Advertisement

Journal of Systems Science and Complexity

, Volume 26, Issue 6, pp 968–977 | Cite as

Pricing convertible bonds and change of probability measure

  • Zhaoli Jia
  • Shuguang ZhangEmail author
Article
  • 169 Downloads

Abstract

The changes of numeraire can be used as a very powerful tool in pricing contingent claims in the context of a complete market. By using the method of numeraire changes to evaluate convertible bonds when the value of firm, and those of zero-coupon bonds follow general adapted stochastic processes in this paper, using It_o theorem and Gisanov theorem. A closed-form solution is derived under the stochastic volatility by using fast Fourier transforms.

Key words

Convertible bonds European option numeraire changes stochastic volatility model 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Black F and Scholes M, The pricing of options and corporate liabilities, Journal of Political Economy, 1973, 81(3): 637–654.CrossRefGoogle Scholar
  2. [2]
    Ingersoll J, A contingent claim valuation of convertible securities, Journal of Financial Economics, 1977, 4: 463–478.CrossRefGoogle Scholar
  3. [3]
    Brennan M J and Schwatz E S, Convertible bonds: Valuation and optimal strategies for call and conversion, Journal of Financial, 1977, 32(5): 1699–1715.Google Scholar
  4. [4]
    Brennan M J and Schwatz E S, Analyzing convertible bonds, Journal of Financial and Quantitative Analysis, 1980, 15(4): 907-929.Google Scholar
  5. [5]
    McConnell J and Schwartz E S, Taming LYONS, The Journal of Finance, 1986, 41(3): 561–576.CrossRefGoogle Scholar
  6. [6]
    Hull J and White A, The pricing of options on assets with stochastic volatilities, Journal of Finance, 1987, 42(2): 281–300.CrossRefGoogle Scholar
  7. [7]
    Geman H, Karoui N E, and Rochet J C, Changes of numeraire: Changes of probability measure and option price, Journal of Applied Probability, 1995, 32: 443–458.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Longstaff F and Schwartz E, A simple approach to valuing risky fixed and floating rate debt, Journal of Finance, 1995, 50(3): 789-821.Google Scholar
  9. [9]
    Jarrow R and Turnbull S, Pricing derivatives on financial securities subject to credit risk, The Journal of Finance, 1995, 50(1): 53–85.CrossRefGoogle Scholar
  10. [10]
    Tsiveriotis K and Fernandes C, Valuing convertible bonds with credit risk, The Journal of Fixed Income, 1998, 9: 95–102.CrossRefGoogle Scholar
  11. [11]
    Hull J, Options, Futures, and Other Derivatives, 6th ed., Prentice Hall, Englewood Cliffs, 2003.zbMATHGoogle Scholar
  12. [12]
    Yigitbasioglu A B, Pricing convertible bonds with interest rate, equity, credit, and FX risk, The Business School for Financial Markets, 2001, 14: 1–10.Google Scholar
  13. [13]
    Sîrbu M, Pikovsky I, and Shreve S E, Perpetual convertible bonds American barrier options using the decomposition technique, SIAM Journal of Control and Optimization, 2004, 43: 58–85.CrossRefzbMATHGoogle Scholar
  14. [14]
    Shreve S E, Stochastic Calculus for Finance II: Continuous Time Models, Springer-Verlag, New York, 2004.zbMATHGoogle Scholar
  15. [15]
    Carr P, Hedging under the Heston model with jump-to-default, International Journal of Theoretical and Applied Finance, 2008, 11(4): 403–414.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Heston S, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies, 1993, 6(2): 327–343.CrossRefGoogle Scholar
  17. [17]
    Girsanov I V, On transforming a certain class of stochastic processes by absolutely continuous substitution of measures, Theory Prob. Appl., 1960, 5: 285–301.CrossRefGoogle Scholar
  18. [18]
    Ash R and Doléans-Dade, Probability and Measure Theory, Academic Press, 2000.zbMATHGoogle Scholar
  19. [19]
    Cox J C, Ingersoll J E, and Ross S A, A theory of the term structure of interest rates, Econometrica, 1985, 53: 385–408.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Schoutens W, Simons E, and Tistaert J, A perfect calibration! Now what?, Wilmott Magazine, March, 2005, 66–78.Google Scholar
  21. [21]
    Albrecher H, Mayer P, Schoutens W, and Tistaert J, The little Heston trap, Wilmott Magazine, January, 2007, 83–92.Google Scholar
  22. [22]
    Carr P and Mafan D, Option valuation using the fast Fourier transform, Journal of Computational Finance, 1999, 6: 61–73.Google Scholar

Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Statistics and FinanceUniversity of Science and Technology of ChinaHefeiChina
  2. 2.School of MathematicsHefei University of TechnologyHefeiChina

Personalised recommendations