Mellin Transform Method for the Valuation of the American Power Put Option with Non-Dividend and Dividend Yields

Affiliation(s)

Department of Mathematics, University of Ibadan, Oyo State, Nigeria.

Department of Mathematical Sciences, Ekiti State University, Ado Ekiti, Nigeria.

Department of Mathematics, University of Ibadan, Oyo State, Nigeria.

Department of Mathematical Sciences, Ekiti State University, Ado Ekiti, Nigeria.

ABSTRACT

In this paper we present the Mellin transform method for the valuation of the American power put option with non-dividend and dividend yields, respectively. We use the Mellin transform method to derive the integral representations for the price and the free boundary of the American power put option. We also extend our results to derive the free boundary and the fundamental analytic valuation formula for perpetual American power put option which has no expiry date. Numerical experiments have shown that the Mellin transform method is a better alternative technique compared to the binomial model (BSM), recursive method (RM) and finite difference method (FDM) for the valuation of the American power put option. In general, the Mellin transform method is accurate, flexible and produces accurate prices for the optimal exercise boundary of the American power put option for a wide range of parameters. Hence the Mellin transform method is mutually consistent and agrees with the values of the analytic option valuation formula called the “Black-Scholes model”.

In this paper we present the Mellin transform method for the valuation of the American power put option with non-dividend and dividend yields, respectively. We use the Mellin transform method to derive the integral representations for the price and the free boundary of the American power put option. We also extend our results to derive the free boundary and the fundamental analytic valuation formula for perpetual American power put option which has no expiry date. Numerical experiments have shown that the Mellin transform method is a better alternative technique compared to the binomial model (BSM), recursive method (RM) and finite difference method (FDM) for the valuation of the American power put option. In general, the Mellin transform method is accurate, flexible and produces accurate prices for the optimal exercise boundary of the American power put option for a wide range of parameters. Hence the Mellin transform method is mutually consistent and agrees with the values of the analytic option valuation formula called the “Black-Scholes model”.

KEYWORDS

American Power Option, Dividend Yield, Mellin Transform Method, Non-Dividend Yield, Perpetual Power Put Option

American Power Option, Dividend Yield, Mellin Transform Method, Non-Dividend Yield, Perpetual Power Put Option

Cite this paper

Fadugba, S. and Nwozo, C. (2015) Mellin Transform Method for the Valuation of the American Power Put Option with Non-Dividend and Dividend Yields.*Journal of Mathematical Finance*, **5**, 249-272. doi: 10.4236/jmf.2015.53023.

Fadugba, S. and Nwozo, C. (2015) Mellin Transform Method for the Valuation of the American Power Put Option with Non-Dividend and Dividend Yields.

References

[1] Samuelson, P.A. (1965) Rational Theory of Warrant Pricing. Industrial Management Review, 6, 13-31.

[2] Panini, R. and Srivastav, R.P. (2005) Pricing Perpetual Options Using Mellin Transforms. Applied Mathematics Letters, 18, 471-474.

http://dx.doi.org/10.1016/j.aml.2004.03.012

[3] Frontczak, R. and Schöbel, R. (2008) Pricing American Options with Mellin Transforms. Working Paper.

[4] Nwozo, C.R. and Fadugba, S.E. (2014) Mellin Transform Method for the Valuation of Some Vanilla Power Options with Non-Dividend Yield. International Journal of Pure and Applied Mathematics, 96, 79-104.

http://dx.doi.org/10.12732/ijpam.v96i1.7

[5] Kim, I. (1990) The Analytic Valuation of American Options. The Review of Financial Studies, 3, 547-572.

http://dx.doi.org/10.1093/rfs/3.4.547

[6] Kim, I. and Yu, G.G. (1996) An Alternative Approach to the Valuation of American Options and Applications. Review of Derivatives Research, 1, 61-85.

http://dx.doi.org/10.1007/BF01536395

[7] Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. (1954) Tables of Integral Transforms, Vol. 1-2. McGraw-Hill, New York.

[8] Fadugba, S.E. and Nwozo, C.R. (2015) Integral Representations for the Price of Vanilla Put Options on a Basket of Two-Dividend Paying Stocks. Applied Mathematics, 6, 783-792.

http://dx.doi.org/10.4236/am.2015.65074

[9] Frontczak, R. and Schöbel, R. (2009) On Modified Mellin Transforms, Gauss-Laguerre Quadrature and the Valuation of American Call Options. Tübinger Diskussionsbeitrüge, No. 320.

