Existence of Weak Solution for Black-Scholes Partial Differential Equation and Application of Energy Estimate Theorem in Sobolev Space
DOI:
https://doi.org/10.38142/ljes.v1i3.170Keywords:
Weak solutions, Black-Scholes equations, Sobolev spaces, Option and Smooth functionsAbstract
Purpose:
This paper aims to solve the BS second-order parabolic equation in Sobolev spaces to obtain weak solutions for financial applications, extending previous work in this field.
Methodology:
This paper constructs a set of functions that transforms the Black-Scholes partial differential equation into weak formulations. This study focuses on the Mathematics of finance, particularly the evolution of option pricing. An option's underlying asset involves agreements to buy or sell at a future strike price. The Black-Scholes (BS) equation, widely used in financial applications, models this.
Findings:
The analytical solutions, existence, uniqueness and other estimates were also obtained in weak form using boundary conditions to establish the effects of their financial implications in Sobolev spaces.
Implication:
The problem's regularity conditions were considered, and the coefficients and boundary of the domain are all smooth functions. To this end, this paper illustrates the definitions and assumptions that led to valuable assertions.
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References
Babasola, O. L., Irakoze, I., & Onoja, A. A. (2008). Valuation of European options within the Black-Scholes framework using the Hermite Polynomial. J Sci Eng Res, 5(2), 200-13.
Black, F., & Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81(3), 637–654. https://doi.org/10.1086/260062
Brezis, H., & Brézis, H. (2011). Functional Analysis, Sobolev Spaces and Partial Differential Equations (Vol. 2, No. 3, p. 5). New York: Springer. https://doi.org/10.1007/978-0-387-70914-7
Bianconi, M., MacLachlan, S., & Sammon, M. (2015). Implied Volatility and the Risk-Free Rate of Return in Options Markets. The North American Journal of Economics and Finance, pp. 31, 1–26. https://doi.org/10.1016/j.najef.2014.10.003
Cont, R. (2006). Model Uncertainty and its Impact on the Pricing of Derivative Instruments. Mathematical finance, 16(3), 519–547. https://doi.org/10.1111/j.1467-9965.2006.00281.x
DiPerna, R. J., & Lions, P. L. (1989). Ordinary Differential Equations, Transport Theory and Sobolev Spaces. Inventiones Mathematicae, 98(3), 511-547. https://doi.org/10.1007/BF01393835
Evans, L. C. (2022). Partial Differential Equations (Vol. 19). American Mathematical Society.
Fatone, L., Mariani, F., Recchioni, M. C., & Zirilli, F. (2008). Calibration of a Multiscale Stochastic Volatility Model Using European Option Prices. Mathematical methods in Economics and Finance, 3(1), 49-61.
Heston, S. L. (1993). A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options. The review of financial studies, 6(2), 327–343. https://doi.org/10.1093/rfs/6.2.327
Higham, D. J. (2004). An Introduction to Financial Option Valuation: Mathematics, Stochastics and Computation (Vol. 13). Cambridge University Press. https://doi.org/10.1017/CBO9780511800948
Huang, X. F., Zhang, T., Yang, Y., & Jiang, T. (2017). Ruin Probabilities in a Dependent Discrete-Time Risk Model with Gamma-Like Tailed Insurance Risks. Risks, 5(1), 14. https://doi.org/10.3390/risks5010014
Imanol, P. A. (2016). Sobolev Spaces and Partial Differential Equations [Final Degree Dissertation in Mathematics]. University of the Basque country.
Lindström, E., Ströjby, J., Brodén, M., Wiktorsson, M., & Holst, J. (2008). Sequential Calibration of Options. Computational Statistics & Data Analysis, 52 (6), 2877-2891. https://doi.org/10.1016/j.csda.2007.08.009
Kasozi, J., & Paulsen, J. (2005). Numerical Ultimate Ruin Probabilities Under Interest Force. Journal of Mathematics and Statistics, 1(3), 246-251. https://doi.org/10.3844/jmssp.2005.246.251
Kasozi, J. (2022). Numerical Ultimate Survival Probabilities in an Insurance Portfolio Compounded by Risky Investments. Applications & Applied Mathematics, 17(1).
Nwobi, F. N., Annorzie, M. N., & Amadi, I. U. (2019). The Impact of Crank-Nicolson Finite Difference Method in Valuation of Options. Communications in Mathematical Finance, 8(1), 93-122.
Osu, B. O., Okoroafor, A. C., & Olunkwa, C. (2009). Stability Analysis of Stochastic model of Stock market price. African Journal of Mathematics and Computer Science, 2(6), 98-103.
Osu, B. O. (2010). A Stochastic Model of the Variation of the Capital Market Price. International Journal of Trade, Economics and Finance, 1(2), 297. https://doi.org/10.7763/IJTEF.2010.V1.53
Osu, B. O., & Olunkwa, C. (2014). Weak Solution of Black-Scholes Equation Option Pricing with Transaction Costs. Int J Appl Math, 1(1), 43-55.
Osu, B. O., Eze, E. O., & Obi, C. N. (2020). The Impact of Stochastic Volatility Process on the Values of Assets. Scientific African, 9, e00513. https://doi.org/10.1016/j.sciaf.2020.e00513
Purnawan, I. P. G., Pardita, D. P. Y., & Aziz, I. S. A. (2024). Analysis of Factors Affecting the Rupiah Exchange Rate Against the US Dollar 2000-2022. Loka: Journal of Environmental Sciences, 1(2), 70-76. https://doi.org/10.38142/ljes.v1i2.153
Rodrigo, M. R., & Mamon, R. S. (2006). An Alternative Approach to Solving the Black–Scholes Equation with Time-Varying Parameters. Applied Mathematics Letters, 19(4), 398-402. https://doi.org/10.1016/j.aml.2005.06.012
Sanjaya, N. L. P. M. U., Dharmanegara, I. B. A., & Sariani, N. K. (2024). The Influence of Menu Variations, Location, and Service Quality on Consumer Purchasing Decisions at Terrace Brasserie in Canggu, Badung Regency. Loka: Journal of Environmental Sciences, 1(2), 50-55. https://doi.org/10.38142/ljes.v1i2.159
Shin, B., & Kim, H. (2016). The Solution of Black-Scholes Terminal Value Problem employing Laplace Transform. Glob J Pure Appl Math, pp. 12, 4153–8.
Wilmott, P., Howison, S., & Dewynne, J. (1995). The Mathematics of Financial Derivatives: A Student Introduction. Cambridge University Press. https://doi.org/10.38142/ljes.v1i2.159
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Creative Commons Attribution-NonCommercial 4.0 International License.