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Bordag, Ljudmila A.
Publications (10 of 12) Show all publications
Bordag, L. A. & Mikaelyan, A. (2011). Models of self-financing hedging strategies in illiquid markets: Symmetry reductions and exact solutions. Letters in Mathematical Physics, 96(1-3), 191-207
Open this publication in new window or tab >>Models of self-financing hedging strategies in illiquid markets: Symmetry reductions and exact solutions
2011 (English)In: Letters in Mathematical Physics, ISSN 0377-9017, E-ISSN 1573-0530, Vol. 96, no 1-3, p. 191-207Article in journal (Refereed) Published
Abstract [en]

We study the general model of self-financing trading strategies inilliquid markets introduced by Schoenbucher and Wilmott, 2000.A hedging strategy in the framework of this model satisfies anonlinear partial differential equation (PDE) which contains somefunction g(alpha). This function is deep connected to anutility function.

We describe the Lie symmetry algebra of this PDE and provide acomplete set of reductions of the PDE to ordinary differentialequations (ODEs). In addition we are able to describe all types offunctions g(alpha) for which the PDE admits an extended Liegroup. Two of three special type functions lead to modelsintroduced before by different authors, one is new. We clarify theconnection between these three special models and the generalmodel for trading strategies in illiquid markets. We study withthe Lie group analysis the new special case of the PDE describingthe self-financing strategies. In both, the general model and thenew special model, we provide the optimal systems of subalgebrasand study the complete set of reductions of the PDEs to differentODEs. In all cases we are able to provide explicit solutions tothe new special model. In one of the cases the solutions describepower derivative products.

Place, publisher, year, edition, pages
Berlin: Springer Berlin/Heidelberg, 2011
Keywords
nonlinear PDEs, illiquid markets, option pricing
National Category
Computational Mathematics
Identifiers
urn:nbn:se:hh:diva-5536 (URN)10.1007/s11005-011-0463-3 (DOI)000289484500011 ()2-s2.0-79954644110 (Scopus ID)
Available from: 2010-09-23 Created: 2010-09-01 Last updated: 2018-03-23Bibliographically approved
Bordag, L. A. (2011). Study of the risk-adjusted pricing methodology model with methods of geometrical analysis. Stochastics: An International Journal of Probablitiy and Stochastic Processes, 83(4-6), 333-345
Open this publication in new window or tab >>Study of the risk-adjusted pricing methodology model with methods of geometrical analysis
2011 (English)In: Stochastics: An International Journal of Probablitiy and Stochastic Processes, ISSN 1744-2508, E-ISSN 1744-2516, Vol. 83, no 4-6, p. 333-345Article in journal (Refereed) Published
Abstract [en]

Families of exact solutions are found to a nonlinear modification of the Black-Scholes equation. This risk-adjusted pricing methodology model (RAPM) incorporates both transaction costs and the risk from a volatile portfolio. Using the Lie group analysis we obtain the Lie algebra admitted by the RAPM equation. It gives us the possibility to describe an optimal system of subalgebras and the corresponding set of invariant solutions to the model. In this way we can describe the complete set of possible reductions of the nonlinear RAPM model. Reductions are given in the form of different second order ordinary differential equations. In all cases we provide exact solutions to these equations in an explicit or parametric form. Each of these solutions contains a reasonable set of parameters which allows one to approximate a wide class of boundary conditions. We discuss the properties of these reductions and the corresponding invariant solutions.

Place, publisher, year, edition, pages
Abingdon: Taylor & Francis, 2011
Keywords
Transaction costs, Invariant reductions, Exact solutions, Singular perturbation
National Category
Computational Mathematics
Identifiers
urn:nbn:se:hh:diva-5535 (URN)10.1080/17442508.2010.489642 (DOI)000299732200003 ()2-s2.0-84859308758 (Scopus ID)
Note

Special Issue: Optimal stopping with Applications

Available from: 2010-09-20 Created: 2010-09-01 Last updated: 2018-03-23Bibliographically approved
Bordag, L. A. (2010). Pricing options in illiquid markets: optimal systems, symmetry reductions and exact solutions. Lobachevskii Journal of Mathematics, 31(2), 90-99
Open this publication in new window or tab >>Pricing options in illiquid markets: optimal systems, symmetry reductions and exact solutions
2010 (English)In: Lobachevskii Journal of Mathematics, ISSN 1995-0802, E-ISSN 1818-9962, Vol. 31, no 2, p. 90-99Article in journal (Refereed) Published
Abstract [en]

We study a class of nonlinear pricing models which involves the feedback effect from the dynamic hedging strategies on the price of asset introduced by  Sircar and Papanicolaou. We are first to study the case of a nonlinear demand function involved in the model. Using a Lie group analysis we investigate the symmetry properties of these nonlinear diffusion equations. We provide the optimal systems of subalgebras and the complete set of non-equivalent reductions of studied PDEs to ODEs. In most cases we obtain families of exact solutions or derive particular solutions to the equations.

