CS 476 | SCS

To provide students with an overview of modern numerical algorithms for use in financial applications.

Intended Audience

CS 476 is normally taken in a student’s fourth year. Modern finance now makes extensive use of sophisticated numerical algorithms for option pricing and hedging, risk management and portfolio optimization. Students in business related programs or having an interest in financial applications will find this course beneficial.

Related Courses

Prerequisites: (AMATH 242/341 or CM 271 or CS 370 or 371) and STAT 231/241. Students who receive a good grade in CS335 may contact the instructor of CS476 to seek admission without the formal prerequisites. No finance or business background is assumed.

Hardware/Software

References

Schedule

Outline

Introduction (3 hours)

The two state lattice for no-arbitrage option pricing, and hedging. Modelling asset prices: Brownian motion.

Lattice Methods (6 hours)

Discrete random walk on a lattice. Convergence to continuous Brownian motion. No-arbitrage lattice methods for pricing European/American option.

Black-Scholes Equation (3 hours)

Derivation of Black-Scholes partial differential equation. Relationship to discrete hedging on a lattice. Newton iteration for implied volatility.

Stochastic Differential Equations (9 hours)

Basic idea of Monte Carlo option pricing. Properties of random number generators, problems in high dimensions, algorithms for normally distributed random numbers. Strong vs. weak convergence for SDEs. Convergence properties of forward Euler, Milstein mdethods. The Brownian Bridge. Correlated random numbers.

Numerical Solution of the Black-Scholes Equation (9 hours)

Basic finite difference methods. Explicit and implicit methods, stability. American options. Equivalence of lattice and explicit finite difference methods. Positive coefficient methods.

Portfolio Optimization (6 hours)

Methods for constructing the efficient frontier. Inequality constraints. Effect of data errors on computed portfolio weights.