Valuing European Option Under Double 3/2-Volatility Jump-Diffusion Model With Stochastic Interest Rate and Stochastic Intensity Under Approximative Fractional Brownian Motion

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Siham Bayad, Abdelmajid El Hajaj, Khalid Hilal


In this study, we propose a more comprehensive and realistic option pricing model based on approximative fractional Brownian motion, building upon recent advancements in this area. Specifically, we utilize the double 3/2-volatility Jump-Diffusion model, which incorporates approximative fractional Brownian motion with 3/2-volatility, stochastic interest rate, and stochastic intensity. To account for the stochastic interest rate, we employ a two-factor Vasicek model. Notably, our model accommodates negative interest rates. Consequently, we develop a multi-factor model with a stochastic interest rate structure for pricing European options and derive a closed-form pricing formula with an analytical solution by applying some algebraic calculations and Lie symmetries. In order to demonstrate the superiority of our proposed model over other classical approaches, we present numerical results that showcase the value of a European call option. This comparative analysis underscores the advantages of our model in comparison to traditional models.

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  1. G. Bakshi, C. Cao, Z. Chen, Empirical Performance of Alternative Option Pricing Models, J. Finance. 52 (1997), 2003–2049.
  2. D.S. Bates, Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options, Rev. Financ. Stud. 9 (1996), 69–107.
  3. T. Björk, H. Hult, A note on Wick products and the fractional Black-Scholes model, Finance Stoch. 9 (2005), 197–209.
  4. F. Black, M. Scholes, The Pricing of Options and Corporate Liabilities, J. Politic. Econ. 81 (1973), 637–654.
  5. J.C. Cox, J.E. Ingersoll, S.A. Ross, A Theory of the Term Structure of Interest Rates, Econometrica. 53 (1985), 385–407.
  6. Y. Chang, Y. Wang, S. Zhang, Option Pricing under Double Heston Jump-Diffusion Model with Approximative Fractional Stochastic Volatility, Mathematics. 9 (2021), 126.
  7. J.C. Duan, J.Z. Wei, Pricing Foreign Currency and Cross-Currency Options Under GARCH. J. Derivat. 7 (1999), 51–63.
  8. C. El-Nouty, The Fractional Mixed Fractional Brownian Motion, Stat. Prob. Lett. 65 (2003), 111–120.
  9. J. Goard, Closed-Form Formulae for European Options Under Three-Factor Models, Commun. Math. Stat. 8 (2019), 379–408.
  10. L.A. Grzelak, C.W. Oosterlee, S. Van Weeren, Extension of Stochastic Volatility Equity Models With the Hull–white Interest Rate Process, Quant. Finance. 12 (2012), 89–105.
  11. Y. Han, Z. Li, C. Liu, Option Pricing Under the Fractional Stochastic Volatility Model, ANZIAM J. 63 (2021), 123–142.
  12. S.L. Heston, A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options, Rev. Financ. Stud. 6 (1993), 327–343.
  13. Y. Hu, B. Øksendal, Fractional White Noise Calculus and Applications to Finance, Infin. Dimens. Anal. Quantum. Probab. Relat. Top. 06 (2003), 1–32.
  14. J. Hull, A. White, The Pricing of Options on Assets with Stochastic Volatilities, J. Finance. 42 (1987), 281–300.
  15. J.C. Hull, A.D. White, Using Hull-White Interest Rate Trees, J. Derivat. 3 (1996), 26–36.
  16. G.J. Jiang, Testing Option Pricing Models with Stochastic Volatility, Random Jumps and Stochastic Interest Rates, Int. Rev. Finance. 3 (2002), 233–272.
  17. J. Cao, B. Wang, W. Zhang, Valuation of European Options With Stochastic Interest Rates and Transaction Costs, Int. J. Computer Math. 99 (2022), 227–239.
  18. J. Goard, New Solutions to the Bond-Pricing Equation via Lie’s Classical Method, Math. Comput. Model. 32 (2000), 299–313.
  19. S.G. Kou, A Jump-Diffusion Model for Option Pricing, Manage. Sci. 48 (2002), 1086–1101.
  20. Y.S. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Springer, Berlin, Heidelberg, 2008.
  21. K. Rindell, Pricing of Index Options When Interest Rates Are Stochastic: An Empirical Test, J. Bank. Finance. 19 (1995), 785–802.
  22. L.O. Scott, Option Pricing when the Variance Changes Randomly: Theory, Estimation, and an Application, J. Financ. Quant. Anal. 22 (1987), 419–438.
  23. L.O. Scott, Pricing Stock Options in a Jump-Diffusion Model with Stochastic Volatility and Interest Rates: Applications of Fourier Inversion Methods, Math. Finance. 7 (1997), 413–426.
  24. R. Schöbel, J. Zhu, Stochastic Volatility With an Ornstein-Uhlenbeck Process: An Extension, Rev. Finance. 3 (1999), 23–46.
  25. E.M. Stein, J.C. Stein, Stock Price Distributions with Stochastic Volatility: An Analytic Approach, Rev. Financ. Stud. 4 (1991), 727–752.
  26. L. Sun, Pricing Currency Options in the Mixed Fractional Brownian Motion, Physica A: Stat. Mech. Appl. 392 (2013), 3441–3458.
  27. T.H. Thao, An Approximate Approach to Fractional Analysis for Finance, Nonlinear Anal.: Real World Appl. 7 (2006), 124–132.
  28. O. Vasicek, An Equilibrium Characterization of the Term Structure, J. Financ. Econ. 5 (1977), 177–188.
  29. J.B. Wiggins, Option Values Under Stochastic Volatility: Theory and Empirical Estimates, J. Financ. Econ. 19 (1987), 351–372.
  30. W. Xiao, W. Zhang, W. Xu, Parameter estimation for fractional Ornstein–Uhlenbeck processes at discrete observation, Appl. Math. Model. 35 (2011), 4196–4207.
  31. W. Xiao, W. Zhang, W. Xu, X. Zhang, The Valuation of Equity Warrants in a Fractional Brownian Environment, Physica A: Stat. Mech. Appl. 391 (2012), 1742–1752.