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Introduction

  • Mathematical Finance I (MATH 515, Fall 2025)
  • Instructor: Tahir Choulli

Course Description

This course gives an introduction to Mathematical Finance. We explain how martingale and other stochastic analysis tools are tailor made for addressing problems in Finance/Financial Economics. This exact fit between Stochastic and Finance/Financial Economics is illustrated on the easiest market models such as discrete and discrete-time market models. Thus, this course is also suitable for students from Math and Stats (with almost all branches), Finance, Management Sciences, Economics, Engineering, and Engineering Management. In this course, we will attempt to cover the following topics:

  • Probability tools and their financial role/interpretations: a review Conditional probabilities and expectations. Filtration, adapted and predictable pro-cesses. Martingales, submartingales and supermartingales in discrete time. Doob-Meyer decomposition for supermartingales.
  • Discrete-Time financial models: (Single period and multi-periods models) Model specifications, Arbitrage, completeness and other economic considerations. Self-financing property, value and gain processes. Valuation of contingent claims.
  • Binomial model: Model specifications. Perfect hedging.
  • Utility functions and consumption/investment problems.
  • Incomplete markets: Imperfect hedging like mean-variance or quantile hedging.
  • American options, futures and forward contracts.
  • Transition to the continuous-time framework (if time allows it).

Textbooks are NOT required, and might not be limited to:

  • Introduction to Mathematical Finance. Discrete Time Models by Stanley Pliska, Black-well, 1997
  • Stochastic Calculus for Finance I: The Binomial Asset Pricing Model by Steven Shreve, Springer, 2004.

Concepts and References

Week 1-2

Concepts: Probability space (\(\sigma\)-algebra, \(\mathcal{F}\)-measurable), (conditional) expectation, stochastic process.