Analytical pricing of maximum and minimum options based on a class of partial differential-integral equations: Applications of Mellin transform

Libin Wang; Yue Zhao; Libin Wang; Yue Zhao

doi:10.3934/era.2025345

Electronic Research Archive

2025, Volume 33, Issue 12: 7810-7840. doi: 10.3934/era.2025345

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Analytical pricing of maximum and minimum options based on a class of partial differential-integral equations: Applications of Mellin transform

Libin Wang ^{1,2
,
,},
Yue Zhao ²

1.
Financial Innovation and Risk Management Research Center, Hebei Finance University, Baoding 071051, China
2.
School of Statistics and Data Science, Hebei Finance University, Baoding 071051, China

Received: 12 October 2025 Revised: 16 December 2025 Accepted: 22 December 2025 Published: 25 December 2025

In this work, the analytical pricing of maximum and minimum options under the integrated scenario of interdependent stochastic volatility and jump, and the stochastic interest rate, were investigated by means of the composite Mellin transform approach. The analytic expressions of Mellin transform functions of the price of the maximum put option, minimum call option, and the exchange option were derived by different partial differential-integral equations (PDIEs). Meanwhile, the explicit price of other maximum and minimum options were obtained by means of the payoff decomposition technique and the parity relation of max-min and exchange options. In addition, the convergence of solutions of PDIEs was further demonstrated by transform techniques and decomposition skills. The simulation of the price process of two underlying assets was given to present the effectiveness and uniqueness of the proposed model. Finally, numerical analysis was implemented to examine the accuracy of the PDIE method and the validity of key parameters.
- Mellin transform,
- PDIE,
- stochastic volatility,
- dependent jump,
- stochastic interest rate
Citation: Libin Wang, Yue Zhao. Analytical pricing of maximum and minimum options based on a class of partial differential-integral equations: Applications of Mellin transform[J]. Electronic Research Archive, 2025, 33(12): 7810-7840. doi: 10.3934/era.2025345

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Abstract

In this work, the analytical pricing of maximum and minimum options under the integrated scenario of interdependent stochastic volatility and jump, and the stochastic interest rate, were investigated by means of the composite Mellin transform approach. The analytic expressions of Mellin transform functions of the price of the maximum put option, minimum call option, and the exchange option were derived by different partial differential-integral equations (PDIEs). Meanwhile, the explicit price of other maximum and minimum options were obtained by means of the payoff decomposition technique and the parity relation of max-min and exchange options. In addition, the convergence of solutions of PDIEs was further demonstrated by transform techniques and decomposition skills. The simulation of the price process of two underlying assets was given to present the effectiveness and uniqueness of the proposed model. Finally, numerical analysis was implemented to examine the accuracy of the PDIE method and the validity of key parameters.