[10] Gradshteyn, I. and Ryshik, I. (2007) Table of Integrals Series and Products. 7th Edition, Academic Press, Waltham.

[11] Panini, R. and Srivastav, R.P. (2004) Option Pricing with Mellin Transforms. Mathematical and Computer Modelling, 40, 43-56.

http://dx.doi.org/10.1016/j.mcm.2004.07.008

[12] Vasilieva, O. (2009) A New Method of Pricing Multi-Options Using Mellin Transforms and Integral Equations. Master’s Thesis in Financial Mathematics, School of Information Science, Computer and Electrical Engineering, Halmstad University, Halmstad.

[13] Nguyen, T.H. and Yakubovich, S.B. (1991) The Double Mellin-Barnes Type Integrals and Their Applications to Convolution Theory. World Scientific, Series on Soviet Mathematics, Vol. 6, Singapore, 295p.

[14] Zemanian, A.H. (1987) Generalized Integral Transformations. Dover Publications, New York.

[15] Zieneb, A.E. and Rokiah, R.A. (2011) Analytical Solution for an Arithmetic Asian Option Using Mellin Transforms. International Journal of Mathematical Analysis, 5, 1259-1265.

[1] Samuelson, P.A. (1965) Rational Theory of Warrant Pricing. Industrial Management Review, 6, 13-31.

[2] Panini, R. and Srivastav, R.P. (2005) Pricing Perpetual Options Using Mellin Transforms. Applied Mathematics Letters, 18, 471-474.

http://dx.doi.org/10.1016/j.aml.2004.03.012

[3] Frontczak, R. and Schöbel, R. (2008) Pricing American Options with Mellin Transforms. Working Paper.

[4] Nwozo, C.R. and Fadugba, S.E. (2014) Mellin Transform Method for the Valuation of Some Vanilla Power Options with Non-Dividend Yield. International Journal of Pure and Applied Mathematics, 96, 79-104.

http://dx.doi.org/10.12732/ijpam.v96i1.7

[5] Kim, I. (1990) The Analytic Valuation of American Options. The Review of Financial Studies, 3, 547-572.

http://dx.doi.org/10.1093/rfs/3.4.547

[6] Kim, I. and Yu, G.G. (1996) An Alternative Approach to the Valuation of American Options and Applications. Review of Derivatives Research, 1, 61-85.

http://dx.doi.org/10.1007/BF01536395

[7] Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. (1954) Tables of Integral Transforms, Vol. 1-2. McGraw-Hill, New York.

[8] Fadugba, S.E. and Nwozo, C.R. (2015) Integral Representations for the Price of Vanilla Put Options on a Basket of Two-Dividend Paying Stocks. Applied Mathematics, 6, 783-792.

http://dx.doi.org/10.4236/am.2015.65074

[9] Frontczak, R. and Schöbel, R. (2009) On Modified Mellin Transforms, Gauss-Laguerre Quadrature and the Valuation of American Call Options. Tübinger Diskussionsbeitrüge, No. 320.

[10] Gradshteyn, I. and Ryshik, I. (2007) Table of Integrals Series and Products. 7th Edition, Academic Press, Waltham.

[11] Panini, R. and Srivastav, R.P. (2004) Option Pricing with Mellin Transforms. Mathematical and Computer Modelling, 40, 43-56.

http://dx.doi.org/10.1016/j.mcm.2004.07.008

[12] Vasilieva, O. (2009) A New Method of Pricing Multi-Options Using Mellin Transforms and Integral Equations. Master’s Thesis in Financial Mathematics, School of Information Science, Computer and Electrical Engineering, Halmstad University, Halmstad.

[13] Nguyen, T.H. and Yakubovich, S.B. (1991) The Double Mellin-Barnes Type Integrals and Their Applications to Convolution Theory. World Scientific, Series on Soviet Mathematics, Vol. 6, Singapore, 295p.

[14] Zemanian, A.H. (1987) Generalized Integral Transformations. Dover Publications, New York.

[15] Zieneb, A.E. and Rokiah, R.A. (2011) Analytical Solution for an Arithmetic Asian Option Using Mellin Transforms. International Journal of Mathematical Analysis, 5, 1259-1265.