IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v4y2016i2p34-d70024.html
   My bibliography  Save this article

Lie Symmetries of (1+2) Nonautonomous Evolution Equations in Financial Mathematics

Author

Listed:
  • Andronikos Paliathanasis

    (Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile, Valdivia 5090000, Chile)

  • Richard M. Morris

    (Department of Mathematics, Institute of Systems Science, Durban University of Technology, PO Box 1334, Durban 4000, South Africa
    These authors contributed equally to this work.)

  • Peter G. L. Leach

    (Department of Mathematics, Institute of Systems Science, Durban University of Technology, PO Box 1334, Durban 4000, South Africa
    School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South Africa
    Department of Mathematics and Statistics, University of Cyprus, Lefkosia 1678, Cyprus
    These authors contributed equally to this work.)

Abstract

We analyse two classes of ( 1 + 2 ) evolution equations which are of special interest in Financial Mathematics, namely the Two-dimensional Black-Scholes Equation and the equation for the Two-factor Commodities Problem. Our approach is that of Lie Symmetry Analysis. We study these equations for the case in which they are autonomous and for the case in which the parameters of the equations are unspecified functions of time. For the autonomous Black-Scholes Equation we find that the symmetry is maximal and so the equation is reducible to the ( 1 + 2 ) Classical Heat Equation. This is not the case for the nonautonomous equation for which the number of symmetries is submaximal. In the case of the two-factor equation the number of symmetries is submaximal in both autonomous and nonautonomous cases. When the solution symmetries are used to reduce each equation to a ( 1 + 1 ) equation, the resulting equation is of maximal symmetry and so equivalent to the ( 1 + 1 ) Classical Heat Equation.

Suggested Citation

  • Andronikos Paliathanasis & Richard M. Morris & Peter G. L. Leach, 2016. "Lie Symmetries of (1+2) Nonautonomous Evolution Equations in Financial Mathematics," Mathematics, MDPI, vol. 4(2), pages 1-14, May.
    • Handle: RePEc:gam:jmathe:v:4:y:2016:i:2:p:34-:d:70024
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/4/2/34/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/4/2/34/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Schwartz, Eduardo S, 1997. "The Stochastic Behavior of Commodity Prices: Implications for Valuation and Hedging," Journal of Finance, American Finance Association, vol. 52(3), pages 923-973, July.
    2. Merton, Robert C, 1974. "On the Pricing of Corporate Debt: The Risk Structure of Interest Rates," Journal of Finance, American Finance Association, vol. 29(2), pages 449-470, May.
    3. Black, Fischer & Scholes, Myron S, 1972. "The Valuation of Option Contracts and a Test of Market Efficiency," Journal of Finance, American Finance Association, vol. 27(2), pages 399-417, May.
    4. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. A. Paliathanasis & K. Krishnakumar & K. M. Tamizhmani & P. G. L. Leach, 2015. "Lie Symmetry Analysis of the Black-Scholes-Merton Model for European Options with Stochastic Volatility," Papers 1508.06797, arXiv.org, revised May 2016.
    2. A. Paliathanasis & R. M. Morris & P. G. L. Leach, 2016. "Lie symmetries of (1+2) nonautonomous evolution equations in Financial Mathematics," Papers 1605.01071, arXiv.org.
    3. Scholes, Myron S, 1998. "Derivatives in a Dynamic Environment," American Economic Review, American Economic Association, vol. 88(3), pages 350-370, June.
    4. Zonggang Ma & Chaoqun Ma & Zhijian Wu, 2022. "Pricing commodity-linked bonds with stochastic convenience yield, interest rate and counterparty credit risk: application of Mellin transform methods," Review of Derivatives Research, Springer, vol. 25(1), pages 47-91, April.
    5. Andronikos Paliathanasis & K. Krishnakumar & K.M. Tamizhmani & Peter G.L. Leach, 2016. "Lie Symmetry Analysis of the Black-Scholes-Merton Model for European Options with Stochastic Volatility," Mathematics, MDPI, vol. 4(2), pages 1-14, May.
    6. Mariano González-Sánchez & Eva M. Ibáñez Jiménez & Ana I. Segovia San Juan, 2022. "Market and model risks: a feasible joint estimate methodology," Risk Management, Palgrave Macmillan, vol. 24(3), pages 187-213, September.
    7. Donders, Pablo & Jara, Mauricio & Wagner, Rodrigo, 2018. "How sensitive is corporate debt to swings in commodity prices?," Journal of Financial Stability, Elsevier, vol. 39(C), pages 237-258.
    8. Gordian Rättich & Kim Clark & Evi Hartmann, 2011. "Performance measurement and antecedents of early internationalizing firms: A systematic assessment," Working Papers 0031, College of Business, University of Texas at San Antonio.
    9. Peter Carr & Liuren Wu, 2014. "Static Hedging of Standard Options," Journal of Financial Econometrics, Oxford University Press, vol. 12(1), pages 3-46.
    10. Jeremy Leake, 2003. "Credit spreads on sterling corporate bonds and the term structure of UK interest rates," Bank of England working papers 202, Bank of England.
    11. Boulanouar, Zakaria & Alqahtani, Faisal & Hamdi, Besma, 2021. "Bank ownership, institutional quality and financial stability: evidence from the GCC region," Pacific-Basin Finance Journal, Elsevier, vol. 66(C).
    12. Richardson, Grant & Taylor, Grantley & Lanis, Roman, 2015. "The impact of financial distress on corporate tax avoidance spanning the global financial crisis: Evidence from Australia," Economic Modelling, Elsevier, vol. 44(C), pages 44-53.
    13. Zhijian (James) Huang & Yuchen Luo, 2016. "Revisiting Structural Modeling of Credit Risk—Evidence from the Credit Default Swap (CDS) Market," JRFM, MDPI, vol. 9(2), pages 1-20, May.
    14. Sandrine Lardic & Claire Gauthier, 2003. "Un modèle multifactoriel des spreads de crédit : estimation sur panels complets et incomplets," Économie et Prévision, Programme National Persée, vol. 159(3), pages 53-69.
    15. Polito, Vito & Wickens, Michael, 2015. "Sovereign credit ratings in the European Union: A model-based fiscal analysis," European Economic Review, Elsevier, vol. 78(C), pages 220-247.
    16. Hilscher, Jens & Raviv, Alon, 2014. "Bank stability and market discipline: The effect of contingent capital on risk taking and default probability," Journal of Corporate Finance, Elsevier, vol. 29(C), pages 542-560.
    17. Andres, Christian & Cumming, Douglas & Karabiber, Timur & Schweizer, Denis, 2014. "Do markets anticipate capital structure decisions? — Feedback effects in equity liquidity," Journal of Corporate Finance, Elsevier, vol. 27(C), pages 133-156.
    18. Anna Kovner & Chenyang Wei, 2012. "The private premium in public bonds," Staff Reports 553, Federal Reserve Bank of New York.
    19. Bujar Huskaj & Marcus Nossman, 2013. "A Term Structure Model for VIX Futures," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 33(5), pages 421-442, May.
    20. Bjork, Tomas, 2009. "Arbitrage Theory in Continuous Time," OUP Catalogue, Oxford University Press, edition 3, number 9780199574742.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:4:y:2016:i:2:p:34-:d:70024. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.