Accurate Formulas for Evaluating Barrier Options with Dividends Payout and the Application in Credit Risk Valuation

Reference work entry


To price the stock options with discrete dividend payout reasonably and consistently, the stock price falls due to dividend payout must be faithfully modeled. However, this will significantly increase the mathematical difficulty since the post-dividend stock price process, the stock price process after the price falls due to dividend payout, no longer follows the lognormal diffusion process. Analytical pricing formulas are hard to be derived even for the simplest vanilla options. This chapter approximates the discrete dividend payout by a stochastic continuous dividend yield, so the post-dividend stock price process can be approximated by another lognormally diffusive stock process with a stochastic continuous payout ratio up to the ex dividend date. Accurate approximation analytical pricing formulas for barrier options are derived by repeatedly applying the reflection principle. Besides, our formulas can be applied to extend the applicability of the first passage model — a branch of structural credit risk model. The stock price falls due to the dividend payout in the option pricing problem is analog to selling the firm’s asset to finance the loan repayment or dividend payout in the first passage model. Thus, our formulas can evaluate vulnerable bonds or the equity values given that the firm’s future loan/dividend payments are known.


Barrier option Option pricing Stock option Dividend Reflection principle Lognormal Credit risk 


  1. Bender, R. & Vorst T. (2001). Options on dividends paying stocks. In Proceeding of the 2001 International Conference on Mathematical Finance, Shanghai.Google Scholar
  2. Black, F. (1975). Fact and fantasy in the use of options. Financial Analysts Journal, 31(36–41), 61–72.Google Scholar
  3. Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637.CrossRefGoogle Scholar
  4. Bos, M., & Vandermark, S. (2002). Finessing fixed dividends. Risk, 15, 157–158.Google Scholar
  5. Chiras, D., & Manaster, S. (1978). The informational content of option prices and a test of market efficiency. Journal of Financial Economics, 6, 213–234.CrossRefGoogle Scholar
  6. Dai, T.-S. (2009). Efficient option pricing on stocks paying discrete or path-dependent dividends with the stair tree. Quantitative Finance, 9, 827–838.CrossRefGoogle Scholar
  7. Dai, T.-S., & Lyuu, Y. D. (2009). Accurate approximation formulas for stock options with discrete dividends. Applied Economics Letters, 16, 1657–1663.CrossRefGoogle Scholar
  8. Figlewski, S., & Gao, B. (1999). The adaptive mesh model: A new approach to efficient option pricing. Journal of Financial Economics, 53, 313–351.CrossRefGoogle Scholar
  9. Frishling, V. (2002). A discrete question. Risk, 15, 115–116.Google Scholar
  10. Gaudenzi, M., & Zanette, A. (2009). Pricing American barrier options with discrete dividends by binomial trees. Decisions in Economics and Finance, 32, 129–148.CrossRefGoogle Scholar
  11. Geske, R. (1979). A note on an analytical valuation formula for unprotected American call options on stocks with known dividends. Journal of Financial Economics, 7(4), 375–380.CrossRefGoogle Scholar
  12. Heath, D., & Jarrow, R. (1988). Ex-dividend stock price behavior and arbitrage opportunities. Journal of Business, 61, 95–108.CrossRefGoogle Scholar
  13. Hull, J. (2003). Options, futures, and other derivatives. Prentice Hall.Google Scholar
  14. Kim, I., Ramaswamy, K., & Sundaresan, S. (1993). Does default risk in coupons affect the valuation of corporate bonds? A contingent claims model. Financial Management, 22, 117–131.CrossRefGoogle Scholar
  15. Lando, D. (2004). Credit risk modeling: Theory and applications. Princeton, NJ: Princeton University Press.Google Scholar
  16. Leland, H. E. (1994). Corporate debt value, bond covenants, and optimal capital structure. Journal of Finance, 49, 157–196.CrossRefGoogle Scholar
  17. Leland, H. E., & Toft, K. (1996). Optimal capital structure, endogenous bankruptcy, and the term structure of credit spreads. Journal of Finance, 51, 987–1019.CrossRefGoogle Scholar
  18. Merton, R. (1973). Theory of rational option pricing. Journal of Economics and Management Science, 4, 141–183.Google Scholar
  19. Merton, R. (1974). On the pricing of corporate debt: The risk structure of interest rates. Journal of Finance, 29, 449–470.Google Scholar
  20. Musiela, M., & Rutkowski, M. (1997). Martingale methods in financial modeling. Sydney: Springer.CrossRefGoogle Scholar
  21. Reiner, E., & Rubinstein, M. (1991). Breaking down the barriers. Risk, 4, 28–35.Google Scholar
  22. Roll, R. (1977). An analytic valuation formula for unprotected American call options on stocks with known dividends. Journal of Financial Economics, 5, 251–258.CrossRefGoogle Scholar
  23. Shreve, E. (2007). Stochastic calculus for finance II: Continuous-time models. New York: Springer Finance.Google Scholar
  24. Vellekoop, M., & Nieuwenhuis, J. (2006). Efficient pricing of derivatives on assets with discrete dividends. Applied Mathematical Finance, 13, 265–284.CrossRefGoogle Scholar

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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.National Chiao–Tung UniversityTaiwanRepublic of China
  2. 2.Institute of Information ManagementNational Chiao Tung UniversityTaiwanRepublic of China

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