Analytic Solution of Black-Scholes-Merton European Power Put Option Model on Dividend Yield with Modified-Log-Power Payoff Function

Main Article Content

S.E. Fadugba, A.A. Adeniji, M.C. Kekana, J.T. Okunlola, O. Faweya


This paper proposes a framework based on the celebrated transform of Mellin type (MT) for the analytic solution of the Black-Scholes-Merton European Power Put Option Model (BSMEPPOM) on Dividend Yield (DY) with Modified-Log-Power Payoff Function (MLPPF) under the geometric Brownian motion. The MT has the capability of tackling complex functions by means of its fundamental properties and it is closely related to other well-known transforms such as Laplace and Fourier types. The main goal of this paper is to use MT to obtain a valuation formula for the European Power Put Option (EPPO) which pays a DY with MLPPF. By means of MT and its inversion formula, the price of EPPO on DY was expressed in terms of integral equation. Moreover, the valuation formula of EPPO was obtained with the help of the convolution property of MT and final time condition. The MT was tested on an illustrative example in order to measure its performance, effectiveness and suitability. The MLPPF was compared with other existing payoff functions. Hence, the effect of DY on the pricing of EPPO with MLPPF was also investigated.

Article Details


  1. F. Black, M. Scholes, The Pricing of Options and Corporate Liabilities, J. Political Econ. 81 (1973), 637-654.
  2. S.E. Fadugba, C.R. Nwozo, Valuation of European Call Options via the Fast Fourier Transform and the Improved Mellin Transform, J. Math. Finance. 06 (2016), 338–359.
  3. R.C. Merton, Option Pricing When Underlying Stock Returns Are Discontinuous, J. Financial Econ. 3 (1976), 125–144.
  4. M.J. Brennan, E.S. Schwartz, Finite Difference Methods and Jump Processes Arising in the Pricing of Contingent Claims: A Synthesis, J. Financial Quant. Anal. 13 (1978), 461-474.
  5. J.C. Cox, S.A. Ross, M. Rubinstein, Option Pricing: A Simplified Approach, J. Financial Econ. 7 (1979), 229–263.
  6. P. Boyle, M. Broadie, P. Glasserman, Monte Carlo Methods for Security Pricing, J. Econ. Dyn. Control. 21 (1997), 1267–1321.
  7. F.S. Emmanuel, The Mellin Transform Method as an Alternative Analytic Solution for the Valuation of Geometric Asian Option, Appl. Comput. Math. 3 (2014), 1-7.
  8. D.J. Manuge, P.T. Kim, A Fast Fourier Transform Method for Mellin-Type Option Pricing, arXiv:1403.3756, (2014).
  9. C.R. Nwozo, S.E. Fadugba, Mellin Transform Method for the Valuation of Some Vanilla Power Options With Non-Dividend Yield, Int. J. Pure Appl. Math. 96 (2014), 79-104.
  10. C.R. Nwozo, S.E. Fadugba, Performance Measure of Laplace Transforms for Pricing Path Dependent Options, Int. J. Pure Appl. Math. 94 (2014), 175-197.
  11. S.E. Fadugba, C.R. Nwozo, Mellin Transform Method for the Valuation of the American Power Put Option with Non-Dividend and Dividend Yields, J. Math. Finance. 05 (2015), 249–272.
  12. S.E. Fadugba, Solution of Fractional Order Equations in the Domain of Mellin Transform, J. Nigerian Soc. Phys. Sci. 1 (2019), 138-142.
  13. S.E. Fadugba, C.R. Nwozo, Closed-Form Solution for the Critical Stock Price and the Price of Perpetual American Call Options via the Improved Mellin Transforms, Int. J. Financial Markets Derivatives. 6 (2018), 269-286.
  14. S.E. Fadugba, C.T. Nwozo, Perpetual American Power Put Options With Non-Dividend Yield in the Domain of Mellin Transforms, Palestine J. Math. 9 (2020), 371-385.
  15. S.E. Fadugba, Laplace Transform for the Solution of Fractional Black-Scholes Partial Differential Equation for the American Put Options With Non-Dividend Yield, Int. J. Stat. Econ. 20 (2019), 10-17.
  16. S.J. Ghevariya, BSM European Option Pricing Formula for ML-payoff Function with Mellin Transform, Int. J. Math. Appl. 6 (2018), 34-36.
  17. S.E. Fadugba, A.A. Adeniji, M.C. Kekana, J.T. Okunlola, O. Faweya, Direct Solution of Black-Scholes-Merton European Put Option Model on Dividend Yield With Modified-Log Payoff Function, Int. J. Anal. Appl. 20 (2022), 54.
  18. P. Flajolet, X. Gourdon, P. Dumas, Mellin Transforms and Asymptotics: Harmonic Sums, Theor. Computer Sci. 144 (1995), 3–58.
  19. G. Fikioris, Mellin-Transform Method for Integral Evaluation: Introduction and Applications to Electromagnetics, Springer International Publishing, Cham, 2007.
  20. A.H. Zemanian, Generalized integral transformation, Dover Publications, New York, 1987.
  21. A. Erdelyi, W. Magnus, F. Oberhettinger, F. Tricomi, Tables of Integral Transforms, Vol. 1-2, First edition, McGraw-Hill, New York, 1954.
  22. P. Wilmott, Paul Wilmott on Quantitative Finance, John Wiley & Sons, Ltd., Second edition, 2006.