• No results found

7. Conclusion 51

7.3. Further Research

This research investigates an propagator model with an assumed VWAP execution strat-egy. It is concluded that such a model can not explain both the empirically observed leptokurtic distribution and the variance scaling with duration. A natural topic for fur-ther research is developing an appropriate theoretical model which is able to capture both. One approach could be to relax the assumption of the assumed VWAP execution

1As this is the only martingale of which the quadratic variation is proportional to volume time.

strategy, but this will likely make analytical derivations much harder. Alternatively, one could possibly also consider a propagator model in which the market impact is consid-ered to be stochastic. The distribution of IS then becomes a mixture of the distributions in market noise and market impact.

Also, this thesis research (implicitly) considers different meta-orders (in different stocks) to be independent of each other. However, it is reasonable to argue that the IS of meta-orders that are executed in the same time period are not independent, as the underlying stocks might not be independent. Appropriately modelling the dependence (or correlation) in IS across different stocks can be useful in modelling total trading costs in large portfolio transitions. A notable mention is the research by (Garcia del Molino et al.,2020), in which the correlation in market impact is investigated in US Treasuries.

Popular summary

In general, there is some time between the moment that investors decide to buy or sell stocks and the moment that they actually do. The relative price difference between the stock prices at these moments (times -1 when selling stocks) is called implementation shortfall (IS). The latter can be interpreted as trading costs. IS is random and can be both positive or negative, because stock prices randomly move up and down. On average however, the costs will be positive, because buying (selling) stocks, on average drives the market price up (down).

This thesis investigates what mathematical model is appropriate for forecasting the dis-tribution IS. That is, predicting the probabilities of all the possible outcomes of IS, before the stocks are bought or sold.

First, it is investigated what is an appropriate mathematical structure for modelling the probability distribution of IS. In particular, it is investigated if the already existing propagator model is appropriate for this. Some important properties of the distribution of IS, such as the variance, are derived in this theoretical model.

Then, the model is tested by checking whether the theoretical model predictions align with observed data. It is found that the average costs are approximately proportional to the square-root of the total order size. Also, it is found that the variance of the costs is approximately proportional to how long it takes to trade the stocks (where time is measured in a non-standard way). Finally, it is shown that the distribution of IS is not well described by a Gaussian distribution, which is predicted by the theoretical model.

Then, the thesis focuses on the development of alternative statistical models that attempt to provide a better description of average costs, cost variance and the cost distribution shape. An important part of this is using the appropriate statistical tools for comparing and assessing the models. The thesis also proposes a new method of forecasting; in which information about the time duration of trades in the past can help provide better forecasts of future trading costs.

Finally, the thesis concludes by noting the theoretical and practical implications of the found results. That is, it outlines how the findings relate to earlier literature and how the findings can be used in practice.

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A. Appendix: Derivations.

A.1. Theoretical Framework Derivations.