Stochastic Processes in Finance I

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Typical Scheduling: 
Every fall semester (ISyE)

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


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