Probabilistic Aspects of Financial Modeling
MATH5110/6010, Spring semester, 2023
Instructor: Yizao Wang
Email: yizao.wang@uc.edu
Office: French Hall 4302.
Office Hours: by appointment.
Class meeting: Hybrid mode
- 1~2 in-person lectures MW 11:15am-12:10pm, OldChem 835, and
- 1~2 asynchronous (via Canvas) ones,
per week. Please follow announcements closely regarding meeting formats of each week.
Textbook
- Lecture slides updated at Canvas. See slides for recommended references for further studies if interested.
- Recommended (not required): Introduction to the Mathematics of Finance, by R.J. Williams, 2006.
Course description
This course covers two fundamental probabilistic models in mathematical finance: the (discrete-time) binomial model and the (continuous-time) Black-Scholes model. The financial applications serve as the motivation of studying these two models mathematically. However, data applications are not discussed in this course. The two models themselves exhibit appealing properties and hence deserve further investigation from probabilistic point of view alone. Our treatment of binomial model will be self-contained, illustrating the role of martingales. For Black-Scholes model, however, we shall spend most of the time on the basics of Brownian motions (staring from random walks), stochastic calculus and very briefly stochastic differential equations (with occasional sketched proofs only). The presentation of the Black-Scholes model will be based on formal calculation only.
No prior knowledge on finance is required. The course would be helpful to students interested in stochastic modeling in general.
If you find the course materials interesting and are looking for an opportunity for capstone projects, please read here first.
Grades
- homework (60%) + midterm (20%) + final (20%) + optional project (10% bonus).
Letter grades: A(90%), A-(85%), B+(80%), B(75%), B-(70%), C+(65%), C(60%), C-(55%).
Tentative Schedule
There will be 6 homework sets, to be posted at Canvas.
- Week 1 (Jan 9, Mon): Review of probability. HW1
- Week 2 (Jan 16): No class on MLK, Monday (Jan 16).
Binomial model. Portfolio, path-space point of view.
- Week 3 (Jan 23): Martingale, risk-neutral probability measure, no arbitrage for binomial model. HW2.
- Week 4 (Jan 30) Options pricing.
- Week 5 (Feb 6): Replicating portfolio. HW3.
- Week 6 (Feb 13): Random walks.
- Week 7 (Feb 20): Midterm.
- Week 8 (Feb 27): Brownian motion. HW4.
- Week 9 (Mar 6):
- Week 10 (Mar 13): Spring break.
- Week 11 (Mar 20): Stochastic calculus.
- Week 12 (Mar 27): HW5.
- Week 13 (Apr 3): Stochastic differential equations.
- Week 14 (Apr 10): Black-Scholes model. HW6.
- Week 15 (Apr 17): Class ends on Apr 21, Fri.
- Week 16 (Apr 24): Final