Selected Topics in Applied Mathematics I (Mathematical Finance)

 

Autumn 2003

Professor Paul Feehan

 

Lectures

 

  1. Introduction (9/2/2003). Examples of stochastic differential equations and their applications, including the Black-Scholes equation [Oksendal, chapter 1]. Introduction to options – European, American, barrier and exotic options, including Asian, lookback options [Wilmott, sections 13.11, 13.12, 13.13, 14.1, 14.2]
  2. Review of probability theory (9/4/2003). Finite probability spaces, examples. Probability measures and sigma algebras.  Filtrations of sigma algebras. Random variables. Sigma algebras generated by random variables. [Shreve, sections 1.1, 1.2. Øksendal, section 2.1].
  3. Review of probability theory, continued (9/9/2003). Measure on the real line (distribution) induced by a probability measure and a random variable. Expected value and variance of a random variable. Introduction to the Lebesgue measure and integral. Borel sigma algebra, Borel measures, Borel-measurable functions, and definition of the Lebesgue integral. Relationship with the Riemann integral. [Shreve, sections 1.2, 1.3. Øksendal, section 2.1].
  4. Review of probability theory, continued (9/11/2003). Advantages of the Lebesgue over Riemann integrals. Examples. Lebesgue dominated convergence theorem. Examples. General probability spaces. Uniform, normal distributions. Radon-Nikodym derivative and probability density functions. Independent sets, sigma algebras, and random variables. Joint distributions for two random variables and independence.  [Shreve, sections 1.3, 1.4. Øksendal, section 2.1].
  5. Review of probability theory, continued. Stochastic processes (9/16/2003). Marginal distributions and densities. Covariance, correlation, and independence. Definition of stochastic processes. Finite-dimensional distributions. Existence of stochastic processes and the Komolgorov Extension Theorem. [Øksendal, section 2.2. Karlin & Taylor vol. 1, sections 1.2, 7.1, 7.2, 7.3].
  6. Definition of Brownian motion (9/18/2003). Existence of the Brownian motion stochastic process via the Komolgorov Extension Theorem. Basic properties: mean and covariance, independent increments, continuity of paths. [Øksendal, section 2.2. Karlin & Taylor vol. 1, sections 1.2, 7.1, 7.2, 7.3].
  7. Definition of the Itô integral (9/23/2003). Equivalence of stochastic processes [Karatzas & Shreve, section 1.1]. Canonical Brownian motion [Øksendal, section 2.2]. Motivation for the Itô integral, white noise, stationary process, independent increments, stationary increments [Øksendal, section 3.1; Karlin & Taylor, chapter 1].
  8. Construction of the Itô integral (9/25/2003). Filtrations of sigma algebras. Elementary functions. Itô isometry. Convergence theorems. Martingales. [Øksendal, section 3.2].
  9. Conditional expectation (9/30/2003). Conditional expectation of a random variable, given a sigma algebra. Conditional density function. [Øksendal, Appendix B; Shreve, sections 11.6, 11.7].
  10. Conditional expectation (continued). Itô process and Itô formula (10/2/2003). Itô processes, Itô formula. Examples. [Øksendal, section 4.1].
  11. Itô integration by parts, martingale representation theorem, geometric Brownian motion (10/7/2003). Integration by parts example, integration by parts theorem, ingredients in the proof of Itô’s formula [Øksendal, section 4.1]. Martingale representation theorem, discrete-time martingales and examples of martingales. [Øksendal, section 4.3; Karlin & Taylor, section 6.1].Geometric Brownian motion [Øksendal, section 5.1]. 
  12. Existence & uniqueness of solutions to stochastic differential equations, solution of linear SDEs (10/9/2003).  Existence and uniqueness theorem [Øksendal, section 5.2]. Ornstein-Uhlenbeck equation, Brownian bridge  [Øksendal, chapter 5; Karlin & Taylor, section 15.14; Karatzas & Shreve, section 5.6]. First derivation of the Black-Scholes PDE [Shreve, section 15.6].
  13. Second derivation of the Black-Scholes PDE. Replicating portfolios (10/14/2003). Derivation of the Black-Scholes PDE following Wilmott, delta-hedging [Wilmott, chapter 5]. Self-financing portfolios and replicating strategies [Baxter & Rennie, section 3.6].
  14. Martingale measures and option prices as conditional expectations (10/16/2003). Self-financing portfolios and replicating strategies, option price as a conditional expectation, Girsanov’s theorem and its converses, Martingale representation theorem review, exponential martingales, construction of replicating portfolios in the case of zero risk-free interest rate. [Baxter & Rennie, section 3.6 and 3.7; Øksendal, section 8.6].
  15. Derivation of the Black-Scholes formula (10/21/2003).  Martingale measures for discounted assets, construction of replicating portfolios with non-zero risk-free interest rate, and derivation of the Black-Scholes formula by evaluation of the conditional expectation. Implied volatility. Formulas for the “Greeks”. [Baxter & Rennie, section 3.6 and 3.7; Wilmott, chapter 7].
  16. Meaning of the “Greeks”. Put-Call parity (10/23/2003). [Wilmott, section 2.12 and chapter 7].
  17. American options. Extensions of the Black-Scholes model I (10/28/2003). American options [Baxter & Rennie, section 3.7], stopping times [Karatzas & Shreve, sections 1.2 & 2.6]; extension of Black-Scholes model to case of stochastic interest rate, drift, and volatility processes [Baxter & Rennie, section 6.1].
  18. Extensions of the Black-Scholes model II. Markov property (10/30/2003). Extension of the Black-Scholes model to the case of multiple stocks, with stochastic interest rate, drift, and volatility processes [Baxter & Rennie, section 6.3]; Itô diffusions and the Markov property [Øksendal, section 7.1].

 

Last updated: October 30, 2003 © Paul Feehan