NEW YORK CITY COLLEGE of TECHNOLOGY the City University of New York

NEW YORK CITY COLLEGE OF TECHNOLOGY The City University of New York

DEPARTMENT: Mathematics

COURSE: MAT 4788

TITLE: Financial Risk Modeling

DESCRIPTION: This course aims to provide an overview of the main concepts underlying the analysis of financial risk and to show how these concepts can be implemented in practice. The topics that are covered include the Black-Scholes-Merton model and the Greeks, Numerical Procedures with Monte Carlo simulations, Estimating Volatilities and Correlations, Volatility Smiles, Value at Risk and Credit Risk. Computer models are used throughout the course.

REQUIRED TEXT: Options, Futures and Other Derivatives, 7th Ed. John C. Hull, Pearson Prentice Hall, 2009

RECOMMENDED TEXTS: 1. Risk Management and Financial Institutions, 3rd Ed., John C. Hull, Wiley, 2012

2. Financial Risk Manager Handbook, 6th Ed. Philippe Jorion, Wiley, 2011

CREDITS: 3

PREREQUISITES: MAT 3672, MAT 3770, MAT 3772, MAT 3788

Prepared by Professor Boyan Kostadinov (Spring 2012)

A. Testing/Assessment Guidelines: The following should be scheduled: 1. A class exam at the end of the First Quarter. 2. A class exam at the end of the Second Quarter. 3. A final project and a final exam.

B. Using a Computer Algebra System (CAS) is required.

1 Course Intended Learning Outcomes/Assessment Methods

Learning Outcomes Assessment Methods

1. Learn the basics of option hedging and risk Classroom activities and discussion, management, using the Greeks of the Black-Scholes- homework, project, exams. Merton model. 2. Learn about the nature of the volatility smiles in Classroom activities and discussion, the options market. homework, project, exams. 3. Learn some basic numerical procedures used in Classroom activities and discussion, finance and implement Binomial Tree and Finite homework, project, exams. Difference methods using a computer algebra system. 4. Learn how to use linear and quadratic models to Classroom activities and discussion, compute the Value at Risk. homework, project, exams. 5. Learn how to use Monte Carlo simulations to Classroom activities and discussion, compute the Value at Risk. homework, project, exams. 6. Learn how to use Principal Component Analysis to Classroom activities and discussion, compute the Value at Risk homework, project, exams. 7. Learn how to estimate volatilities and correlations Classroom activities and discussion, from market data using a Moving Average model and homework, project, exams. a GARCH model. 8. Learn the basics of Credit Risk and how to Classroom activities and discussion, estimate default probabilities from bond and equity homework, project, exams. prices. 9. Learn the Gaussian Copula model for time to Classroom activities and discussion, default, extensively used in the financial industry in homework, project, exams. many Credit Risk models. 10. Learn how to use computer technology to assist Classroom activities and discussion, in the above objectives. homework, project, exams.

General Education Learning Outcomes/Assessment Methods

Learning Outcomes Assessment Methods

1. Gather, interpret, evaluate, and apply information Classroom activities and discussion, discerningly from a variety of sources. homework, project, exams. 2. Understand and employ both quantitative and Classroom activities and discussion, qualitative analysis to solve problems. homework, project, exams. 3. Employ scientific reasoning and logical thinking. Classroom activities and discussion, homework, project, exams. 4. Communicate effectively using written and oral Classroom activities and discussion, means. homework, project, exams. 5. Utilize computer based technology in accessing Classroom activities and discussion, information, solving problems and communicating. homework, project, exams. 6. Work with teams. Build consensus and use Classroom activities and discussion, creativity. project. 7. Acquire tools for lifelong learning. Classroom activities and discussion, homework, project, exams.

New York City College of Technology Policy on Academic Integrity

Students and all others who work with information, ideas, texts, images, music, inventions, and other intellectual property owe their audience and sources accuracy and honesty in using, crediting, and citing sources. As a community of intellectual and professional workers, the College recognizes its responsibility for providing instruction in information literacy and academic integrity, offering models of good practice, and responding vigilantly and appropriately to infractions of academic integrity. Accordingly, academic dishonesty is prohibited in The City University of New York and at New York City College of Technology and is punishable by penalties, including failing grades, suspension, and expulsion. The complete text of the College policy on Academic Integrity may be found in the catalog.

