Differential Calculus and Optimization Basics › Differential Calculus and Optimization Basics

Quantitative finance often asks two practical questions: how does a modeled value respond when an input changes, and which feasible choice best meets a stated objective? A trader asking how an option's model value responds to a small stock-price move is asking a sensitivity question: calculus gives a local approximation, while repricing gives the finite change. A portfolio manager choosing an allocation to maximize a specified risk-adjusted objective is posing an optimization problem. Differential calculus describes local rates of change, while optimization compares feasible choices under stated constraints.

Sensitivity analysis and optimization often complement one another. Derivatives describe local changes, while first- and second-order conditions use those changes to characterize candidate optima when the required smoothness and constraint conditions hold. The gradient gives local directional information that a gradient-based method can use to search for improvement. Derivative-free and discrete methods handle problems where that information is unavailable or insufficient.

This chapter builds a calculus foundation for the later pricing and portfolio chapters. We begin with derivatives as measures of sensitivity, then extend to functions of multiple variables using partial derivatives and gradients. From there, we develop the machinery for finding optimal solutions, both unconstrained and constrained, ending with the Lagrange multiplier technique used in modern portfolio theory. Finally, we examine convexity. When minimizing a convex objective over a convex feasible set, every local minimum is global, although a numerical candidate still requires feasibility, convergence, and optimality checks.