PDE for Finance

Course Details and Schedule

Prerequisites

Students must have completed the classes Derivative Securities and Stochastic Calculus in the Mathematics in Finance program or have the equivalent elsewhere. Students wishing to take PDE for Finance without having taken the prerequisite classes in the Mathematics in Finance program must contact the instructor before registering. Students also should be familiar with basic ideas of expected utility maximization.

Computing

Some of the assignments will involve light computation. Some of those will call for the FFT and all will require some kind of visualization. Matlab and R are suitable. If you use a generic programming language such as C++ or Java, you need to find FFT and plotting software. Excel/VBA is not suitable.

Grading

The final grade will be determined by the grades on the homework assignments and the final exam, each counting for about half the total. Weights will be adjusted in a certain range for individual students to maximize their weighted total. Homework grades will be posted on the nyuHome web site. Only registered students may submit homeworks for grading. There will be a penalty for assignments submitted late, which is an increasing but moderate and unspecified. Within reason, it is better to submit an assignment late but complete rather than on the due date but incomplete.

Communication

There is a message board at the nyuHome web site. Sign in with your NYU netid and password, then click on the "Academics" tab, then on the class "PDE FOR FINANCE" link (Warning: this will not work until you register for the class.), then the "Communication" button on the left, then (finally) the "Discussion board" link. Please post all academic questions or comments on the message board (questions about an assignment, answers to questions or other comments, announcements of study sessions, etc.). Always check the message board before working on an assignment, as there often will be corrections or hints. Please email the instructor or TA only for personal matters (schedule an appointment, request to submit an assignment late, etc.).

Collaboration

Students are encouraged to discuss homework exercises with each other. Each student must write the solutions himself or herself. Copying of solutions or allowing others to copy your solutions is considered cheating and will be handled according to NYU cheating policies.


Weekly schedule

Date Lecture topics Readings Due this class
Jan. 28 The heat equation, evolution equations, superposition principle, fundamental solution Kohn’s notes: Sections 2 and 3
Goodman’s notes: Section 1 		none
Feb. 4 The Fourier transform, boundary conditions and the method of images Kohn’s notes, Sections 2 and 3
Goodman’s notes, Section 2 		HW 1
Feb. 11 Qualitative properties: comparison principles and smoothing Kohn’s notes, Sections 2 and 3
Goodman’s notes, Section 3 		HW 2
Feb. 25 Explicit solutions, interest rate models and the Heston model Goodman’s notes, Section 4 HW 3
March 3 The relationship between PDE and SDE, backward and forward equations Kohn’s notes, Section 1 HW 4
March 10 Stopping times, hitting times, boundary conditions Kohn’s notes, Sections 1 and 2 HW 5
March 24 Deterministic optimal control, the maximum principle, and the Hamilton Jacobi equation Kohn’s notes, Section 4 HW 6
March 31 Jump conditions, viscosity solutions, numerical solution Kohn’s notes, Section 4
Goodman ’s notes, Section 5 		HW 7
Apr. 7 Stochastic optimal control, second order Hamilton Jacobi equations Kohn’s notes, Section 5 HW 8
Apr. 14 Examples: Merton optimal consumption and investment models, Almgren and Chriss optimal execution Kohn’s notes, Section 5
Goodman’s notes, Section 6 		HW 9
Apr. 21 Optimal decision boundaries, boundary conditions, execution and transaction cost problems Kohn’s notes, Section 7
Goodman’s notes, Section 7 		HW 10
Apr. 28 Jump diffusions Kohn’s notes, Section 8 HW 11
May 5 Perturbation theory and analytical approximation methods Goodman’s notes, Section 8 HW 12
May 12 Final exam, same room (1302), same time (5:10 - 7 pm.)