Place, publisher, year, edition, pages
Moscow: Maik Nauka/Interperiodica, 2010
Keywords
Illiquid market, Nonlinearity, Explicit solutions, Lie group analysis
National Category
Mathematical Analysis Computational Mathematics
Identifiers
urn:nbn:se:hh:diva-5530 (URN)10.1134/S1995080210020022 (DOI)2-s2.0-77952977517 (Scopus ID)
Available from: 2010-09-01 Created: 2010-09-01 Last updated: 2018-03-23Bibliographically approved
Bordag, L. A. & Matveev, S. K. (2009). Berechnungen der Gleichgewichtslage der Welle unterpräzisierten geometrischen Voraussetzungen der Druckfluidströmung in LHS 750. Hamburg: HYDROS
Open this publication in new window or tab >>Berechnungen der Gleichgewichtslage der Welle unterpräzisierten geometrischen Voraussetzungen der Druckfluidströmung in LHS 750
2009 (German)Report (Other academic)
Place, publisher, year, edition, pages
Hamburg: HYDROS, 2009. p. 31
National Category
Physical Sciences
Identifiers
urn:nbn:se:hh:diva-5534 (URN)
Available from: 2010-09-01 Created: 2010-09-01 Last updated: 2018-03-23Bibliographically approved
Bordag, L. A. (2009). Symmetry reductions and exact solutions for nonlinear diffusion equations. International Journal of Modern Physics A, 24(8/9), 1713-1716
Open this publication in new window or tab >>Symmetry reductions and exact solutions for nonlinear diffusion equations
2009 (English)In: International Journal of Modern Physics A, ISSN 0217-751X, E-ISSN 1793-656X, Vol. 24, no 8/9, p. 1713-1716Article in journal (Refereed) Published
Abstract [en]

The symmetry properties of nonlinear diffusion equations are studied using a Lie group analysis. Reductions and families of exact solutions are found for some of these equations.

© 2009 World Scientific Publishing Company.

Place, publisher, year, edition, pages
Singapore: World Scientific, 2009
Keywords
Diffusion, Nonlinearity, Explicit solutions
National Category
Mathematical Analysis Computational Mathematics
Identifiers
urn:nbn:se:hh:diva-5526 (URN)10.1142/S0217751X09045285 (DOI)000265092600048 ()2-s2.0-65249183062 (Scopus ID)
Available from: 2010-09-01 Created: 2010-09-01 Last updated: 2018-03-23Bibliographically approved
Bordag, L. A. & Matveev, S. K. (2008). Erste  Berechnungen der Druckfluidströmung. Hamburg: HYDROS
Open this publication in new window or tab >>Erste  Berechnungen der Druckfluidströmung
2008 (German)Report (Other academic)
Place, publisher, year, edition, pages
Hamburg: HYDROS, 2008. p. 27
National Category
Computational Mathematics
Identifiers
urn:nbn:se:hh:diva-5532 (URN)
Available from: 2010-09-01 Created: 2010-09-01 Last updated: 2018-03-23Bibliographically approved
Bordag, L. A. (2008). On Option-Valuation in Illiquid Markets: Invariant Solutions to a Nonlinear Model. In: A. Sarychev A. Shiryaev, M. Guerra and M. R. Grossinho (Ed.), Andrey Sarychev; et al (Ed.), Mathematical control theory and finance: . Paper presented at Workshop on Mathematical Control Theory and Finance, Lisabon, PORTUGAL, 10-14 April, 2007 (pp. 71-94). Berlin: Springer Berlin/Heidelberg
Open this publication in new window or tab >>On Option-Valuation in Illiquid Markets: Invariant Solutions to a Nonlinear Model
2008 (English)In: Mathematical control theory and finance / [ed] Andrey Sarychev; et al, Berlin: Springer Berlin/Heidelberg, 2008, p. 71-94Conference paper, Published paper (Other academic)
Abstract [en]

The present model describes a perfect hedging strategy for a large trader. In this case the hedging strategy affects the price of the underlying security. The feedback-effect leads to a nonlinear version of the Black-Scholes partial differential equation. Using Lie group theory we reduce in special cases the partial differential equation to some ordinary differential equations. The Lie group found for the model equation gives rise to invariant solutions. Families of exact invariant solutions for special values of parameters are described. © 2008 Springer-Verlag Berlin Heidelberg.

