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Optimal trading with transaction costs using a PMP gradient method
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
2016 (English)Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesisAlternative title
Optimal handel med transaktionskostnader genom en gradientmetod baserad på PMP (Swedish)
Abstract [en]

This thesis considers a portfolio optimization problem with linear transaction costs, as interpreted by Ampfield Aktiebolag, and analyses it by using a gradient method based on Pontryagin's maximum principle (PMP). First the problem is outlined and afterwards it turns out that a gradient PMP method is easy to employ and gives reasonable solutions. As with many gradient methods the convergence is very slow, but a good estimate could possibly be found in sub-second time with the right implementation and computer.

The strength of the method is the good complexity, linear in the number of time steps and quadratic in the number of dimensions for each iteration. This is compared with quadratic and dynamic programming which have polynomial and exponential complexity respectively.

The main weakness, apart from slow convergence, lies in the assumptions that have to be made. All functions, such as the volatility and transaction costs, are considered to only depend on time, not the transactions made. Using the method in this thesis on a more realistic problem would be difficult, why the PMP gradient method is most suited for a preliminary analysis of the problem.

Abstract [sv]

Detta examensarbete analyserar ett portföljoptimeringsproblem med linjära transaktionskostnader, såsom det är tolkat av Ampfield Aktiebolag, med hjälp av en gradient metod baserad på Pontryagins maximumprincip, eller PMP. Först presenteras problemet och efteråt visar det sig att en gradientmetod är enkel att applicera och ger rimliga lösningar. Som för många gradientmetoder är konvergensen väldigt långsam, men en rimlig approximation kan möjligen hittas på under en sekund med rätt realisation och dator.

Styrkan hos metoden är den goda komplexiteten, linjär i antalet tidssteg och kvadratisk i antalet dimensioner per iteration. Detta jämförs med kvadratisk och dynamisk programmering, som respektive har polynomiell och exponentiell komplexitet.

Den största svagheten, förutom långsam konvergens, ligger i antagandena som måste göras. Alla funktioner, såsom volatiliteten och transaktionskostnaderna, antas bara bero på tiden, inte transaktionerna som gjorts. Att använda metoden i detta arbete på ett mer realistiskt problem skulle vara svårt, varför gradientmetoden lämpar sig bäst för en preliminär analys av problemet.

Place, publisher, year, edition, pages
TRITA-MAT-E, 2016:34
National Category
Computational Mathematics
URN: urn:nbn:se:kth:diva-188811OAI: diva2:939248
Subject / course
Optimization and Systems Theory
Educational program
Master of Science - Applied and Computational Mathematics
Available from: 2016-06-18 Created: 2016-06-18 Last updated: 2016-06-18Bibliographically approved

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