Stochastic Processes in Finance I

Department: 
MATH
Course Number: 
6759
Hours - Lecture: 
3
Hours - Lab: 
0
Hours - Recitation: 
0
Hours - Total Credit: 
3
Typical Scheduling: 
Every fall semester

Mathematical modeling of financial markets, derivative securities pricing, and portfolio optimization. Concepts from probability and mathematics are introduced as needed. Crosslisted with ISYE 6759.

Prerequisites: 

MATH 3215 and some knowledge of computer programming

Course Text: 

Class notes are used for this course; these notes are at the level of Introduction to Mathematical Finance: Discrete Time Models by S. Pliska, published by Blackwell Publishers

Topic Outline: 
  • Discussion of prerequisites, including basic probability background and linear systems of equations
  • Some probability background, including Riemann-Stieltjes integrals and conditional probabilities and conditional expectations. Definitions of some financial terms.
  • The Binomial Market Model and its use in pricing and hedging claims. European style options, some exotic options.
  • Model implementation, implied volatility. Probability background: convergence in distribution, and a central limit theorem
  • Convergence of Binomial option prices to Black-Scholes option prices, and a sketch of the Black-Scholes Market Model
  • Use of derivative securities, and strategies for trading. The greeks, and their use in option trading.
  • Probability and mathematics background: martingales, separating hyperplanes, linear programming and duality
  • The Discrete-time, Stochastic Market Model, conditions of no-arbitrage and completeness, and pricing and hedging claims
  • Variations of the basic models: American style options, foreign exchange derivatives, derivatives on stocks paying dividends, and forward prices and futures prices
  • Probability background: Markov chains. Stochastic volatility and implied trees.
  • Stochastic interest rates, bonds and interest rate derivatives. Model implementation.
  • Mathematical background: optimization techniques. Incomplete Market Models. Utility-based pricing in complete markets and in incomplete markets. Portfolio optimization