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On Discrete Probability Approximations for Transaction Cost Problems

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Abstract

In this paper we consider a discrete-time formulation of dynamic transaction cost problems. We examine applicability of numerical discrete probability approximation as an alternative simplistic approach to solve dynamic transaction cost problems. We provide a computational study of a lattice-based heuristic method on simple transaction cost models and highlight its many advantages. The solution of these problems provides a dynamic investor with important insights as to how the portfolio should be re-balanced when faced with transaction costs.

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Notes

  1. Reasonable dimensions for lattice based methods are generally less than or equal to three because of the curse of dimensionality.

  2. Note \((1+g)\) is the continuous-time analogue of discrete time risky growth \(r_{i,k}\).

  3. However, by using suitable penalty adjustments in our dynamic programming formalism, we could accommodate generalized liquidity/portfolio constraints of the form \(-a \le {\mathcal {A}}_{i,k}^- \le a\) for some \(a \in {\mathcal {R}}\). Our model could easily accommodate constraints of the type \(Y_{k+1}^{-}+(1-\lambda _{k}) \sum X_{i,k+1}^- \ge\)0 for more generalized transaction cost structures. This could be handled by suitable adjustments to the dynamic programming methodology by invoking some constraints or applying a penalty to the value function. In theory by forward evolving the discrete probability approximation under the state equations we get an evolution of the state variable domain relevant for dynamic programming.

  4. GBM only for illustrative purposes. In general, the approximation schemes work for any arbitrary distribution.

  5. By \(\gamma _\ell =\frac{1}{\ell }\) we refer to as the deformation parameter. The deformation parameter going to zero helps us get the limiting case of a continuous distribution.

  6. Scaling by time horizon does not impact the optimal controls obtained as we shall see.

  7. Assuming a logarithmic utility and choosing \(u=e^{m\varDelta T+\sigma \sqrt{\varDelta T}}\), \(d=e^{m \varDelta T -\sigma \sqrt{\varDelta T}}\) and \(p=1/2\) we create an approximation for Merton point using dynamic programming.

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Acknowledgements

The ideas and methodologies in this paper were developed during the author’s PhD studies as in Butt (2012). The author will also like to acknowledge the valuable comments of the referees that has led to a much improved version of this paper.

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  1. Department of Business Administration, Lahore School of Economics, Lahore, Pakistan

    Nabeel Butt

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Butt, N. On Discrete Probability Approximations for Transaction Cost Problems. Asia-Pac Financ Markets 26, 365–389 (2019). https://doi.org/10.1007/s10690-019-09270-8

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