## Short Course: Numerical Solution of Hamilton Jacobi Bellman PDEs in Finance

#### November 12-14, 2012

#### Times: 16:00-19:00

#### Location: Aula de Grados, Facultad de InformĂˇtica, Universidade da CoruĂ±a

### Lecturer: Peter Forsyth, Cheriton
School of Computer Science, University of Waterloo

Many problems in finance can be posed as non-linear Hamilton
Jacobi Bellman (HJB) Partial Integro Differential Equations (PIDEs).
Examples of such problems include: dynamic asset allocation for
pension plans, optimal operation of natural gas storage facilities,
optimal execution of trades, and pricing of variable annuity products
(e.g. Guaranteed Minimum Withdrawal Benefit).

This course will
discuss general numerical methods for solving the HJB PDEs which
arise from these types of problems. After an introductory lecture,
we will give an example where seemingly reasonable methods do not
converge to the correct (viscosity) solution of a nonlinear HJB
equation. A set of general guidelines is then established which will
ensure convergence of the numerical method to the viscosity solution.
Emphasis will be placed on methods which are straightforward to
implement. We then illustrate these techniques on some of the
problems mentioned above.

### Day 1

- Lecture 1: Examples of HJB Equations, Viscosity Solutions
(1 hr)
- Lecture 2: Sufficient Conditions for Convergence to the
Viscosity Solution (1 hr)
- Lecture 3 (Part I): Pension Plan Asset Allocation, Passport Options (.5 hr)

### Day 2

- Lecture 3 (Part II): Pension Plan Asset Allocation, Passport Options (.5 hr)
- Lecture 4: Guaranteed Minimum Withdrawal Benefit (GMWB)
Variable Annuity: Impulse Control Formulation (1 hr)
- Lecture 5: Gas Storage (1 hr)

### Day 3

- Lecture 6: Continuous Time Mean Variance Asset Allocation (1 hr)
- Lecture 7: Optimal Trade Execution (1 hr)
- Lecture 8: Summary (.5 hr)

### Further details

Please contact
Carlos Vazquez Cendon

Slides, 2012
Day 1
and
Day 2
and
Day 3