On Pricing Equity-Linked Investment Products with a Threshold Expense Structure

Bae, Tae-Han;Ko, Bang-Won

  • Received : 20100400
  • Accepted : 20100400
  • Published : 2010.08.31


This paper considers a certain expense structure where a vendor of equity-linked investment product will collect its expenses continuously from the investor's account whenever the investment performance exceeds a certain threshold level. Under the Black-Scholes framework, we derive compact convolution formulas for evaluating the total expenses to be collected during the investment period by using the joint Laplace transform of the Brownian motion and its excursion time. We provide numerical examples for illustration.


Threshold expense structure;joint Laplace transform;Brownian motion;excursion time


  1. Black, F. and Scholes, M. (1973). The pricing of options and corporate liability, Journal of Political Economy, 81, 637-654.
  2. Borodin, A. and Salminen, P. (2002). Handbook on Brownian Motion-Facts and Formulae, Birkhauser, Bael.
  3. Gerber, H. U. and Shiu, E. W. W. (2006a). Optimal dividend strategies in the compound Poisson model, North American Actuarial Journal, 10, 76-92.
  4. Gerber, H. U. and Shiu, E. S. W. (2006b). On optimal dividends: From re ection to refraction, Journal of Computational and Applied Mathematics, 186, 4-22.
  5. Hugonnier, J. N. (1999). The Feynman-Kac formula and pricing occupation time derivatives, International Journal of Theoretical and Applied Finance, 2, 155-178.
  6. Jeanblanc, M., Pitman, J. and Yor, M. (1997). The Feynman-Kac formula and decomposition of Brownian paths, Computational and Applied Mathematics, 6, 27-52.
  7. Jeanblanc, M., Yor, M. and Chesney, M. (2009). Mathematical Methods for Financial Markets, Springer.
  8. Kac, M. (1949). On the distribution of certain Wiener functionals, Transactions of the AMS, 65, 1-13.
  9. Karatzas, I. and Shreve, S. E. (1991). Brownian Motions and Stochastic Calculus, Springer, New York.
  10. Meyer, M. (2001). Continuous Stochastic Calculus with Applications to Finance, Chapman & Hall/CRC.
  11. Milevsky, M. A. and Posner, S. E. (2001). The Titanic option: Valuation of the guaranteed minimum death benefit in variable annuities and mutual funds, Journal of Risk and Insurance, 68, 93-128.
  12. Shiryayev, A. N. (1999). Essentials of Stochastic Finance: Facts, Models, heory, World Scientific, Singapore.
  13. Shreve, S. E. (2004). Stochastic Calculus for Finance II: Continuous-Time Models, Springer, New York.

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