Meotdi Traykov, Miglena Trencheva, Radoslav Mavrevski, Anton Stoilov, Ivan Trenchev


This article is based on the methodology of comparative analysis, using an innovative approach for pricing of various goods and services. Benchmarking is the continuous search to find and adapt better pricing methods that leading to increased profits. We will consider the numerical solution of partial differential equations, based on Black-Scholes model for pricing of goods and services within European option. Also, we will present formulation and numerical behavior of explicit and implicit methods that can be use in pricing for company assets within European option.


partial differential equations, economics, pricing, goods, services

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Black, F., and Scholes, M., 1973. The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81(3), 637-654. DOI:

Boyle, P. P., and Tian, Y., 1998. An explicit finite difference approach to the pricing of barrier options. Applied Mathematical Finance, 5(1), 17-43. DOI:

Brandimarte, P., 2002. Numerical Methods in Finance and Economics: A MATLAB-Based Introduction. Torino, Italy: Politecnico di Torino.

Brennan, M. J., and Schwartz, E. S., 1977. Convertible Bonds: Valuation and Optimal Strategies for Call and Conversion. The Journal of Finance, 32(5), 1699-1715. DOI:

Clemen, R. T., and Reilly, T., 1996. Making Hard Decisions: An Introduction to Decision Analysis (2nd ed.). Pacific Grove, CA: Duxbury.

Cvitanic, J., and Zapatero, F., 2004. Introduction to the Economics and Mathematics of Financial Markets. Cambridge, MA: MIT Press.

Elton, E. J., and Gruber, M., 1995. Modern Portfolio Theory and Investment Analysis (5th ed.). New York: John Wiley & Sons.

Fabozzi, F. J., 1996. Bond Markets: Analysis and Strategies. Upper Saddle River, NJ: Prentice Hall.

Figlewski, S., and Gao, B., 1999. The adaptive mesh model: A new approach to efficient option pricing. Journal of Financial Economics, 53(3), 313-351. DOI: 405X(99)00024-0

Hull, J., 2003. Options, Futures and Other Derivatives (5th ed.). Upper Saddle River, NJ: Prentice Hall.

Hull, J. C., and White, A. D., 1993. Efficient Procedures for Valuing European and American Path- Dependent Options. Journal of Derivatives, 1(1), 21-31. DOI:

Kaas, R., Goovaerts, M., Dhaene, J., and Denuit, M., 2008. Modern Actuarial Risk Theory. Using R (2nd ed.). Berlin: Springer-Verlag.

Kemna, A. G. Z., and Vorst, A. C. F., 1990. A pricing method for options based on average values. Journal of Banking & Finance, 14(1), 113-129. DOI: 4266(90)90039-5

Kunitomo, N., and Ikeda, M., 1992. Pricing options with curved boundaries. Mathematical Finance, 2(4), 275-298. DOI:

Merton, R. C., 1973. Theory of rational option pricing. The Bell Journal of Economics and Management Science, 4(1), 141-183. DOI:

Neftci, S. N., 2000. An Introduction to the Mathematics of Financial Derivatives (2nd ed.). San Diego, CA: Academic Press.

Pliska, S. R., 1997. Introduction to Mathematical Finance: Discrete Time Models. Malden, MA: Blackwell Publishers.

Rich, D., 1996. The valuation and behavior of Black-Scholes options subject to intertemporal default risk. Review of Derivatives Research, 1(1), 25-59. DOI:

Saaty, T., 1994. Fundamentals of Decision Making. Pittsburg: RWS Publications.

Taleb, N., 1996. Dynamic Hedging: Managing Vanilla and Exotic Options. New York: Wiley.

Tavella, D., and Randall, C., 2000. Pricing Financial Instruments: The Finite Difference Method. New York: Wiley.

Topper, J., 2005. Financial Engineering with Finite Elements. New York: Wiley.

Wilmott, P., 1999. Derivatives: The Theory and Practice of Financial Engineering. Chichester, West Sussex: Wiley.

Wilmott, P., Dewynne, J., and Howison, S., 1993. Option Pricing: Mathematical Models and Computation. Oxford: Oxford Financial Press.

Zvan, R., Forsyth, P., and Vetzal, K., 1997. Robust Numerical Methods for PDE Models of Asian Options. Journal of Computational Finance, 1(2), 39-78. DOI:

Zvan, R., Vetzal, K. R., and Forsyth, P. A., 2000. PDE Methods for Pricing Barrier Options. Journal of Economic Dynamics and Control, 24(11-12), 1563-1590. DOI:



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