Practice of Financial Engineering

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Institutional Citation

Reference this proof in your academic research or publications.

NICEFA Visual Mathematics. (2026). The Greeks: Sensitivity Analysis via PDE Differentiation: Visual Proof & Intuition. Retrieved from https://www.nicefa.org/library/advanced-stochastic-processes/the-greeks--sensitivity-analysis-via-pde-differentiation

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