Accurate delta hedging of european options using conformable calculus
DOI:
https://doi.org/10.18381/eq.v21i1.7324Keywords:
option pricing; delta hedging; conformable calculus; risk managementAbstract
Objective: we aim to develop a method for delta hedging portfolios of European options based on the theory of conformable calculus which improves accuracy of risk management of listed options in a first-order approximation.Methodology: we allow the time derivative in the classic Black-Scholes-Merton model to have a fractional order0 ≤ α ≤ 1 and calculate the corresponding delta of a portfolio of listed options as a function of this conformableparameter.Results: applying this method to a portfolio consisting of eight European options on the SPX index, we find that conformable delta hedging offers more accurate average predictions than classical delta hedging.Limitations: this method is applicable for delta hedging in European options only.Originality: this is the first successful application of conformable calculus to delta hedging in European options.Conclusions: application of Conformable Calculus allows for a greater flexibility in the local approximation to price in delta-hedging European options and offers a new and more precise methodology to this objective.Downloads
References
Abdeljawad, T. (2015). On conformable fractional calculus. Journal of Computational and Applied Mathematics, 279: 57–66. https://doi.org/10.1016/j.cam.2014.10.016
Anderson, D. R. & Ulness, D. J. (2015). Newly defined conformable derivatives. Advances in Dynamical Systems and Applications, 10(2): 109–137. http://campus.mst.edu/adsa
Anderson, D. R. & Camrud, D. J. (2019). On the nature of the conformable derivative and its applications to physics. Journal of Fractional Calculus and Applications, 10(2): 92-135. https://doi:10.21608/jfca.2019.308538
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
Bloomberg (2022). Bloomberg Professional. Accessed from April to November 2022.
Chung, W. S. (2015). Fractional newton mechanics with conformable fractional derivative. Journal of Computational and Applied Mathematics, 290: 150–158. https://doi.org/10.1016/j.cam.2015.04.049
El-Ajou, A. (2020). A modification to the conformable fractional calculus with some applications. Alexandria Engineering Journal, 59(4): 2239–2249. https://doi.org/10.1016/j.aej.2020.02.003
Hull, J. C. (2018). Options, Futures, and Other Derivatives (9th ed.). Harlow, England: Pearson Educational.
Hull, J., & White, A. (1987). The pricing of options on assets with stochastic volatilities. Journal of Finance, 42(2): 281–300. https://doi.org/10.1111/j.1540-6261.1987.tb02568.x
Hull, J., & White, A. (2017). Optimal delta hedging for options. Journal of Banking & Finance, 82: 180–190. https://doi.org/10.1016/j.jbankfin.2017.05.006
Khalil, R., Al Horani, M., Yousef, A., & Sababheh, M. (2014). A new definition of fractional derivative. Journal of Computational and Applied Mathematics, 264: 65–70. https://doi.org/10.1016/j.cam.2014.01.002
Kilbas, A. A., Srivastava, H. M. & Trujillo, J. J. (2006). Theory and applications of fractional differential equations (North-Holland mathematics studies; v. 204). Amsterdam: Elsevier.
Martynyuk, A. A. (2018). On the stability of solutions of fractional-like equations of perturbed motion. Dopovidi Natsional’noi Akademii Nauk Ukrainy. Matematyka, Pryrodoznavstvo, Tekhnichni Nauky, (6): 9–16. https://doi.org/10.15407/dopovidi2018.06.009
Martynyuk, A., Stamov, G., & Stamova, I. (2019). Practical stability analysis with respect to manifolds and boundedness of differential equations with fractional-like derivatives. Rocky Mountain Journal of Mathematics, 49(1): 211–233. https://doi.org/10.1216/rmj-2019-49-1-211
Martynyuk, A., & Stamova, I. (2018). Fractional-like derivative of Lyapunov-type functions and applications to stability analysis of motion. Electronic Journal of Differential Equations. (62): 1-12. https://hdl.handle.net/10877/15203
Merton, R. C. (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4(1): 141-183. https://doi.org/10.2307/3003143
Xia, K., Yang, X., & Zhu, P. (2023). Delta hedging and volatility-price elasticity: A two-step approach. Journal of Banking & Finance, 153: 106898. https://doi.org/10.1016/j.jbankfin.2023.106898
Zhao, D., & Luo, M. (2017). General conformable fractional derivative and its physical interpretation. Calcolo. A Quarterly on Numerical Analysis and Theory of Computation, 54(3): 903–917. https://doi.org/10.1007/s10092-017-0213-8
Zhou, H. W., Yang, S., & Zhang, S. Q. (2018). Conformable derivative approach to anomalous diffusion. Physica A: Statistical Mechanics and its Applications, 491: 1001–1013. https://doi.org/10.1016/j.physa.2017.09.101
Published
How to Cite
Issue
Section
License
El contenido publicado en EconoQuantum se encuentra bajo una Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
