Application of the Trefftz method for option pricing

• Autorzy: Brzozowska-Rup Katarzyna , Hożejowska Sylwia , Hożejowski Leszek

Identyfikatory

DOI

10.29119/1641-3466.2020.146.3

• Języki publikacji: EN

Abstrakty

EN

Purpose: Option pricing is hardly a new topic, however, in many cases they lack an analytical solution. The article proposes a new approach to option pricing based on the semi-analytical Trefftz method. Design/methodology/approach: An appropriate transformation makes it possible to reduce the Black-Scholes equation to the heat equation. This admits the Trefftz method (which has shown its effectiveness in heat conduction problems) to be employed. The advantage of such an approach lies in its computational simplicity and in the fact that it delivers a solution satisfying the governing equation. Findings: The theoretical option pricing problem being considered in the paper has been solved by means of the Trefftz method, and the results achieved appear to comply with those taken from the Black-Scholes formula. Numerical simulations have been carried out and compared, which has confirmed the accuracy of the proposed approach. Originality/value: Although a number of solutions to the Black-Scholes model have appeared, the paper presents a thoroughly novel idea of implementation of the Trefftz method for solving this model. So far, the method has been applied to problems having nothing in common with finance. Therefore the present approach might be a starting point for software development, competitive to the existing methods of pricing options.

Słowa kluczowe

EN

contract options; option pricing; Black-Scholes model; Trefftz method

PL

opcje kontraktowe; wycena opcji; model Blacka-Scholesa; Metoda Trefftza

• Wydawca: Wydawnictwo Politechniki Śląskiej
• Czasopismo: Zeszyty Naukowe. Organizacja i Zarządzanie / Politechnika Śląska
• Rocznik: 2020
• Tom: z. 146
• Strony: 37--49
• Opis fizyczny: Bibliogr. 25 poz.

Twórcy

autor

Brzozowska-Rup Katarzyna

• Kielce University of Technology, Faculty of Management and Computer Modeling, Kielce

autor

Hożejowska Sylwia

• Kielce University of Technology, Faculty of Management and Computer Modeling, Kielce

autor

Hożejowski Leszek

• Kielce University of Technology, Faculty of Management and Computer Modeling, Kielce

