Contents |
Course Code
IAM 611 (9700611)
Credit
(3-0) 3
Prerequisites
Some understanding of the following subjects will be necessary: probability theory, optimal control, martingale theory, stochastic integration
Content
Three of the main tasks in financial data analysis are:
- Estimation of probabilistic models from real data, i.e., statistics
- Computation of probabilities (e.g., value at risk, probability of ruin) and expactations (e.g, derivative pricing) in the estimated models
- Optimization (e.g., american option pricing, portfolio optimization)
This course looks at these problems and the mathematical and computational methods to approach them.
Aims/Learning Outcomes
The course will emphapsize both theory and practice. The aim is for the student to build a unified and intuitive understanding of the above problems and have at least some practical understanding and experience about how to approach them.
Regular homeworks and programming problems will be assigned. Programming assignments will be coded in MATLAB. There will be an inclass midterm and a take home final exam. The homeworks will constitute 40% of the grade, the midterm and the final 30% each.
Outline
In all the problems we will study, the course will always begin from the simplest of models and gradually look at models of increasing complexity.
Review
Throughout the course we will review briefly the following topics as we need them:
- Continuous stochastic processes (Stochastic integration, SDEs, Martingale theory, Girsanov's theorem and the related PDE theory),
- Optimal Control,
- Discontinous stochastic proccesses,
- Fundamental concepts, ideas and models in Finance (replication based pricing, hedging, complete/incomplete markets, several important interest rate and market models).
Statistics
- Value at risk, model building approach, [JCH] John C. Hull, Options, Futures and Other Derivatives, Section 18.3
- Estimating volatilities and correlations [JCH], Chapter 19
- Statistics for Diffusion Processes, Lapeyre, Sulem and Talay, Understanding Numerical Analysis for Financial Models, preprint, Chapter 10
- Inverse problems and Model calibration, Rama Cont and Peter Tankov, Financial Modeling with Jump processes, Chapter 13
- Calibration of an HJM model
Time permitting, we would like to also study Bayesian methods for model estimation.
Computation and Optimization
For this part of the course we will mainly follow the preprint version of the book by Lapeyre, Sulem and Talay and Glasserman's Monte Carlo Methods in Financial Engineering:
- Option Pricing and PDEs
- Finite Difference Methods for option prices
- Finite Difference Methods for stochastic optimal control problems
- Tree Methods for option prices
- Monte Carlo and Importance Sampling for option pricing and risk estimation
- Monte Carlo methods for pricing american options
Textbooks
We will use parts of the following books. The instructor will provide lecture notes and other neccessary course material.
- Paul Glasserman, Monte Carlo Methods in Financial Engineering
- John C Hull, Options, Futures and Other derivatives
- Lapeyre, Sulem and Talay, Understanding Numerical Analysis for Financial Models
- Cont and Tankov, Financial Modelling with Jump Processes
- Lipster and Shiryaev, Statistics of Random Processes
- Dupuis and Kushner, Numerical Methods for Stochastic Control Problems in Continuous Time
Additional reference books
- Fleming and Richel, Deterministic and Stochastic Optimal Control
- Evans, Partial Differential Equations
- Soner and Fleming, Controlled Markov Processes and Viscosity Solutions
- Karatzas and Shreve, Brownian Motion and Stochastic Calculus
- Karatzas and Shreve, Methods of Mathematical Finance
- Protter, Stochastic Integration and Differential Equations