From approximate to exact probability models in dynamic portfolio decision theory

Abstract

In the academic literature dynamic portfolio theory under transaction cost literature is restricted to a continuous time framework which results in quasi-variational Hamilton-Jacobi-Bellman (HJB) free boundary Partial Differential Equations (PDEs). The objective of this article is to create a generic, robust, and efficient framework that could handle both discrete and continuous models simultaneously. The discrete time formulation is a special case of continuous time formulation. The article proposes probability deformation solution schemes and examines their efficiency. Analysis is restricted to the popular transaction cost frameworks introduced by Davis and Norman in 1990 and Taksar et al. in 1988. In contrast to continuous version of the model the discrete version of the model is intuitive and easy to implement. Easy to implement heuristics to solve dynamic portfolio problems are very valuable in providing insights in to re-balancing portfolios when faced with transaction costs.

Keywords:

• Dynamic programming
• dynamic portfolio management
• transaction cost
• discrete probability approximation

Acknowledgements

The ideas and methodologies in this article were developed during the author’s PhD studies as in Butt, 2012. The author would also like to acknowledge the valuable comments of the referees that have led to a much improved version of this article.

Disclosure statement

The author has nothing to disclose and has no conflict of interest.

Notes

1 The word “deformed” would be used a lot by us. Basically we term the approximation of a continuous distribution as a deformation process.

2 Even though the deformation schemes solve for discrete time the continuous time solution is also obtained as a limit.

3 Forward looking scenarios are very vital in regulatory calculations in the financial industry.

4 Suitable grid space setting so as to ensure that the interpolation and extrapolation errors are virtually zero.

5 So if the joint random variables were (x, y) we would set $E[x]=0,E[y]=0$

6 So if the joint random variables were (x, y) we would set $E[x^2]=0,E[y^2]=0,E[xy]=0$