Analytical and numerical solutions for the pricing of a combination of two financial derivatives in a market under Hull-White model
محورهای موضوعی : Financial Mathematics
Hossein Sahebi Fard
1
(Department of Mathematics, Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Semnan, Iran.)
Elham Dastranj
2
(Department of Mathematics, Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Semnan, Iran.)
Abdolmajid Abdolbaghi Ataabadi
3
(Department of Management, Faculty of Industrial Engineering and Management, Shahrood University of Technology, Shahrood, Semnan, Iran.)
کلید واژه: Hull-White model, parabolic differential equation, zero-coupon bond option,
چکیده مقاله :
In this paper‎ a combination of two financial derivatives in financial markets modelled of future interest rates is presented and evaluated. In fact ‎European option pricing is driven when zero-coupon bond is considered as underlying asset in a market under Hull-White model‎. ‎For this purpose, the exact solutions of the valuation of this bond option are driven, using Lie group symmetries method. Then in the next part, the finite difference method is applied to find numerical solutions for assumed bond option pricing. Then the significance and usefulness of this approximated method is comparing with the exact solutions by some plotted graphs.
[1] Bjork, T., Arbitrage Theory in Continuous Time, 2nd edn, Oxford University Press, 2004.
[2] Dastranj, E., Latifi, R., A comparison of option pricing models, International Journal of Financial Engineering, 2017, 4.02n03, 1750024. Doi: 10.1142/S2424786317500244
[3] Sahebi Fard, H., Dastranj, E., Hejazi, S.R., Power Option Pricing Under Fractional Double Heston model, Annual Iranian Mathematics Conference, 2018, P.3911–3916.
[4] Dastranj, E., Hejazi, S.R., Exact solutions for Fokker-Plank equation of geometric Brownian motion with Lie point symmetries, Computational Methods for Differential Equations, 2018, 6.3, P.372-379.
[5] Dastranj, E., Sahebi Fard, H., Abdolbaghi, A., Latifi, R., A comparison between Capped and power option pricing with prevention the arbitrage opportunity: Evidence from a stochastic market with double stochastic volatility, double jump and stochastic intensity. Asset Management and Financing, 2020, Doi: 10.22108/amf.2020.119750.1479.
[6] Dastranj, E., Sahebi Fard, H., Abdolbaghi, A., Hejazi, S.R.,Power option pricing under the unstable conditions (Evidence of power option pricing under fractional Heston model in the Iran gold market), Physica A: Statistical Mechanics and its Applications, 2020, 537, 122690. Doi: 10.1016/j.physa.2019.122690.
[7] Dastranj, E., Sahebi Fard, H., Exact solutions and numerical simulation for Bakstein-Howison model, Computational Methods for Differential Equations, 2021 Apr 24.
[8] Dura, G., and Moşneagu, A. M., Numerical approximation of Black-Scholes equation, Annals of the AlexandruIoanCuza University-Mathematics, 2010, 56(1), P.39-64. Doi: 10.2478/v10157-010-0004-x
[9] Elliott, R. J.,van der Hoek,J., An application of hidden Markov models to asset allocation problems, Finance and Stochastics, 1997, 3, P.229-–238. Doi: 10.1007/s007800050022
[10] Gerebrink, A., Lundgren, J., Malmström, F., Thorén, O., Maximum Likelihood calibration of the Vasicek model to the Swedish interest rate market,2018, BS thesis.
[11] Habibi, N., Lashkarian, E., Dastranj, E.,Hejazi, S.R., Lie symmetry analysis, conservation laws and numerical approximations of time-fractional Fokker–Planck equations for special stochastic process in foreign exchange markets, Physica A, 2019, 513, P.750–766. Doi: 10.1016/j.physa.2018.08.155
[12] Hejazi, S. R., Lie group analysis, Hamiltonian equations and conservation laws of Born-Infeld equation, Asian-European Journal of Mathematics, 2014, 7(3), 1450040. Doi: 10.1142/S1793557114500405
[13] Heynen, R. C., Kat, H. M., Pricing and hedging power options, Financial Engineering and the Japanese Markets, 1996, 3(3), P. 253–261. Doi: 10.1007/BF02425804
[14] Hull, J.,White, A., Pricing interest-rate-derivative securities, The review of financial studies, 1990, 3(4), P.573-592. Doi: 10.1093/rfs/3.4.573
[15] Øksendal, B., Stochastic Differential Equations, Springer, New York, NY, USA, 2003. Doi: 10.1007/BF02925504
[16] Pan, J., Zhou, X., Pricing for options in a mixed fractional Hull-White interest rate model, international Journal of Financial Engineering, 2017, 4(01), 1750011. Doi: 10.1142/S2424786317500116
[17] Tong, Z., Option pricing with long memory stochastic volatility models, Diss. Universitéd’Ottawa/University of Ottawa, 2012.
[18] Mpanda, M.M., Pricing European and American Bond Options under the Hull-White extended Vasicek Model, Diss. 2013.
[19] Sahebi Fard, H., Easapour, H., Dastranj, E., Stochastic analysis to find exact solutions of rainbow option under 2-D Black-Scholes model, Journal of Interdisciplinary Mathematics, accepted.
[20] Yang, S., Siu, T., Pricing bond options under a Markovian regime-switching Hull–White model, Economic Modelling, 2013, 30, 933940. Doi: 10.1016/j.econmod.2012.09.041