Bibliografia

• 1. Available online: https://www.gpw.pl/pub/GPW/files/PDF/standardy_pl/Standard_opcje_ WIG20_1.12.201.pdf, 26.05.2020.
• 2. Beck, J.V., and Woodbury, K.A. (2016). Inverse heat conduction problem: Sensitivity coefficient insights, filter coefficients and intrinsic verification. International Journal Heat and Mass Transfer, Vol. 97, pp. 578-588, doi:10.1016/j.ijheatmasstransfer.2016.02.034.
• 3. Boyle, P.P. (1977). Options: a Monte Carlo approach. Journal of Financial Economics, Vol. 4, pp. 323-338. doi: 10.1016/0304-405X(77)90005-8.
• 4. Brennan, M.J., and Schwartz, E.S. (1977). The Valuation of American Put Options. The Journal of Finance, Vol. 32, No. 2, pp. 449-462. doi: 10.2307/2326779.
• 5. Cervera, J.A.G. (2019). Solution of the Black-Scholes equation using artificial neural networks. Journal of Physics, Conf. Ser. 1221 012044. doi:10.1088/1742-6596/1221/1/012044.
• 6. Courtadon, G. (1982). A More accurate Finite Difference Approximation for the Valuation of Options. Journal of Financial and Quantitative Analysis, Vol. 17, pp.697-70. doi: 10.2307/2330857.
• 7. Duffie, D., and Glynn, P. (1996). Efficient Monte Carlo estimation of security prices. Annals of Applied Probability, Vol. 5, pp. 897-905.
• 8. Gong, H., and Thavaneswaran, A., Sing, J. (2010). A Black-Scholes Model with GARCH Volatility. The Mathematical Scientist, No. 35, pp. 37-42.
• 9. Grabowski, M., Hożejowska, S., Pawińska, A., Poniewski, M.E., and Wernik, J. (2018). Heat transfer coefficient identification in minichannel flow boiling with hybrid Picard–Trefftz method. Energies, Vol. 11, No. 2057, pp. 1-13. doi:10.3390/en11082057.
• 10. Grysa, K. (2010). Funkcje Trefftza i ich zastosowania w rozwiązywaniu zagadnień odwrotnych. Kielce: Wydawnictwo Politechniki Świętokrzyskiej.
• 11. Hadamard, J. (1902). Sur Les Problèmes Aux DérivéesPartielles Et Leur Signification Physique. Princeton, USA: Princeton University Bulletin, pp. 49-52.
• 12. Hożejowska, S. (2015). Homotopy perturbation method combined with Trefftz method in numerical identification of temperature fields in flow boiling. Journal of Theoretical and Applied Mechanics, Vol. 53, Iss. 4, pp. 969-980. doi: 10.15632/jtam-pl.53.4.969.
• 13. Hull, J., and White, A. (1990). Valuing derivative securities using the explicit finite difference method, Journal of Financial and Quantitative Analysis. Vol. 25, pp. 87-100. doi: 10.2307/2330889.
• 14. Jakubowski, J., and Sztencel, R. (2001). Wstęp do teorii prawdopodobieństwa. Script.
• 15. Kamiński, S. (2013). The pricing of options on WIG20 using GARCH models. Working Papers, No. 6(91). University of Warsaw.
• 16. Maciąg, A. (2009). Funkcje Trefftza dla wybranych prostych i odwrotnych zagadnień mechaniki. Kielce: Wydawnictwo Politechniki Świętokrzyskiej.
• 17. Maciejewska, B. (2017). The application of Beck`s method combined with the FEM and Trefftz functions to determine the heat transfer coefficient in minichannel. Journal of Theoretical and Applied Mechanics, Vol. 55, Iss. 1, pp. 103-11. doi:10.15632/jtam-pl.55.1.103.
• 18. Napiórkowski, A. (2002). Charakterystyka, wycena i zastosowanie wybranych opcji egzotycznych. Warszawa: Narodowy Bank Polski.
• 19. Ozisik, M.N., and Orlande, H.R.B. (2000). Inverse Heat Transfer: Fundamentals and Applications. New York, NY, USA: Taylor & Francis.
• 20. Piontek, K. Zmienność implikowana instrumentów finansowych – wprowadzenie Katedra Inwestycji Finansowych i Ubezpieczeń. Akademia Ekonomiczna we Wrocławiu. Available online: http://credit.ae.wroc.pl/-kpiontek/implik.pdf, 26.05.2020.
• 21. Rosenbloom, P.C., and Widder, D.V. (1959). Expansions in terms of heat polynomials and associated functions. Transactions of the American Mathematical Society, Vol. 92, pp. 220-266. doi: 10.1090/S0002-9947-1959-0107118-2.
• 22. Rubaszek, M. (2012). Modelowanie polskiej gospodarki z pakietem R. Oficyna Wydawnicza Szkoła Główna Handlowa w Warszawie.
• 23. Sawangtong, P., Trachoo, K., Sawangtong, W., and Wiwattanapataphee, B. (2018). The Analytical Solution for the Black-Scholes Equation with Two Assets in the Liouville-Caputo Fractional Derivative Sense. Mathematics, Vol. 6, No. 129, pp. 1-14. doi:10.3390/math6080129.
• 24. Trefftz, E. (1926). Ein Gegenstück zum Ritzschen Verfahren. 2 International Kongress für Technische Mechanik. Zürich, Switzerland, pp. 131-137.
• 25. Weron, R., and Weron, A. (2008). Inżynieria finansowa. Wycena instrumentów pochodnych, symulacje komputerowe, statystyka rynku. Warszawa: WNT.

Uwagi

PL

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).