3 MAT 4788 Financial Risk Modeling, Text: Options, Futures and Other Derivatives, Prentice Hall, 7th ed., by John C. Hull. Session Chapter and section from the text Homework from the text 1 Ch.13 The Black-Scholes-Merton Model, 13.1 Lognormal property of stock prices p. 303: # 1 and 13.2 The distribution of the rate of return. 2 13.4 Volatility and 13.5 The idea underlying the Black-Scholes-Merton equation p. 303: # 2 3 13.7 Risk-Neutral valuation and 13.8 Black-Scholes pricing formulas pp. 303-305: # 3, 4, 13, 14, 16 4 Ch.17 The Greek Letters, 17.4 Delta hedging p. 384: # 2, 3 5 17.6 Gamma and 17.8 Vega pp. 384-386: # 5, 9, 24, 25 6 Ch.18 Volatility Smiles, 18.1 Why the volatility smile is the same for calls and puts p. 401: # 1, 2 and 18.3 Equity Options 7 18.5 The volatility term structure and volatility surfaces and pp. 401-402: # 3, 4, 5, 6, 8, 13 18.7 When a single large jump is anticipated 8 First Examination 9 Ch.19 Basic Numerical Procedures, 19.1 Binomial trees p. 447: # 2 10 19.6 Monte Carlo Simulation p. 448: # 9 11 19.8 Finite difference methods p. 448: # 18 12 Ch.20 Value at Risk, 20.1 The VaR measure 13 20.3 Model-building approach p. 471: # 1 14 20.4 Linear Model p. 471: # 3, 4, 5 15 20.5 Quadratic Model pp. 472-473: # 8, 9, 16, 17 16 20.6 Monte Carlo Simulation p. 473: # 18 17 20.9 Principal Component Analysis p. 473: # 19 18 Midterm Review 19 Second Examination 20 Ch.21 Estimating Volatilities and Correlations, 21.1 Estimating volatility 21 21.2 The exponentially weighted moving average model pp. 494-495: # 1, 3, 7 22 21.3 The GARCH(1,1) model pp. 494-495: # 2, 8 23 21.5 Maximum likelihood methods p. 496: # 17 24 21.7 Correlations pp. 495-496: # 9, 11 25 Ch.22 Credit Risk, 22.4 Estimating default probabilities from bond prices p. 521: # 14, 15 26 22.6 Using equity prices to estimate default probabilities p. 523: # 31 27 22.9 Default correlation. The Gaussian Copula model for time to default p. 521: # 11 28 Student Project Presentations 29 Final Exam Review 30 Final Examination

4 MAT 4788 Financial Risk Modeling, Text: Options, Futures and Other Derivatives, Prentice Hall, 7th ed., by John C. Hull. Financial Risk Modeling Topics Homework Ch.13 The Black-Scholes-Merton Model, 13.1 Lognormal property of stock prices, p. 303: # 1 pp. 277-279 and 13.2 The distribution of the rate of return, pp. 279-280 13.4 Volatility, pp. 282-285 and 13.5 The idea underlying the BSM equation, pp. 285-287 p. 303: # 2 13.7 Risk-Neutral valuation, pp. 289-291and 13.8 Black-Scholes pricing formulas, pp. 291-293 pp. 303-305: # 3, 4, 13, 14,16,18 Ch.17 The Greek Letters, 17.4 Delta hedging, pp. 360-367 p. 384: # 2, 3 17.6 Gamma, pp. 369-373 and 17.8 Vega, pp. 373-375 pp. 384-386: # 5, 9, 24, 25 Ch.18 Volatility Smiles, 18.1 Why the volatility smile is the same for calls and puts, pp. 389-390 and p. 401: # 1, 2 18.3 Equity Options, pp. 393-395 18.5 The volatility term structure and volatility surfaces, pp. 396-397 and pp. 401-402: # 3, 4, 5, 6, 8, 13 18.7 When a single large jump is anticipated, pp. 398-400 First Examination Ch.19 Basic Numerical Procedures, 19.1 Binomial trees, pp. 407-414 p. 447: # 2 19.6 Monte Carlo Simulation, pp. 426-432 p. 448: # 9 19.8 Finite difference methods, pp. 435-445 p. 448: # 18 Ch.20 Value at Risk, 20.1 The VaR measure, pp. 451-454 20.3 Model-building approach, pp. 456-458 p. 471: # 1 20.4 Linear Model, pp. 458-462 p. 471: # 3, 4, 5 20.5 Quadratic Model, pp. 462-464 pp. 472-473: # 8, 9, 16, 17 20.6 Monte Carlo Simulation, pp. 464-465 p. 473: # 18 20.9 Principal Component Analysis, pp. 466-470 p. 473: # 19 Midterm Review Second Examination Ch.21 Estimating Volatilities and Correlations, 21.1 Estimating volatility, pp. 477-479 21.2 The exponentially weighted moving average model, pp. 479-481 pp. 494-495: # 1, 3, 7 21.3 The GARCH(1,1) model, pp. 481-482 pp. 494-495: # 2, 8 21.5 Maximum likelihood methods, pp. 483-487 p. 496: # 17 21.7 Correlations, pp. 491-493 pp. 495-496: # 9, 11 Ch.22 Credit Risk, 22.4 Estimating default probabilities from bond prices, pp. 500-503 p. 521: # 14, 15 22.6 Using equity prices to estimate default probabilities, pp. 506-507 p. 523: # 31 22.9 Default correlation. The Gaussian Copula model for time to default, pp. 512-517 p. 521: # 11 Student Project Presentations Final Exam Review Final Examination