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# Applied Mathematics & Statistics

AMS 514, Computational Finance
Review of foundations: stochastic calculus, martingales, pricing, and arbitrage. Basic principles of Monte Carlo and the efficiency and effectiveness of simulation estimators. Generation of pseudo- and quasi-random numbers with sampling methods and distributions. Variance reduction techniques such as control variates, antithetic variates, stratified and Latin hypercube sampling, and importance sampling. Discretization methods including first and second order methods, trees, jumps, and barrier crossings. Applications in pricing American options, interest rate sensitive derivatives, mortgage-backed securities and risk management. Whenever practical, examples will use real market data. Extensive numerical exercises and projects in a general programming environment will also be assigned.
Prerequisite:  AMS 512 and  513

Instructor Consent Required for Registration

Required Textbook s for Fall 2018:

None

Recommended Textbooks for Fall 2018 :

"Finance with Monte Carlo" by Ronald W. Shonkwiler; 2013 Springer; ISBN: 9781461485117

Learning Outcomes:

1) Demonstrate mastery of basic concepts:
* Limitations of models;
* Design, testing and evaluation of models;
* How to build realistic applications using Matlab;
* Theoretical framework for pair trading.

2) Implement in Matlab a pair trading application in the S&P 500 universe
* Testing and evaluating the results;
* 3 Regime shifting models.

3) Demonstrate mastery of Hidden Markov Models
*The theoretical framework of HMMs;
* Estimation of HMMs;
* Implementation of an application in Matlab to reveal the regimes of the S&P 500 index;
* Testing and evaluation of results.

4)  Demonstrate an understanding of derivative pricing
* Risk neutral pricing;
* Pricing with transforms;
* Implementation of Matlab programs using FFT.

5) Demonstrate an understanding of finite difference methods
* Solving PDEs with finite difference methods;
* Different approximation schemes;
* Implementation in Matlab of solutions of PDEs with finite difference methods.

6) Demonstrate an understanding of frequency domain
* Introduction to time series analysis in the frequency domain. 