IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1309.6505.html
   My bibliography  Save this paper

General Properties of Solutions to Inhomogeneous Black-Scholes Equations with Discontinuous Maturity Payoffs and Application

Author

Listed:
  • Hyong-Chol O
  • Ji-Sok Kim

Abstract

We provide representations of solutions to terminal value problems of inhomogeneous Black-Scholes equations and studied such general properties as min-max estimates, gradient estimates, monotonicity and convexity of the solutions with respect to the stock price variable, which are important for financial security pricing. In particular, we focus on finding representation of the gradient (with respect to the stock price variable) of solutions to the terminal value problems with discontinuous terminal payoffs or inhomogeneous terms. Such terminal value problems are often encountered in pricing problems of compound-like options such as Bermudan options or defaultable bonds with discrete default barrier, default intensity and endogenous default recovery. Our results are applied in pricing defaultable discrete coupon bonds.

Suggested Citation

  • Hyong-Chol O & Ji-Sok Kim, 2013. "General Properties of Solutions to Inhomogeneous Black-Scholes Equations with Discontinuous Maturity Payoffs and Application," Papers 1309.6505, arXiv.org, revised Sep 2013.
    • Handle: RePEc:arx:papers:1309.6505
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1309.6505
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Bergman, Yaacov Z & Grundy, Bruce D & Wiener, Zvi, 1996. "General Properties of Option Prices," Journal of Finance, American Finance Association, vol. 51(5), pages 1573-1610, December.
    2. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    3. Nicole El Karoui & Monique Jeanblanc‐Picquè & Steven E. Shreve, 1998. "Robustness of the Black and Scholes Formula," Mathematical Finance, Wiley Blackwell, vol. 8(2), pages 93-126, April.
    4. 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.
    5. Peter Buchen, 2004. "The pricing of dual-expiry exotics," Quantitative Finance, Taylor & Francis Journals, vol. 4(1), pages 101-108.
    6. Erik Ekstrom & Johan Tysk, 2005. "Properties of option prices in models with jumps," Papers math/0509232, arXiv.org, revised Nov 2005.
    7. Yaacov Z. Bergman & Bruce D. Grundy & Zvi Wiener, "undated". "General Properties of Option Prices (Revision of 11-95) (Reprint 058)," Rodney L. White Center for Financial Research Working Papers 1-96, Wharton School Rodney L. White Center for Financial Research.
    8. Jagannathan, Ravi, 1984. "Call options and the risk of underlying securities," Journal of Financial Economics, Elsevier, vol. 13(3), pages 425-434, September.
    9. Cox, John C & Ross, Stephen A, 1976. "A Survey of Some New Results in Financial Option Pricing Theory," Journal of Finance, American Finance Association, vol. 31(2), pages 383-402, May.
    10. 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.
    11. Rossella Agliardi, 2011. "A comprehensive structural model for defaultable fixed-income bonds," Quantitative Finance, Taylor & Francis Journals, vol. 11(5), pages 749-762.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Vahidreza Yousefi & Siamak Haji Yakhchali & Jolanta Tamošaitienė, 2019. "Application of Duration Measure in Quantifying the Sensitivity of Project Returns to Changes in Discount Rates," Administrative Sciences, MDPI, vol. 9(1), pages 1-14, February.
    2. Hyong-Chol O & Dae-Sung Choe, 2019. "Pricing Formulae of Power Binary and Normal Distribution Standard Options and Applications," Papers 1903.04106, arXiv.org.
    3. Hyong Chol O & Tae Song Kim, 2020. "Analysis on the Pricing model for a Discrete Coupon Bond with Early redemption provision by the Structural Approach," Papers 2007.01511, arXiv.org.
    4. Hyong-Chol O. & Jong-Chol Kim & Il-Gwang Jon, 2017. "Numerical analysis for a unified 2 factor model of structural and reduced form types for corporate bonds with fixed discrete coupon," Papers 1709.06517, arXiv.org, revised Aug 2018.

