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Hybrid Equity Warrants Pricing Formulation under Stochastic Dynamics

Authors: Teh Raihana Nazirah Roslan, Siti Zulaiha Ibrahim, Sharmila Karim


A warrant is a financial contract that confers the right but not the obligation, to buy or sell a security at a certain price before expiration. The standard procedure to value equity warrants using call option pricing models such as the Black–Scholes model had been proven to contain many flaws, such as the assumption of constant interest rate and constant volatility. In fact, existing alternative models were found focusing more on demonstrating techniques for pricing, rather than empirical testing. Therefore, a mathematical model for pricing and analyzing equity warrants which comprises stochastic interest rate and stochastic volatility is essential to incorporate the dynamic relationships between the identified variables and illustrate the real market. Here, the aim is to develop dynamic pricing formulations for hybrid equity warrants by incorporating stochastic interest rates from the Cox-Ingersoll-Ross (CIR) model, along with stochastic volatility from the Heston model. The development of the model involves the derivations of stochastic differential equations that govern the model dynamics. The resulting equations which involve Cauchy problem and heat equations are then solved using partial differential equation approaches. The analytical pricing formulas obtained in this study comply with the form of analytical expressions embedded in the Black-Scholes model and other existing pricing models for equity warrants. This facilitates the practicality of this proposed formula for comparison purposes and further empirical study.

Keywords: Cox-Ingersoll-Ross model, equity warrants, Heston model, hybrid models, stochastic.

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[1] M. Alhagyan, M. Masnita and Z. Omar, "Estimation of geometric fractional Brownian motion perturbed by stochastic volatility model," Far East J. Math. Sci., vol. 99, no. 2, pp. 221, Jan. 2016.
[2] F. Black and M. Scholes, "The pricing of options and corporate liabilities," Political Economy J., pp. 637-654, May. 1973.
[3] X. J. He and S.-P. Zhu, "A closed-form pricing formula for European options under the Heston model with stochastic interest rate," J. Comput. Appl. Math., pp. 323-333, 2018.
[4] W. Wattanatorn, "Beyond Black-Scholes: The Stochastic Volatility Option Pricing Model and Empirical Evidence from Thailand," in 2014. Professor Sangvien Indaravijiya Conf.
[5] L. Teng, M. Ehrhardt and M. Günther, "Option Pricing with dynamically correlated stochastic interest rate," Acta Math. Univ. Comenianae, pp. 179-190, Sept. 2015.
[6] C. R. Nwozo and S. E. Fadugba, "On stochastic volatility in the valuation of European options," Mathematics & Computer Science British J., pp. 104-127, Oct. 2015.
[7] J. Eisenberg, "Optimal dividends under a stochastic interest rate," Insur. Math. Econ., vol. 65, pp. 259-266, Nov. 2015.
[8] X. J. He and W. Chen, "A semianalytical formula for European options under a hybrid Heston–Cox–Ingersoll–Ross model with regime switching," Finance & Economics International J., Oct. 2019.
[9] S. L. Heston, "A closed-form solution for options with stochastic volatility with applications to bond and currency options," Rev. Financ. Stud., vol. 6, no. 2, pp. 327-343, Apr. 1993.
[10] M. K. M. Al Hagyan, "Modeling financial environments using geometric fractional Brownian motion model with long memory stochastic volatility," Universiti Utara Malaysia, 2018.
[11] J. Cao, T. R. N. Roslan and W. Zhang, "The Valuation of Variance Swaps under Stochastic Volatility, Stochastic Interest Rate and Full Correlation Structure," Korean Mathematical Society J., vol. 57, no. 5, pp. 1167-1186, Jun. 2020.
[12] T. R. N. Roslan, "On the pricing of forward-start variance swaps with stochastic volatility and stochastic interest rate," Far East J. Math. Sci, vol. 102, pp. 3223-3240, July 2017.
[13] L. A. Grzelak and C. W. Oosterlee, "On the Heston model with stochastic interest rates," Financial Mathematics SIAM J., vol. 2, no. 1, pp. 255-286, Feb. 2011.
[14] K. Rindell, "Pricing of index options when interest rates are stochastic: An empirical test," Banking & Finance J., vol. 19, no. 5, pp. 785-802, Aug. 1995.
[15] M. Abudy and Y. Izhakian, "Pricing stock options with stochastic interest rate," Portfolio Analysis and Management International J., vol. 1, no. 3, pp. 250, Jan. 2013.
[16] T. R. N. Roslan, A. F. Jameel and S. Z. Ibrahim, "Modeling the price of hybrid equity warrants under stochastic volatility and stochastic interest rates," Advanced Computer Technology International J., vol, 9, no. 3, pp. 3586-3589, Mar. 2020.
[17] W. Xiao, W. Zhang, X. Zhang and X. Chen, "The valuation of equity warrants under the fractional Vasicek process of the short-term interest rate," Physica A, vol. 394, pp. 320-337, Jan. 2014.
[18] J. Cao, G. Lian and T. R. N. Roslan, "Pricing variance swaps using stochastic volatility and stochastic interest rate," Appl. Math. Comput., vol. 277, pp. 72-81, Mar. 2016.