Unit 8 of 14
Numerical methods for finance
- 8 lessons
- Videos 57 to 64
Most pricing problems are solved numerically, and every method has its own error. This unit covers root finding, Monte Carlo simulation, SDE discretization, finite differences and Fourier pricing.
Conditioning belongs to the problem and stability to the algorithm, and bisection and Newton's method converge under stated conditions. A Monte Carlo standard error measures sampling error, not model or discretization error. Euler and Milstein schemes differ in strong and weak order. Antithetic and control variates and importance sampling can reduce variance without bias, and randomized quasi-Monte Carlo adds an error estimate to evenly spread points. Finite differences solve pricing PDEs on stable grids, regression handles early exercise, and characteristic functions give Fourier prices.
Every lesson will be one video. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.
Root finding, Monte Carlo and discretization
Numerical conditioning and root finding
How the conditioning of an equation differs from the stability of an algorithm, why floating-point cancellation loses digits, and when bisection and Newton's method converge, quadratically or not at all.
Video 57
Monte Carlo pricing and sampling error
How a Monte Carlo estimator averages discounted payoffs drawn under a pricing measure, how its standard error and confidence interval measure sampling error, what coverage means, and which errors they leave out.
Video 58
SDE discretization and weak convergence
How the Euler-Maruyama scheme and the Milstein correction approximate an SDE, how strong and weak orders of convergence differ, why payoff regularity affects the weak rate, and how Euler breaks square-root positivity.
Video 59
Variance reduction and quasi-Monte Carlo
Variance reduction and unbiased estimators
How antithetic variates, control variates with the optimal control coefficient and importance sampling with an importance weight lower Monte Carlo variance while keeping the estimator unbiased.
Video 60
Quasi Monte Carlo and effective dimension
How quasi-Monte Carlo integrates with low-discrepancy points measured by star discrepancy, how randomized quasi-Monte Carlo gives an honest standard error, and how the Brownian bridge construction changes effective dimension.
Video 61
PDE, regression and Fourier methods
Finite difference PDE methods
How a pricing PDE becomes the heat equation and is solved on a spatial grid by the finite difference method, and how explicit, implicit and Crank-Nicolson schemes differ in stability and the discrete maximum principle.
Video 62
Least squares Monte Carlo for American options
How least-squares Monte Carlo fits continuation values by regression, turns them into a Bermudan exercise policy, and why independent evaluation gives a policy lower bound while fitting bias can inflate in-sample values.
Video 63
Fourier transform pricing
How a characteristic function gives call prices through the Fourier transform of a damped call and its inversion, why the damping parameter must lie in the moment strip, and where truncation error and aliasing come from.
Video 64