Foundations priced options on grids and trees — methods that excel in one dimension and choke in many. This topic builds the engine that survives high dimension and rich dynamics: Monte Carlo simulation, from pseudo-random uniforms through correlated multi-asset paths, SDE discretisation and its convergence theory, variance reduction, and the regression trick that lets a forward-simulating method price backward-looking American options. The second half upgrades the model: constant Black–Scholes volatility gives way to the Heston stochastic-variance process and the volatility-smile machinery; and because calibration requires pricing thousands of vanillas per optimiser iteration, the topic closes with Fourier methods built on characteristic functions.
By the end of this note you should be able to:
- explain how pseudo-random numbers are manufactured, what can go wrong (lattice structure, RANDU), and how normals and correlated normals are generated;
- write the European Monte Carlo pricing algorithm with confidence intervals, and deploy antithetic and control variates;
- discretise an SDE with Euler or Milstein, state strong/weak convergence orders, and handle the CIR positivity problem;
- choose between pseudo- and quasi-random sequences, and set up Sobol + inversion + Brownian bridge correctly;
- run the Longstaff–Schwartz regression algorithm for American options;
- write the Heston SDEs, map each parameter to its smile effect, simulate with full truncation, and calibrate to a surface;
- price European options from a characteristic function via Carr–Madan FFT and the COS method.
In this lesson
- Contents
- The big picture
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