Mellin transform in higher dimensions for the valuation of the European basket put option with multi-dividend paying stocks
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Numerical approximations and analytical techniques have been proposed for the pricing of basket put option but there is no known integral equation for the valuation of European basket put option with multi-dividend yields. Mellin transform is useful when dealing with the unstable mathematical system. This paper presents the integral equation for the price of the European basket put option which pays multi-dividend yields by means of the Mellin transform in higher dimensions that enables option equations to be solved directly in terms of market prices rather than log-prices, providing a more natural setting to the problem of pricing. The expression for the integral equation for the valuation of the European basket put option was obtained by solving the multi-dimensional partial differential equation for the price of the option via the multi-dimensional Mellin transform. The analytical solution to the derived integral equation for the case of two-dividend paying stocks was obtained. Also the effect of the correlation coefficients on the price of the European basket put option was considered. A comparative study of the Mellin transform, Monte Carlo method and implied binomial model for the valuation of the option in the case of was considered. The numerical results showed that negatively correlated assets are more sensitive to correlation changes than positively correlated assets as shown in Tables 1 and 2. Also the numerical evaluation of our expression is more efficient and produces a comparable result than the other methods. Hence the Mellin transform is a good approach for the valuation of European basket put option with multi-dividend yields.
-  Aniela, K.I. Numerical methods for pricing basket options, Department of Mathematics, The Ohio State University, 2004.
-  Barraquand, J. and Martineau, D. Numerical valuation of high dimensional multivariate American securities. The Journal of Financial and Quantitative Analysis, 30 (1995) 383-405.
-  Black, F. and Scholes, M. The pricing of options and corporate liabilities. Journal of Political Economy 81 (1973) 637-654.
-  Fadugba, S.E. and Nwozo, C.R. Integral representations for the price of vanilla put options on a basket of two-dividend paying stocks. Applied Mathematics, 2015; 6: 783-792. http://dx.doi.org/10.4236/am.2015.65074
-  Flajolet, P., Gourdon, X. and Dumas P. Mellin transforms and asymptotics: Harmonic sums. Theoretical Computer Science 1995, 144: 3-58.
-  JU, N. Pricing Asian and basket options via Taylor expansion. Journal of Computational Finance, 2002, 5: 79-103.
-  Manuge D.J. Basket option pricing and the Mellin transform. M.Sc. Thesis, The University of Guelph, Ontario, Canada, 2013.
-  Nwozo, C.R. and Fadugba, S.E. Mellin transform method for the valuation of some vanilla power options with non-dividend yield. International Journal of Pure and Applied Mathematics, 2014; 96: 79-104. (dx.doi.org/10.12732/ijpam.v96i1.7)
-  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, Sweden.
-  Panini, R. and Srivastav, R.P. Option pricing with Mellin transforms. Mathematical and Computer Modelling. 2004; 40: 43-56. (DOI:10.1016/j.mcm.2004.07.008)
-  Reed I.S. The Mellin type of double integral. Duke Mathematical Journal. 1944; 11: 565-572.
-  Schneggenburger C, A Monte Carlo pricing tool for American basket put options on two assets, Diploma thesis in Mathematical Finance, Oxford University, 2002.
-  Henry Wan, Pricing American-style basket options by implied binomial tree. Applied Finance Project, University of California, Berkeley, 2002.
-  Xu, G. and Zheng, H. Basket options valuation for a local volatility jump-diffusion model with asymptotic expansion method. Insurance: Mathematics and Economics, 2010; 47: 415-422.
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