    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. Antonio Mele, 2003. "Fundamental Properties of Bond Prices in Models of the Short-Term Rate," The Review of Financial Studies, Society for Financial Studies, vol. 16(3), pages 679-716, July.
    2. Hyong-Chol O & Tae-Song Choe, 2022. "General properties of the Solutions to Moving Boundary Problems for Black-Sholes Equations," Papers 2203.05726, arXiv.org.
    3. José Fajardo & Ernesto Mordecki, 2006. "Skewness Premium with Lévy Processes," IBMEC RJ Economics Discussion Papers 2006-04, Economics Research Group, IBMEC Business School - Rio de Janeiro.
    4. Lim, Terence & Lo, Andrew W. & Merton, Robert C. & Scholes, Myron S., 2006. "The Derivatives Sourcebook," Foundations and Trends(R) in Finance, now publishers, vol. 1(5–6), pages 365-572, April.
    5. Eric Rasmusen, 2004. "When Does Extra Risk Strictly Increase the Value of Options?," Finance 0409004, University Library of Munich, Germany.
    6. Norden, Lars, 2001. "Hedging of American equity options: do call and put prices always move in the direction as predicted by the movement in the underlying stock price?," Journal of Multinational Financial Management, Elsevier, vol. 11(4-5), pages 321-340, December.
    7. Chuang Yuang Lin & Dar Hsin Chen & Chin Yu Tsai, 2011. "The limitation of monotonicity property of option prices: an empirical evidence," Applied Economics, Taylor & Francis Journals, vol. 43(23), pages 3103-3113.
    8. Mykland, Per Aslak, 2019. "Combining statistical intervals and market prices: The worst case state price distribution," Journal of Econometrics, Elsevier, vol. 212(1), pages 272-285.
    9. Jérôme Detemple & Carlton Osakwe, 2000. "The Valuation of Volatility Options," Review of Finance, European Finance Association, vol. 4(1), pages 21-50.
    10. Stephane Crepey, 2004. "Delta-hedging vega risk?," Quantitative Finance, Taylor & Francis Journals, vol. 4(5), pages 559-579.
    11. Mele, Antonio, 2004. "General Properties of Rational Stock-Market Fluctuations," Economics Series 153, Institute for Advanced Studies.
    12. Antonio Mele, 2003. "Fundamental Properties of Bond Prices in Models of the Short-Term Rate," Review of Financial Studies, Society for Financial Studies, vol. 16(3), pages 679-716, July.
    13. GARCIA, René & RENAULT, Éric, 1998. "Risk Aversion, Intertemporal Substitution, and Option Pricing," Cahiers de recherche 9801, Universite de Montreal, Departement de sciences economiques.
    14. Constantinides, George M. & Jackwerth, Jens Carsten & Perrakis, Stylianos, 2005. "Option pricing: Real and risk-neutral distributions," CoFE Discussion Papers 05/06, University of Konstanz, Center of Finance and Econometrics (CoFE).
    15. Eric Rasmusen, 2007. "When Does Extra Risk Strictly Increase an Option's Value?," Review of Financial Studies, Society for Financial Studies, vol. 20(5), pages 1647-1667, 2007 14.
    16. Alfredo Ibáñez, 2005. "Option-Pricing in Incomplete Markets: The Hedging Portfolio plus a Risk Premium-Based Recursive Approach," Computing in Economics and Finance 2005 216, Society for Computational Economics.
    17. Yue, Tian & Zhang, Jin E. & Tan, Eric K.M., 2020. "The Chinese equity index options market," Emerging Markets Review, Elsevier, vol. 45(C).
    18. Minqiang Li, 2010. "A quasi-analytical interpolation method for pricing American options under general multi-dimensional diffusion processes," Review of Derivatives Research, Springer, vol. 13(2), pages 177-217, July.
    19. Robert R. Bliss, 2000. "The pitfalls in inferring risk from financial market data," Working Paper Series WP-00-24, Federal Reserve Bank of Chicago.
    20. Rasmussen, Nicki Søndergaard, 2002. "Hedging with a Misspecified Model," Finance Working Papers 02-15, University of Aarhus, Aarhus School of Business, Department of Business Studies.

    More about this item

    Statistics

    Access and download statistics

    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:arx:papers:1309.6505. 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: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    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.