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Financial Stochastic Calculus

  • This course should be useful for well-prepared students who are in the fields of finance, economics, statistics, or mathematics, but it is definitely directed toward students who also have a genuine interest in fundamental mathematics. Naturally, we deal with financial theory to a serious extent.

  • Our work requires a high level of comfort with the tools of real analysis, including uniform continuity, Cauchy's convergence criterion, notions of integrability, and calculations in inner product spaces. Knowledge of function spaces (L-one, L-two, Hilbert space, etc) may not be explicitly assumed, but many function spaces will be introduced and used in the course and students who have not seen these before face heavy sledding.

  • "Measure-theoretic probability theory" enters the conversation regularly, but, with a reasonable amount of work, it can also be picked up as the course progresses. Basic mathematical analysis is the core prerequisite, there is no denying that some knowledge of measure theory will be useful --- at least to the level of having understood the Borel-Cantelli lemmas, the definition of convergence with probability one, and the Dominated Convergence Theorem.

  • Students who have learned time series analysis are perfectly well prepared, as are students with a graduate course in real analysis. Many students with lesser backgrounds have taken the course and done well. It is substantially a matter of priorities and motivation.

  • We are after the stochastic calculus with finance in view, and we are going after it in the simplest way that we can possibly muster. If we are honest at each turn, this challenge is plenty hard enough.

      

Course Topics

Binomial Derivative Pricing, Brownian motion, Martingales, Winner integration, Ito integration, Localization and Ito's integral, Ito's formula, Stochastic differential equations, Arbitrage and SDEs, The diffusion equation, Representation theorems,Girsanov theory
Arbitrage and martingales, Continuous Derivative Pricing, The Feynman-Kac connection.

 

Textbook: Stochastic Calculus for Finance I, II, Shreve

 

               

Prerequisites: 

Measure Theory, Stochastic Processes, Time Series, Elementary Functional Analysis.

Grading Policy: 

Homework and Project -------- %30

Midterm Exam ------------------  %30

Final Exam ----------------------    %40

Teacher Assistants: 

N/A

Time: 

To Be Announced.

Term: 
DONE
Grade: 
Graduate

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