References

[1]	F. Black, M. S. Scholes, The pricing of options and corporate liablities, J. Polit. Econ., 81 (1973), 637–654. https://doi.org/10.1086/260062 doi: 10.1086/260062
[2]	R. Stulz, Options on the Minimum or the Maximum of two risky Assets, J. Financ. Econ., 10 (1982), 161–185. https://doi.org/10.1016/0304-405X(82)90011-3 doi: 10.1016/0304-405X(82)90011-3
[3]	H. Johnson, Options on the Minimum or the Maximum of several Assets, J. Financ. Quant. Anal., 22 (1987), 277–283. https://doi.org/10.2307/2330963 doi: 10.2307/2330963
[4]	G. Kim, E. Koo, Closed-form pricing formula for exchange option with credit risk, Chaos, Solitons Fractals, 91 (2016), 221–227. https://doi.org/10.1016/j.chaos.2016.06.005 doi: 10.1016/j.chaos.2016.06.005
[5]	R. C Merton, Option pricing when underlying stock returns are discontinuous, J. Financ. Econ., 3 (1976), 125–144. https://doi.org/10.1016/0304-405X(76)90022-2 doi: 10.1016/0304-405X(76)90022-2
[6]	W. Li, L. Liu, G. Lv, C. Li, Exchange option pricing in jump-diffusion models based on esscher transform, Commun. Stat.- Theory Methods, 47 (2018), 4661–4672. https://doi.org/10.1080/03610926.2018.1444180 doi: 10.1080/03610926.2018.1444180
[7]	S. Heston, A closed-form solutions for options with stochastic volatility with applications to bond and currency options, Rev. Financ. Stud., 6 (1993), 327–343. https://doi.org/10.1093/rfs/6.2.327 doi: 10.1093/rfs/6.2.327
[8]	D. Duffie, J. Pan, K. Singleton, Transform analysis and asset pricing for affine jump-diffusions, Econometrica, 68 (1987), 1343–1376.
[9]	Q. Zhou, J. J. Yang, W. X. Wu, Pricing vulnerable options with correlated credit risk under jump-diffusion processes when corpoate liabilities are random, Acta Math. Appl. Sin. Engl. Ser., 35 (2019), 305–318. https://doi.org/10.1007/s10255-019-0821-y doi: 10.1007/s10255-019-0821-y
[10]	P. Carr, D. B. Madan, Option valuation using the fast Fourier transform, J. Comput. Finance, 2 (1999), 61–73. https://doi.org/10.21314/JCF.1999.043 doi: 10.21314/JCF.1999.043
[11]	S. Zhang, L. Wang, Fast Fourier transform option pricing with stochastic interest rate, stochastic volatility and double jumps, Appl. Math. Comput., 219 (2013), 10928–10933. https://doi.org/10.1016/j.amc.2013.05.008 doi: 10.1016/j.amc.2013.05.008
[12]	S. A. Lin, X. He, A close-form pricing formula for foward start options under a Markov regime-switching stochastic volatility model, Chaos, Solutions Fractals, 144 (2021), 110644. https://doi.org/10.1016/j.chaos.2020.110644 doi: 10.1016/j.chaos.2020.110644
[13]	Y. S. Xie, G. Deng, Vulnerable European option Pricing in a Markov regime-switching stochastic volatility model with stochastic interest rate, Chaos, Solutions Fractals, 156 (2022), 111896. https://doi.org/10.1016/j.chaos.2022.111896 doi: 10.1016/j.chaos.2022.111896
[14]	Y. A. Brychkow, H. J. Glaeske, A. P. Prudnikov, V. K. Tuan, Multidimensional Integral Transformations, 2$^{nd}$ edition, Gordon and Breach Science Publishers, Philadelphia, 1992.
[15]	R. Frontczak, Pricing options in jump diffusion models using Mellin transforms, Math. Finance, 3 (2013), 366–373. https://doi.org/10.4236/jmf.2013.33037 doi: 10.4236/jmf.2013.33037
[16]	J. H. Yoon, Mellin transform method for European option with Hull-White stochastic interest rate, J. Appl. Math., 2014 (2014), 759562. https://doi.org/10.1155/2014/759562 doi: 10.1155/2014/759562
[17]	J. H. Yoon, J. H. Kim, The pricing of vulnerable options with double Mellin transforms, J. Math. Anal. Appl, 422 (2015), 838–857. https://doi.org/10.1016/j.jmaa.2014.09.015 doi: 10.1016/j.jmaa.2014.09.015
[18]	J. Jeon, J. H. Yoon, A closed-form solution for lookback options using Mellin transform approach, East Asian Math. J., 32 (2016), 301–310. https://doi.org/10.7858/eamj.2016.022 doi: 10.7858/eamj.2016.022
[19]	C. Li, H. Liu, L. Liu, Q. Yao, Pricing vulnerable options under jump diffusion processes using double Mellin transform, Commun. Stat.- Simul. Comput., 52 (2023), 703–716. https://doi.org/10.1080/03610918.2020.1864645 doi: 10.1080/03610918.2020.1864645
[20]	S. Kou, A jump-diffusion model for option pricing, Manage. Sci., 48 (2002), 1086–1101. https://doi.org/10.1287/mnsc.48.8.1086.166 doi: 10.1287/mnsc.48.8.1086.166
[21]	M. Ruijter, C. W. Oosterlee, Two dimensional fourier cosine series expansion method for pricing financial options, SIAM J. Sci. Comput., 34 (2012), 642–671. https://doi.org/10.1137/12086205 doi: 10.1137/12086205
[22]	M. Mininni, G. Orlando, G. Taglialatela, Challenges in approximating the Black and Scholes call formula with hyperbolic tangents, Decisions Econ. Finance, 44 (2021), 73–100. https://doi.org/10.1007/s10203-020-00305-8 doi: 10.1007/s10203-020-00305-8
[23]	G. Orlando, G. Taglialatela, A review on implied volatility calculation, J. Comput. Appl. Math., 320 (2017), 202–220. https://doi.org/10.1016/j.cam.2017.02.002 doi: 10.1016/j.cam.2017.02.002

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