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Two frameworks for pricing defaultable derivatives

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  • Zaevski, Tsvetelin S.
  • Kounchev, Ognyan
  • Savov, Mladen

Abstract

The purpose of this paper is to present two essentially different schemes for deriving the partial differential equations (PDE) for the price of the so-called defaultable derivatives. In the first one the asset price is represented as a solution of a stochastic differential equation (SDE), stopped at a stochastic time. The second one explores the idea of adding a jump process assuming that the stopping time is the moment of its first jump. We investigate also the role of the loss rate, which represents the loss of the asset at the default moment. In both cases we examine various assumptions and dependencies between the underlying asset, the stopping time, and the loss rate. We examine separately the cases when the underlying asset price is driven by a Brownian motion or by a Lévy process.

Suggested Citation

  • Zaevski, Tsvetelin S. & Kounchev, Ognyan & Savov, Mladen, 2019. "Two frameworks for pricing defaultable derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 123(C), pages 309-319.
    • Handle: RePEc:eee:chsofr:v:123:y:2019:i:c:p:309-319
    DOI: 10.1016/j.chaos.2019.04.025
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    as
    1. Koo, Eunho & Kim, Geonwoo, 2017. "Explicit formula for the valuation of catastrophe put option with exponential jump and default risk," Chaos, Solitons & Fractals, Elsevier, vol. 101(C), pages 1-7.
    2. Han, Xingyu, 2018. "Pricing and hedging vulnerable option with funding costs and collateral," Chaos, Solitons & Fractals, Elsevier, vol. 112(C), pages 103-115.
    3. Vidal Nunes, João Pedro & Ruas, João Pedro & Dias, José Carlos, 2015. "Pricing and static hedging of American-style knock-in options on defaultable stocks," Journal of Banking & Finance, Elsevier, vol. 58(C), pages 343-360.
    4. Xu, Guangli & Shao, Xinjian & Wang, Xingchun, 2019. "Analytical valuation of power exchange options with default risk," Finance Research Letters, Elsevier, vol. 28(C), pages 265-274.
    5. Lee, Min-Ku & Kim, Jeong-Hoon, 2018. "Pricing of defaultable options with multiscale generalized Heston’s stochastic volatility," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 144(C), pages 235-246.
    6. Leland, Hayne E, 1994. "Corporate Debt Value, Bond Covenants, and Optimal Capital Structure," Journal of Finance, American Finance Association, vol. 49(4), pages 1213-1252, September.
    7. Duffie, Darrell & Singleton, Kenneth J, 1997. "An Econometric Model of the Term Structure of Interest-Rate Swap Yields," Journal of Finance, American Finance Association, vol. 52(4), pages 1287-1321, September.
    8. Robert A. Jarrow & Stuart M. Turnbull, 2008. "Pricing Derivatives on Financial Securities Subject to Credit Risk," World Scientific Book Chapters, in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 17, pages 377-409, World Scientific Publishing Co. Pte. Ltd..
    9. Jaimungal, Sebastian & Wang, Tao, 2006. "Catastrophe options with stochastic interest rates and compound Poisson losses," Insurance: Mathematics and Economics, Elsevier, vol. 38(3), pages 469-483, June.
    10. Longstaff, Francis A & Schwartz, Eduardo S, 1995. "A Simple Approach to Valuing Risky Fixed and Floating Rate Debt," Journal of Finance, American Finance Association, vol. 50(3), pages 789-819, July.
    11. Xiao, Tim, 2013. "A Simple and Precise Method for Pricing Convertible Bond with Credit Risk," EconStor Open Access Articles and Book Chapters, ZBW - Leibniz Information Centre for Economics, vol. 19(4), pages 259-277.
    12. 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.
    13. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    14. Wang, Guanying & Wang, Xingchun & Zhou, Ke, 2017. "Pricing vulnerable options with stochastic volatility," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 485(C), pages 91-103.
    15. Pavel V. Gapeev & Monique Jeanblanc, 2019. "Defaultable Claims In Switching Models With Partial Information," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(04), pages 1-18, June.
    16. Hull, John & White, Alan, 1995. "The impact of default risk on the prices of options and other derivative securities," Journal of Banking & Finance, Elsevier, vol. 19(2), pages 299-322, May.
    17. Lee, Min-Ku & Yang, Sung-Jin & Kim, Jeong-Hoon, 2016. "A closed form solution for vulnerable options with Heston’s stochastic volatility," Chaos, Solitons & Fractals, Elsevier, vol. 86(C), pages 23-27.
    18. Wang, Xingchun, 2018. "Valuing executive stock options under correlated employment shocks," Finance Research Letters, Elsevier, vol. 27(C), pages 38-45.
    19. Ian Cooper & Marcel Martin, 1996. "Default risk and derivative products," Applied Mathematical Finance, Taylor & Francis Journals, vol. 3(1), pages 53-70.
    20. Tomas Björk & Yuri Kabanov & Wolfgang Runggaldier, 1997. "Bond Market Structure in the Presence of Marked Point Processes," Mathematical Finance, Wiley Blackwell, vol. 7(2), pages 211-239, April.
    21. Black, Fischer & Cox, John C, 1976. "Valuing Corporate Securities: Some Effects of Bond Indenture Provisions," Journal of Finance, American Finance Association, vol. 31(2), pages 351-367, May.
    22. Duffie, Darrell & Singleton, Kenneth J, 1999. "Modeling Term Structures of Defaultable Bonds," Review of Financial Studies, Society for Financial Studies, vol. 12(4), pages 687-720.
    23. Chunsheng Zhou, 1997. "A jump-diffusion approach to modeling credit risk and valuing defaultable securities," Finance and Economics Discussion Series 1997-15, Board of Governors of the Federal Reserve System (U.S.).
    24. Yu, Jun, 2015. "Catastrophe options with double compound Poisson processes," Economic Modelling, Elsevier, vol. 50(C), pages 291-297.
    25. Ye, Jinchun, 2019. "Random distribution kernels and three types of defaultable contingent payoffs," Insurance: Mathematics and Economics, Elsevier, vol. 85(C), pages 198-204.
    26. Kim, Geonwoo & Koo, Eunho, 2016. "Closed-form pricing formula for exchange option with credit risk," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 221-227.
    27. Jeon, Junkee & Kim, Geonwoo, 2019. "Pricing of vulnerable options with early counterparty credit risk," The North American Journal of Economics and Finance, Elsevier, vol. 47(C), pages 645-656.
    28. 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.
    29. In Joon Kim & Krishna Ramaswamy & Suresh Sundaresan, 1993. "Does Default Risk in Coupons Affect the Valuation of Corporate Bonds?: A Contingent Claims Model," Financial Management, Financial Management Association, vol. 22(3), Fall.
    30. Agostino Capponi & Stefano Pagliarani & Tiziano Vargiolu, 2014. "Pricing vulnerable claims in a Lévy-driven model," Finance and Stochastics, Springer, vol. 18(4), pages 755-789, October.
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    More about this item

    Keywords

    Stopping times; Default; Risk-neutral measure; Asset pricing; Derivative pricing; Convertible bonds;
    All these keywords.

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • C25 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Discrete Regression and Qualitative Choice Models; Discrete Regressors; Proportions; Probabilities
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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