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Algorithmic trading with Markov chains
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.ORCID iD: 0000-0001-9210-121X
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
(English)Manuscript (preprint) (Other academic)
Abstract [en]

An order book consists of a list of all buy and sell offers, represented by price and quantity, available to a market agent. The order book changes rapidly, within fractions of a second, due to new orders being entered into the book. The volume at a certain price level may increase due to limitorders, i.e. orders to buy or sell placed at the end of the queue, or decrease because of market orders or cancellations.

In this paper a high-dimensional Markov chain is used to represent the state and evolution of the entire order book. The design and evaluation of optimal algorithmic strategies for buying and selling is studied within the theory of Markov decision processes. General conditions are provided that guarantee the existence of optimal strategies. Moreover, a value-iteration algorithm is presented that enables finding optimal strategies numerically.

As an illustration a simple version of the Markov chain model is calibrated to high-frequency observations of the order book in a foreign exchange market. In this model, using an optimally designed strategy for buying one unit provides a significant improvement, in terms of the expected buy price, over a naive buy-one-unit strategy.

National Category
Other Mathematics
URN: urn:nbn:se:kth:diva-25075OAI: diva2:355640
QC 20101007Available from: 2010-10-07 Created: 2010-10-07 Last updated: 2010-10-08Bibliographically approved
In thesis
1. Approximation and Calibration of Stochastic Processes in Finance
Open this publication in new window or tab >>Approximation and Calibration of Stochastic Processes in Finance
2010 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis is a study of approximation and calibration of stochastic processes with applications in finance. It consists of an introduction and four research papers.

The introduction is as an overview of the role of mathematics incertain areas of finance. It contains a brief introduction to the mathematicaltheory of option pricing, as well as a description of a mathematicalmodel of a financial exchange. The introduction also includessummaries of the four research papers.

In Paper I, Markov decision theory is applied to design algorithmic trading strategies in an order driven market. A high dimensional Markov chain is used to model the state and evolution of the limit order book. Trading strategies are formulated as optimal decision problems. Conditions that guarantee existence of optimal strategies are provided, as well as a value-iterative algorithm that enables numerical construction of optimal strategies. The results are illustrated with numerical experiments on high frequency data from a foreign exchange market.

Paper II focuses on asset pricing with Lévy processes. The expected value E[g(XT )] is estimated using a Monte Carlo method, when Xt is a d-dimensional Lévy process having infinite jump activity and a smooth density. Approximating jumps smaller then a parameter ε > 0 by diffusion results in a weak approximation, Xt, of Xt. The main result of the paper is an estimate of the resulting model error E[g(XT )] − E[g(T )], with a computable leading order term.

Option prices in exponential Lévy models solve certain partia lintegro-differential equations (PIDEs). A finite difference scheme suitable for solving such PIDEs is studied in Paper III. The main resultsare estimates of the time and space discretization errors, with leading order terms in computable form. If the underlying Lévy process has infinite jump activity, the jumps smaller than some ε > 0 are replacedby diffusion. The size of this diffusion approximation is estimated, as well as its effect on the space and time discretization errors. Combined, the results of the paper are enough to determine how to jointly choose the grid size and the parameter ε.

In Paper IV it is demonstrated how optimal control can be used to calibrate a jump-diffusion process to quoted option prices. The calibration problem is formulated as an optimal control problem with the model parameter as a control variable. The corresponding regularized Hamiltonian system is solved with a symplectic Euler method.

Place, publisher, year, edition, pages
Stockholm: KTH, 2010. viii, 45 p.
Trita-MAT. MA, ISSN 1401-2278 ; 10:05
National Category
urn:nbn:se:kth:diva-25090 (URN)978-91-7415-741-3 (ISBN)
Public defence
2010-10-18, Sal F3, Lindstedtsvägen 26, KTH, Stockholm, 13:00 (English)
QC 20101008Available from: 2010-10-08 Created: 2010-10-08 Last updated: 2010-10-08Bibliographically approved

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Hult, HenrikKiessling, Jonas
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