Research on the Application of Mathematical Models in Option Pricing
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Research on the Application of Mathematical Models in Option Pricing

Jiayi Ji 1*
1 New Channel-UIBE Qingdao A-level Centre
*Corresponding author: DoraJi1018@outlook.com
Published on 6 August 2025
Volume Cover
TNS Vol.132
ISSN (Print): 2753-8826
ISSN (Online): 2753-8818
ISBN (Print): 978-1-80590-305-5
ISBN (Online): 978-1-80590-306-2
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Abstract

Option pricing is one of the core problems in modern financial mathematics. This paper systematically reviews the mathematical models used in option pricing, including classical models (Black-Scholes model, binomial tree model), modern stochastic models (Heston model, Merton jump-diffusion model), numerical methods (Monte Carlo simulation, finite difference method), and machine learning techniques. Through theoretical analysis and empirical comparisons, the study reveals the mathematical principles, applicability, and limitations of these models. Furthermore, the study discusses model optimization directions in the context of real financial markets, particularly for special cases such as China's A-share market. The research shows that the evolution of mathematical models has always balanced market incompleteness and computational efficiency. Future trends will focus on hybrid models integrating stochastic analysis and data science.

Keywords:

Option pricing, Black-Scholes model, stochastic volatility, Monte Carlo simulation, machine learning

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Ji,J. (2025). Research on the Application of Mathematical Models in Option Pricing. Theoretical and Natural Science,132,16-22.

References

[1]. 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

[2]. Derman, E. (1999). Regimes of volatility. Goldman Sachs Quantitative Strategies Research Notes, April.

[3]. Merton, R.C. (1976). Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics, 3(1-2), 125-144. https: //doi.org/10.1016/0304-405X(76)90022-2

[4]. Cont, R. (2001). Empirical properties of asset returns: Stylized facts and statistical issues. Quantitative Finance, 1(2), 223-236. https: //doi.org/10.1080/713665670

[5]. Zhang, L., Wang, H., & Li, J. (2017). Institutional characteristics and option pricing in emerging markets: The case of China. Emerging Markets Review, 31, 1-15. https: //doi.org/10.1016/j.ememar.2016.12.001

[6]. Heston, S.L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies, 6(2), 327-343. https: //doi.org/10.1093/rfs/6.2.327

[7]. Bates, D.S. (1996). Jump and stochastic volatility: Exchange rate processes implicit in deutsche mark options. Review of Financial Studies, 9(1), 69-107. https: //doi.org/10.1093/rfs/9.1.69

[8]. Gatheral, J. (2006). The Volatility Surface: A Practitioner's Guide. Wiley Finance.

[9]. Liu, X., Zhang, Y., & Chen, W. (2019). Neural stochastic differential equations for option pricing. Journal of Computational Finance, 23(3), 1-28. https: //doi.org/10.21314/JCF.2019.358

[10]. Ruf, J., & Wang, W. (2020). Neural networks for option pricing and hedging: A literature review. Journal of Computational Finance, 24(1), 1-46. https: //doi.org/10.21314/JCF.2020.409

[11]. 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

[12]. Hull, J.C. (2018). Options, Futures and Other Derivatives (10th ed.). Pearson.

[13]. Billingsley, P. (1999). Convergence of Probability Measures (2nd ed.). Wiley.

[14]. Shreve, S.E. (2004). Stochastic Calculus for Finance II: Continuous-Time Models. Springer. https: //doi.org/10.1007/978-1-4757-4296-1

[15]. Duffie, D., Pan, J., & Singleton, K. (2000). Transform analysis and asset pricing for affine jump-diffusions. Econometrica, 68(6), 1343-1376. https: //doi.org/10.1111/1468-0262.00164

[16]. Cont, R., & Voltchkova, E. (2005). Finite difference methods for option pricing in jump-diffusion and exponential Lévy models. SIAM Journal on Numerical Analysis, 43(4), 1596-1626. https: //doi.org/10.1137/030600810

[17]. Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering. Springer. https: //doi.org/10.1007/978-0-387-21617-1

[18]. Longstaff, F.A., & Schwartz, E.S. (2001). Valuing American options by simulation: A simple least-squares approach. Review of Financial Studies, 14(1), 113-147. https: //doi.org/10.1093/rfs/14.1.113

[19]. Wilmott, P., Dewynne, J., & Howison, S. (1995). The Mathematics of Financial Derivatives: A Student Introduction. Cambridge University Press.

[20]. Çetin, U., Jarrow, R.A., & Protter, P. (2004). Liquidity risk and arbitrage pricing theory. Finance and Stochastics, 8(3), 311-341. https: //doi.org/10.1007/s00780-004-0123-x

[21]. Luo, R., Zhang, Y., & Chen, W. (2018). GANs for financial volatility surfaces. Proceedings of the 32nd International Conference on Neural Information Processing Systems, 1027-1038.

