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Approximation and Calibration of Stochastic Processes in Finance
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
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.
Series
Trita-MAT. MA, ISSN 1401-2278 ; 10:05
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-25090ISBN: 978-91-7415-741-3 (print)OAI: oai:DiVA.org:kth-25090DiVA: diva2:355746
Public defence
2010-10-18, Sal F3, Lindstedtsvägen 26, KTH, Stockholm, 13:00 (English)
Opponent
Supervisors
Note
QC 20101008Available from: 2010-10-08 Created: 2010-10-08 Last updated: 2010-10-08Bibliographically approved
List of papers
1. Algorithmic trading with Markov chains
Open this publication in new window or tab >>Algorithmic trading with Markov chains
(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
Identifiers
urn:nbn:se:kth:diva-25075 (URN)
Note
QC 20101007Available from: 2010-10-07 Created: 2010-10-07 Last updated: 2010-10-08Bibliographically approved
2. Diffusion approximation of Lévy processes with a view towardsfinance
Open this publication in new window or tab >>Diffusion approximation of Lévy processes with a view towardsfinance
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Let the (log-)prices of a collection of securities be given by a d–dimensional Lévy process Xt having infinite activity and a smooth density. The value of a European contract with payoff g(x) maturing at T is determined by E[g(XT )]. Let ¯XT be a finite activity approximation to XT , where diffusion is introduced to approximate jumps smaller than a given truncation level ε > 0. The main result of this work is a derivationof an error expansion for the resulting model error, E[g(XT )−g( ¯XT )], with computable leading order term. Our estimate depends both on the choice of truncation level ε and the contract payoff g, and it is valid even when g is not continuous. Numerical experiments confirm that the error estimate is indeed a good approximation of the model error.

Using similar techniques we indicate how to construct an adaptive truncation type approximation. Numerical experiments indicate that a substantial amount of work is to be gained from such adaptive approximation. Finally, we extend the previous model error estimates to the case of Barrier options, which have a particular path dependent structure.

National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-25078 (URN)
Note
QC 20101007Available from: 2010-10-07 Created: 2010-10-07 Last updated: 2010-10-08Bibliographically approved
3. Computable error estimates of a finite difference scheme for option pricing in exponential Lévy models
Open this publication in new window or tab >>Computable error estimates of a finite difference scheme for option pricing in exponential Lévy models
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Option prices in exponential L´evy models solve certain partial integrodifferential equations (PIDEs). This work focuses on a finite difference scheme that issuitable for solving such PIDEs. The scheme was introduced in [Cont and Voltchkova, SIAM J. Numer. Anal., 43(4):1596–1626, 2005]. The main results of this work are new estimates of the dominating error terms, namely the time and space discretization errors. In addition, the leading order terms of the error estimates are determined in computable form. The payoff is only assumed to satisfy an exponential growth condition, it is not assumed to be Lipschtitz continuous as in previous works.

If the underlying Lévy process has infinite jump activity, then the jumps smallerthan some ε> 0 are approximated by diffusion. The resulting diffusion approximationerror is also estimated, with leading order term in computable form, as well as its effecton the space and time discretization errors. Consequently, it is possible to determine how to jointly choose the space and time grid sizes and the parameter ε.

National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-25081 (URN)
Note
QC 20101007Available from: 2010-10-07 Created: 2010-10-07 Last updated: 2010-10-08Bibliographically approved
4. Calibration of a Jump-Diffusion Process Using Optimal Control
Open this publication in new window or tab >>Calibration of a Jump-Diffusion Process Using Optimal Control
2012 (English)In: Numerical Analysis Of Multiscale Computations / [ed] Engquist, B; Runborg, O; Tsai, YHR, Springer Berlin/Heidelberg, 2012, 259-277 p.Conference paper, Published paper (Refereed)
Abstract [en]

A method for calibrating a jump-diffusion model to observed option prices is presented. The calibration problem is formulated as an optimal control problem, with the model parameters as the control variable. It is well known that such problems are ill-posed and need to be regularized. A Hamiltonian system, with non-differentiable Hamiltonian, is obtained from the characteristics of the corresponding Hamilton-Jacobi-Bellman equation. An explicit regularization of the Hamiltonian is suggested, and the regularized Hamiltonian system is solved with a symplectic Euler method. The paper is concluded with some numerical experiments on real and artificial data.

Place, publisher, year, edition, pages
Springer Berlin/Heidelberg, 2012
Series
Lecture Notes in Computational Science and Engineering, ISSN 1439-7358 ; 82
Keyword
Stochastic Volatility, Options
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-25087 (URN)10.1007/978-3-642-21943-6_12 (DOI)000310180800012 ()978-3-642-21942-9 (ISBN)
Conference
Workshop on Numerical Analysis and Multiscale Computations Location: Banff Int Res Stn, Banff, Canada Date: DEC 06-11, 2009
Note

QC 20101008. Updated from manuscript to conference paper.

Available from: 2010-10-08 Created: 2010-10-08 Last updated: 2012-11-23Bibliographically approved

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