Place, publisher, year, edition, pages
Berlin: Springer Berlin/Heidelberg, 2008
Keywords
Hedging strategy, Control theory, Finance, mathematical models
National Category
Computational Mathematics
Identifiers
urn:nbn:se:hh:diva-5856 (URN)10.1007/978-3-540-69532-5_5 (DOI)000260160700005 ()2-s2.0-84895313351 (Scopus ID)978-3-540-69531-8 (ISBN)3-540-69531-1 (ISBN)978-3-540-69532-5 (ISBN)
Conference
Workshop on Mathematical Control Theory and Finance, Lisabon, PORTUGAL, 10-14 April, 2007
Available from: 2010-09-22 Created: 2010-09-21 Last updated: 2018-03-23Bibliographically approved
Bordag, L. A. & Frey, R. (2008). Pricing options in illiquid markets: symmetry reductions and exact solutions. In: Matthias Ehrhardt (Ed.), Nonlinear Models in Mathematical Finance: New Research Trends in Option Pricing (pp. 103-130). New York: Nova Science Publishers, Inc.
Open this publication in new window or tab >>Pricing options in illiquid markets: symmetry reductions and exact solutions
2008 (English)In: Nonlinear Models in Mathematical Finance: New Research Trends in Option Pricing / [ed] Matthias Ehrhardt, New York: Nova Science Publishers, Inc., 2008, p. 103-130Chapter in book (Other academic)
Abstract [en]

The present paper is concerned with nonlinear Black Scholes equations arising in certain option pricing models with a large trader and/or transaction costs. In the first part we give an overview of existing option pricing models with frictions. While the financial setup differs between models, it turns out that in many of these models derivative prices can be characterized by fully nonlinear versions of the standard parabolic Black-ScholesPDE. In the second part of the paper we study a typical nonlinear Black-Scholes equation using methods from Lie group analysis. The equation possesses a rich symmetry group. By introducing invariant variables,  invariant solutions can therefore be characterized in terms of solutions to ordinary differential equations. Finally we discuss properties and applications of these solutions.

Place, publisher, year, edition, pages
New York: Nova Science Publishers, Inc., 2008
Keywords
Nonlinear Black Scholes equations, Option pricing models, Illiquid markets
National Category
Computational Mathematics
Identifiers
urn:nbn:se:hh:diva-5531 (URN)978-1-60456-931-5 (ISBN)
Available from: 2010-09-01 Created: 2010-09-01 Last updated: 2018-03-23Bibliographically approved
Bordag, L. A. & Matveev, S. K. (2008). Präzisierte Berechnungender Druckfluidströmung in LHS 750. Hamburg: HYDROS
Open this publication in new window or tab >>Präzisierte Berechnungender Druckfluidströmung in LHS 750
2008 (German)Report (Other academic)
Place, publisher, year, edition, pages
Hamburg: HYDROS, 2008. p. 19
National Category
Physical Sciences
Identifiers
urn:nbn:se:hh:diva-5533 (URN)
Available from: 2010-09-01 Created: 2010-09-01 Last updated: 2018-03-23Bibliographically approved
Bordag, L. A. & Chmakova, A. Y. (2007). Explicit solutions for a nonlinear model of financial derivatives. International Journal of Theoretical and Applied Finance, 10(1), 1-21
Open this publication in new window or tab >>Explicit solutions for a nonlinear model of financial derivatives
2007 (English)In: International Journal of Theoretical and Applied Finance, ISSN 0219-0249, Vol. 10, no 1, p. 1-21Article in journal (Refereed) Published
Abstract [en]

Families of explicit solutions are found to a nonlinear Black-Scholes equation which incorporates the feedback-effect of a large trader in case of market illiquidity. The typical solution of these families will have a payoff which approximates a strangle. These solutions were used to test numerical schemes for solving a nonlinear Black-Scholes equation. © World Scientific Publishing Company.

Place, publisher, year, edition, pages
Singapore: World Scientific, 2007
Keywords
Black–Scholes model, illiquidity, nonlinearity, explicit solutions
National Category
Computational Mathematics
Identifiers
urn:nbn:se:hh:diva-37585 (URN)10.1142/S021902490700407X (DOI)2-s2.0-33846441215 (Scopus ID)
Note

Funding: The work of the second author was kindly supported by the HWP-project, grant number 02014 of the Brandenburg, MWFK and by the grant of Halmstad University, Sweden.

Available from: 2018-07-13 Created: 2018-07-13 Last updated: 2018-07-13Bibliographically approved
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