[22]. Bayer, C., Gatheral, J., & Karlsmark, M. (2019). Fast neural network approach for direct covar estimation. Journal of Computational Finance, 22(3), 1-28. https: //doi.org/10.21314/JCF.2019.358

[23]. Rebentrost, P., Gupt, B., & Bromley, T.R. (2018). Quantum computational finance: Monte Carlo pricing of financial derivatives. Physical Review A, 98(2), 022321. https: //doi.org/10.1103/PhysRevA.98.022321

[24]. Barberis, N., Huang, M., & Santos, T. (2001). Prospect theory and asset prices. Quarterly Journal of Economics, 116(1), 1-53. https: //doi.org/10.1162/003355301556310

References

[1]. 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

[2]. Derman, E. (1999). Regimes of volatility. Goldman Sachs Quantitative Strategies Research Notes, April.

[3]. Merton, R.C. (1976). Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics, 3(1-2), 125-144. https: //doi.org/10.1016/0304-405X(76)90022-2

[4]. Cont, R. (2001). Empirical properties of asset returns: Stylized facts and statistical issues. Quantitative Finance, 1(2), 223-236. https: //doi.org/10.1080/713665670

[5]. Zhang, L., Wang, H., & Li, J. (2017). Institutional characteristics and option pricing in emerging markets: The case of China. Emerging Markets Review, 31, 1-15. https: //doi.org/10.1016/j.ememar.2016.12.001

[6]. Heston, S.L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies, 6(2), 327-343. https: //doi.org/10.1093/rfs/6.2.327

[7]. Bates, D.S. (1996). Jump and stochastic volatility: Exchange rate processes implicit in deutsche mark options. Review of Financial Studies, 9(1), 69-107. https: //doi.org/10.1093/rfs/9.1.69

[8]. Gatheral, J. (2006). The Volatility Surface: A Practitioner's Guide. Wiley Finance.

[9]. Liu, X., Zhang, Y., & Chen, W. (2019). Neural stochastic differential equations for option pricing. Journal of Computational Finance, 23(3), 1-28. https: //doi.org/10.21314/JCF.2019.358

[10]. Ruf, J., & Wang, W. (2020). Neural networks for option pricing and hedging: A literature review. Journal of Computational Finance, 24(1), 1-46. https: //doi.org/10.21314/JCF.2020.409

[11]. 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

[12]. Hull, J.C. (2018). Options, Futures and Other Derivatives (10th ed.). Pearson.

[13]. Billingsley, P. (1999). Convergence of Probability Measures (2nd ed.). Wiley.

[14]. Shreve, S.E. (2004). Stochastic Calculus for Finance II: Continuous-Time Models. Springer. https: //doi.org/10.1007/978-1-4757-4296-1

[15]. Duffie, D., Pan, J., & Singleton, K. (2000). Transform analysis and asset pricing for affine jump-diffusions. Econometrica, 68(6), 1343-1376. https: //doi.org/10.1111/1468-0262.00164

[16]. Cont, R., & Voltchkova, E. (2005). Finite difference methods for option pricing in jump-diffusion and exponential Lévy models. SIAM Journal on Numerical Analysis, 43(4), 1596-1626. https: //doi.org/10.1137/030600810

[17]. Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering. Springer. https: //doi.org/10.1007/978-0-387-21617-1

[18]. Longstaff, F.A., & Schwartz, E.S. (2001). Valuing American options by simulation: A simple least-squares approach. Review of Financial Studies, 14(1), 113-147. https: //doi.org/10.1093/rfs/14.1.113

[19]. Wilmott, P., Dewynne, J., & Howison, S. (1995). The Mathematics of Financial Derivatives: A Student Introduction. Cambridge University Press.

[20]. Çetin, U., Jarrow, R.A., & Protter, P. (2004). Liquidity risk and arbitrage pricing theory. Finance and Stochastics, 8(3), 311-341. https: //doi.org/10.1007/s00780-004-0123-x

[21]. Luo, R., Zhang, Y., & Chen, W. (2018). GANs for financial volatility surfaces. Proceedings of the 32nd International Conference on Neural Information Processing Systems, 1027-1038.

[22]. Bayer, C., Gatheral, J., & Karlsmark, M. (2019). Fast neural network approach for direct covar estimation. Journal of Computational Finance, 22(3), 1-28. https: //doi.org/10.21314/JCF.2019.358

[23]. Rebentrost, P., Gupt, B., & Bromley, T.R. (2018). Quantum computational finance: Monte Carlo pricing of financial derivatives. Physical Review A, 98(2), 022321. https: //doi.org/10.1103/PhysRevA.98.022321

[24]. Barberis, N., Huang, M., & Santos, T. (2001). Prospect theory and asset prices. Quarterly Journal of Economics, 116(1), 1-53. https: //doi.org/10.1162/003355301556310

Cite this article

Ji,J. (2025). Research on the Application of Mathematical Models in Option Pricing. Theoretical and Natural Science,132,16-22.

Data availability

The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.

About volume

Volume title: Proceedings of CONF-APMM 2025 Symposium: Simulation and Theory of Differential-Integral Equation in Applied Physics

ISBN: 978-1-80590-305-5(Print) / 978-1-80590-306-2(Online)
Editor: Marwan Omar, Shuxia Zhao
Conference website: https://www.confapmm.org/dalian.html
Conference date: 27 September 2025
Series: Theoretical and Natural Science
Volume number: Vol.132
ISSN: 2753-8818(Print) / 2753-8826(Online)

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