The purpose of this course is to prepare students for potential future employment or research that requires deep understanding of certain topics in financial mathematics, and proficiency in applying this understanding in real-world settings. "Deep" here means that we'll not only learn what the theory says, but also why it is true, by studying proofs and justifications; this is important because real-world settings rarely match the theory exactly and deep knowledge is needed to know how to adapt. "Proficiency" here means that we will learn to apply the theory both accurately and efficiently; this is important because real-world settings are generally intolerant of mistakes, and pervaded by time constraints.

Consequently, after completing this course, students should have acquired the understanding of certain content and the proficiency of certain skills. 

Content

• Basic definitions and facts in probability, conditional expectation, and martingales
• Mathematical formulations of financial markets (binomial, finite market model, American and European contingent claims)
• Theory explaining the arbitrage-free pricing of contingent claims in viable and complete markets, including the proofs that justify this theory
• Methods of implementing the theory in examples, to calculate prices, determine viability, completeness, and other market characteristics

Skills

• Read, understand, and memorize abstract financial mathematics theorems and proofs within a reasonable time constraint
  
• Present written abstract mathematical arguments and proofs in financial mathematics, from memory, within a reasonable time constraint
• Solve computational problems and produce written calculations on derivative pricing and market analysis, within a reasonable time constraint