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Calculus for Quants

Research interviews rarely ask for calculus as calculus. They ask for a sensitivity, an approximation, an optimal weight or a half-life, and each one is a short calculus step. Candidates who have not used it recently lose time re-deriving the basics. This lesson collects the calculus that comes up, with the worked version of each.

Derivatives are sensitivities

A derivative says how much an output moves per unit move in an input. In finance most derivatives have names: an option's delta is ∂V/∂S\partial V / \partial S, a bond's modified duration is minus the derivative of its log price with respect to the yield, and a portfolio's exposure to a factor is a derivative of its return.

The chain rule connects them. If a quantity depends on a price through another variable, multiply the sensitivities. It also explains log returns: ddSln⁡S=1S\frac{d}{dS}\ln S = \frac{1}{S}, so a small change in ln⁡S\ln S is the percentage change dS/SdS / S.

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