Numerical Methods and Algorithms in Finance › Summary

This chapter introduced three fundamental categories of numerical methods that form the computational foundation of quantitative finance.

Root-finding algorithms invert price equations numerically. For a continuous function with a valid sign-changing bracket, bisection converges through interval halving in exact arithmetic; finite implementations also need stopping and stagnation checks. Newton-Raphson has at least quadratic local convergence at a simple root with a locally Lipschitz derivative, but can fail outside its basin. Safeguarded hybrids retain a bracket and force sufficient interval progress.

Interpolation techniques construct continuous functions from discrete inputs. Linear interpolation is fast and simple but has derivative discontinuities at knots where adjacent segment slopes differ. Cubic splines provide smoother derivatives, while two-dimensional interpolators can fit a surface; neither choice automatically enforces economic or no-arbitrage constraints.

Numerical integration approximates definite integrals for pricing and risk calculations. The trapezoidal rule and Simpson's rule are composite polynomial rules. Gaussian quadrature can attain high accuracy with few evaluations for sufficiently smooth integrands, while known kinks should be split or handled adaptively.

These techniques appear throughout the remaining chapters. Option pricing uses root-finding to invert prices for implied volatility. For a Greek computed numerically, one option is to reprice after a small input change and form a finite difference. Curve construction and valuation often need values between quoted tenors, obtained by interpolation or parametric models. Monte Carlo directly estimates expectations for path-dependent pricing and may use quadrature for subproblems.