MATH 86: MATHEMATICAL FINANCE

Syllabus: Winter 2023 Instructor: John W. Welborn

Location: Kemeny Hall 004 Email: John.W.Welborn@dartmouth.edu

Tue/Thu: 2:25 – 4:15 PM Office Hours: By Request

Wed X-Hour: 5:30 – 6:20 PM

COURSE DESCRIPTION

Financial derivatives can be thought of as wagers on uncertain future financial events. This course will take a mathematically rigorous approach to understanding the Black-Scholes-Merton model and its applications to pricing financial derivatives and risk management. Topics will include arbitrage-free pricing, binomial tree models, measure theory, Ito calculus, the Black-Scholes analysis, derivatives pricing, volatility modeling, and hedging.

PREREQUISITES

MATH 20 and MATH 40, or MATH 60; MATH 23; and COSC 1 or the equivalent.

COURSE TEXTBOOKS

1. Baxter, Martin., and Andrew Rennie. Financial Calculus: An Introduction to Derivative Pricing. Cambridge: Cambridge University Press, 1996. Available online.

2. Wilmott, Paul. Paul Wilmott Introduces Quantitative Finance. 2nd ed. Chichester, West Sussex, England: J. Wiley & Sons Ltd., 2007. Available online.

• Problem Sets: 20%

• Midterm Exam: 25% (2/7/23)

• Final Project: 20% (3/2/23)

• Final Exam: 35% (3/10/23)

EXAMS

The midterm (25%) and final exam (35%) will be open book and open note. Students must do their own work and adhere to the Academic Honor Principle. Students who require testing accommodations must contact me as soon as possible and provide the appropriate documentation.

FINAL PROJECT

Your final project (20%) may be on any topic related to mathematical finance. Students are encouraged to consider either an empirical or theoretical project. Potential topics include local volatility modeling, exotic options pricing formulae, numerical methods, and jump diffusion processes. Projects will be graded on novelty, quality, technical proficiency, and research.

PROBLEM SETS

There will be 4 problem sets due throughout the term. Each assignment is worth 5% of your final grade. For each assignment, you may work in groups of up to 3-4 other students. To complete the assignment, upload a single, clear, and legible PDF document to Canvas.

COURSE SCHEDULE

Fundamental to the principle of independent learning are the requirements of honesty and integrity in the performance of academic assignments, both in and out of the classroom. Dartmouth operates on the principle of academic honor, without proctoring of examinations. Any student who submits work which is not his or her own, or commits other acts of academic dishonesty, violates the purposes of the college and is subject to disciplinary actions, up to and including suspension or separation. All students must follow the Academic Honor Principle.

MENTAL HEALTH

The academic environment at Dartmouth is challenging, our terms are intensive, and classes are not the only demanding part of your life. There are a number of resources available to you on campus to support your wellness, including your undergraduate dean (http://www.dartmouth.edu/~upperde/), Counseling and Human Development (http://www.dartmouth.edu/~chd/), and the Student Wellness Center (http://www.dartmouth.edu/~healthed/).

STUDENT ACCESSIBILITY NEEDS

Students requesting disability-related accommodations and services for this course are required to register with Student Accessibility Services (SAS; Apply for Services webpage; student.accessibility.services@dartmouth.edu; 1-603-646-9900) and to request that an accommodation email be sent to me in advance of the need for an accommodation. Then, students should schedule a follow-up meeting with me to determine relevant details such as what role SAS or its Testing Center may play in accommodation implementation. This process works best for everyone when completed as early in the quarter as possible. If students have questions about whether they are eligible for accommodations or have concerns about the implementation of their accommodations, they should contact the SAS office. All inquiries and discussions will remain confidential.

RELIGIOUS OBSERVANCES

Dartmouth has a deep commitment to support students’ religious observances and diverse faith practices. Some students may wish to take part in religious observances that occur during this academic term. If you have a religious observance that conflicts with your participation in the course, please meet with me as soon as possible—before the end of the second week of the term at the latest—to discuss appropriate course adjustments.

PAPERS AND JOURNAL ARTICLES

• Bachelier, Louis et al. Louis Bachelier’s Theory of Speculation: The Origins of Modern Finance. Princeton: Princeton University Press, 2006. Online.

• Black, E. and M. Scholes. 1973. The valuation of options and corporate liabilities. Journal of Political Economy. 81: 637-54.

• Einstein, Albert, R. Fürth, and A. D. Cowper. Investigations on the Theory of the Brownian Movement. London: Methuen & co. ltd., 1926. Print.

• Harrison, J. M. and D. M. Kreps. 1979. Martingales and arbitrage in the multi-period securities markets. Journal of Economic Theory 20: 381-408.

• Harrison, J. M. and S. R. Pliska. 1981. Martingales and stochastic integration in the theory of continuous trading. Stochastic Processes and Applications. 11: 215-60.

• Markowitz, H. 1952. Portfolio selection. Journal of Finance 19: 425-42.

• Merton, R. C. 1973. Theory of rational option pricing. Bell Journal of Economics 4.

• Sharpe, W. 1964. Capital asset prices: a theory of market equilibrium under conditions of risk. Journal of Finance 19: 425-42.

• Thorpe, E.O. 1969. Optimal gambling systems for favorable games. Review of the International Statistics Institute 37(3).

BOOKS

• Bergomi, Lorenzo. Stochastic Volatility Modeling. First edition. Boca Raton, FL: Chapman and Hall/CRC, an imprint of Taylor and Francis, 2015. Print.

• Cox, John C., and Mark. Rubenstein. Options Markets. Englewood Cliffs, N.J: Prentice-Hall, 1985. Print.

• Derman, Emanuel, and Michael B. Miller. The Volatility Smile : an Introduction for Students and Practitioners. Hoboken, New Jersey: Wiley, 2016. Print.

• Gatheral, Jim. The Volatility Surface a Practitioner’s Guide. Hoboken, N.J: John Wiley & Sons, 2006. Print.

• Haug, Espen Gaarder. The Complete Guide to Option Pricing Formulas. 2nd ed. New York: McGraw-Hill, 2007. Print.

• Hull, John. Options, Futures, and Other Derivatives. Eleventh edition. Boston: Pearson, 2021. Print.

• Musiela, Marek, and Marek Rutkowski. Martingale Methods in Financial Modelling. 2nd ed. Berlin ;: Springer, 2007. Print.

• Natenberg, Sheldon. Option Volatility and Pricing: Advanced Trading Strategies and Techniques. 2nd edition. McGraw-Hill, 2014. Print.

• Shreve, Steven. Stochastic Calculus for Finance | The Binomial Asset Pricing Model. 1st ed. 2004. New York, NY: Springer New York, 2004. Web.

• Taleb, Nassim Nicholas. Dynamic Hedging: Managing Vanilla and Exotic Options. New York: Wiley, 1997. Print.

• Wilmott, Paul, Sam Howison, and Jeff Dewynne. The Mathematics of Financial Derivatives: A Student Introduction. Cambridge University Press, 2012. Print.