Differential Calculus and Optimization Basics › Summary

This chapter established the calculus and optimization foundations for quantitative finance:

Derivatives as sensitivities: A derivative gives a local rate of change in a differentiable model. Marginal profit, option Greeks, and differentiable portfolio-model sensitivities use this concept; not every financial exposure is a smooth derivative.

Multivariable calculus: Partial derivatives measure held-fixed-input sensitivities. For a differentiable function, a nonzero gradient gives steepest first-order ascent among Euclidean unit directions in the chosen coordinate scaling. A zero gradient has no unique first-order direction.

Unconstrained optimization: Differentiable interior stationary points have zero gradient. Under the stated second-order smoothness conditions, a positive- or negative-definite Hessian gives a strict extremum, and mixed signs give a saddle even when other eigenvalues are zero. Semidefinite zero-eigenvalue cases without mixed signs need further analysis. Gradient descent seeks minima under suitable objective and step conditions.

Constrained optimization: Lagrange multipliers incorporate constraints into a stationarity system. Under regularity conditions, their signed values give local first-order objective sensitivity to the specified constraint right-hand sides, with units and signs determined by the formulation.

Convexity: Every local minimum of a convex objective over a convex feasible set is global. Portfolio variance is convex when its covariance matrix is positive semidefinite, so mean-variance minimization is a convex program when its feasible set is also convex. For a twice-differentiable function on an open convex domain, the Hessian must be positive semidefinite throughout the domain; eigenvalues at one point classify only local curvature.

These tools appear throughout quantitative finance. The next chapter develops integral calculus and differential equations, building from local rates of change to accumulated quantities and continuous-time dynamics. Later chapters reuse optimization in portfolio construction and parameter estimation.