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)

More Information

Further details

Please contact Carlos Vazquez Cendon

Slides, 2012 Day 1 and Day 2 and Day 3