# ACM30080 PDEs in Financial Maths

The aim of the course is to study the three main types of partial differential equations: parabolic (diffusion equation), elliptic (Laplace equation), and hyperbolic (wave equation), and the techniques of solving these for various initial and boundary value problems on bounded and unbounded domains, using eigenfunction expansions (separation of variables, and elementary Fourier series), integral transform methods (Fourier and Laplace transforms). Applications and examples, such as the solution technique for Black-Scholes option pricing, will be discussed throughout the course

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Curricular information is subject to change

Learning Outcomes:

Upon completion of this module the student should be able to:
Explain the terms existence, uniqueness and stability for PDEs and how this relates to initial and boundary conditions.
Explain the difference between, and be able to give examples of, Dirichlet and Neumann boundary conditions.
Determine whether a PDE is linear, nonlinear, homogeneous, inhomogeneous for given examples.
Solve ﬁrst order PDEs using the method of characteristics.
Classify a second order partial differential equation by type (Hyperbolic, Elliptic, Parabolic) and classify it by order or linearity.
Complete (given a partial proof) proofs for uniqueness of solutions for the wave, diffusion and Laplace equations.
Demonstrate the use of energy methods and the maximum principle to determine uniqueness and stability for a given PDE.
Construct Fourier series expansions for a given periodic function.
Construct half period expansions, both sine and cosine.
Apply the method of separation of variables to solve the diffusion equation, wave equation and Laplace equation with both Dirichlet and Neumann conditions.
Extend the method of separation of variables to inhomogeneous PDEs and apply in applications.
Complete in clearly argued mathematics (given a partial proof) the proof of convergence of Fourier series This includes the Riemann Lebesque Lemma.
Determine the Fourier transform of basic functions.
Determine the Fourier transform of a derivative and the convolution of two functions.
Apply the Fourier transform to the diffusion equation and interpret the solutions generated.
Model the pricing of a European option by providing a commentary on the Black-Scholes equation. Reduce the Black-Scholes equation to the diffusion equation and solve for a call and put option.
Apply the Fourier transform to Laplace and wave equation and for the latter derive the D’Alembert form of the solution to the wave equation.
Implement the Laplace transform to solve the diffusion equation and wave equation for a semi-inﬁnite domain with standard initial conditions.
Construct a proof of the Fourier Integral Theorem.
Express the solution of boundary value problems for the Laplacian in two and three dimensions using Green’s functions.
Prove that the Laplacian of the free-space Green’s functions in two and three dimensions equals the Dirac delta function, and apply the free-space Green’s functions and the method of images to solve boundary value problems.

Student Effort Hours:
Student Effort Type Hours
Specified Learning Activities

36

Autonomous Student Learning

36

Lectures

30

Total

102

Approaches to Teaching and Learning:
Lectures, Problem Classes, Assignments
Requirements, Exclusions and Recommendations
Learning Recommendations:

It is recommended that students should be familiar with material in vector integral and differential calculus.

Module Requisites and Incompatibles
Incompatibles:
ACM30120 - PDEs for Fin. Maths (online), ACM30220 - Partial Differential Equations

Assessment Strategy
Description Timing Open Book Exam Component Scale Must Pass Component % of Final Grade
Class Test: Two class tests Throughout the Trimester n/a Standard conversion grade scale 40% No

40

Continuous Assessment: Weekly homework question Throughout the Trimester n/a Standard conversion grade scale 40% No

20

Examination: End of trimester examination 2 hour End of Trimester Exam No Standard conversion grade scale 40% No

40

Carry forward of passed components
No

Resit In Terminal Exam
Spring Yes - 2 Hour
Feedback Strategy/Strategies

• Group/class feedback, post-assessment

How will my Feedback be Delivered?

Not yet recorded.

Name Role
Dr José Miguel Zapata García Lecturer / Co-Lecturer
Constantinos Menelaou Tutor
Timetabling information is displayed only for guidance purposes, relates to the current Academic Year only and is subject to change.

Autumn

Tutorial Offering 1 Week(s) - Autumn: Weeks 2-12 Fri 13:00 - 13:50
Lecture Offering 1 Week(s) - Autumn: All Weeks Tues 12:00 - 12:50
Lecture Offering 1 Week(s) - Autumn: All Weeks Wed 13:00 - 13:50
Autumn