On the pricing of double barrier options under stochastic volatility models: A probabilistic approach

Jerome Detemple; Yerkin Kitapbayev; Danila Shabalin; Jerome Detemple; Yerkin Kitapbayev; Danila Shabalin

doi:10.3934/math.20251007

AIMS Mathematics

2025, Volume 10, Issue 9: 22622-22649. doi: 10.3934/math.20251007

Previous Article Next Article

Research article Special Issues

On the pricing of double barrier options under stochastic volatility models: A probabilistic approach

1.
Questrom School of Business, Boston University, Boston, MA 02215, USA
2.
Mathematics Department, Khalifa University of Science and Technology, P.O. Box 127788, Abu Dhabi, United Arab Emirates
3.
Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia
4.
Vega Institute Foundation, Moscow 119234, Russia

Received: 21 July 2025 Revised: 07 September 2025 Accepted: 22 September 2025 Published: 29 September 2025
MSC : 60H30, 60J55, 91G20, 91G60

We study the pricing of double barrier knock-out options under stochastic volatility using a conditional Monte Carlo method based on the local time-space formula of Peskir. A valuation formula including an early knock-out discount is provided, where the discount depends on the local time of the underlying stochastic processes and the deltas of the option at the barriers. The latter solve a system of coupled Volterra integral equations of the first kind. This characterization leads to an efficient numerical method for general volatility diffusion models. An algorithm for numerical implementation, based on a conditional quasi-Monte Carlo simulation method, is presented and shown to converge numerically to the true value of the claim. A numerical study is performed to illustrate properties of double barrier knock-out calls in the Heston stochastic volatility model. In our calibration, we find that short-dated (long-dated) at-the-money (ATM) knock-out call prices increase (decrease) when the speed of mean reversion increases, long run mean volatility increases, and vol-of-vol decreases. We also find that the deltas and vegas of short-dated ATM double barrier calls can be very sensitive to volatility in contrast to long-dated ones.
- barrier option,
- Volterra integral equation,
- local time,
- stochastic volatility model,
- Monte-Carlo method
Citation: Jerome Detemple, Yerkin Kitapbayev, Danila Shabalin. On the pricing of double barrier options under stochastic volatility models: A probabilistic approach[J]. AIMS Mathematics, 2025, 10(9): 22622-22649. doi: 10.3934/math.20251007

Related Papers:

Abstract

We study the pricing of double barrier knock-out options under stochastic volatility using a conditional Monte Carlo method based on the local time-space formula of Peskir. A valuation formula including an early knock-out discount is provided, where the discount depends on the local time of the underlying stochastic processes and the deltas of the option at the barriers. The latter solve a system of coupled Volterra integral equations of the first kind. This characterization leads to an efficient numerical method for general volatility diffusion models. An algorithm for numerical implementation, based on a conditional quasi-Monte Carlo simulation method, is presented and shown to converge numerically to the true value of the claim. A numerical study is performed to illustrate properties of double barrier knock-out calls in the Heston stochastic volatility model. In our calibration, we find that short-dated (long-dated) at-the-money (ATM) knock-out call prices increase (decrease) when the speed of mean reversion increases, long run mean volatility increases, and vol-of-vol decreases. We also find that the deltas and vegas of short-dated ATM double barrier calls can be very sensitive to volatility in contrast to long-dated ones.

References

[1]	G. Agazzotti, J. P. Aguilar, C. Aglieri Rinella, J. L. Kirkby, Calibration and option pricing with stochastic volatility and double exponential jumps, J. Comput. Appl. Math., 465 (2025), 116563. https://doi.org/10.1016/j.cam.2025.116563 doi: 10.1016/j.cam.2025.116563
[2]	K. E. Atkinson, An existence theorem for Abel integral equations, SIAM J. Math. Anal., 5 (1974), 729–736. https://doi.org/10.1137/0505071 doi: 10.1137/0505071
[3]	D. S. Bates, Jumps and stochastic volatility: Exchange rate processes implicit in deutsche mark options, Rev. Financ. Stud., 9 (1996), 69–107. https://doi.org/10.1093/rfs/9.1.69 doi: 10.1093/rfs/9.1.69
[4]	M. Boyarchenko, S. Levendorski, Valuation of continuously monitored double barrier options and related securities, Math. Financ., 22 (2012), 419–444. http://doi.org/10.1111/j.1467-9965.2010.00469.x doi: 10.1111/j.1467-9965.2010.00469.x
[5]	H. Brunner, Collocation methods for Volterra integral and related functional equations, Cambridge: Cambridge University Press, 2004. https://doi.org/10.1017/CBO9780511543234
[6]	R. E. Caflisch, B. Moskowitz, Modified Monte Carlo methods using quasi-random sequences, In: Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing. Lecture Notes in Statistics, Springer, New York, 1995. https://doi.org/10.1007/978-1-4612-2552-2_1
[7]	P. Carr, A. Itkin, Semi-closed form solutions for barrier and American options written on a time-dependent Ornstein-Uhlenbeck process, J. Deriv., 29 (2021), 9–26. https://doi.org/10.3905/jod.2021.1.133 doi: 10.3905/jod.2021.1.133
[8]	P. Carr, A. Itkin, D. Muravey, Semi-closed form prices of barrier options in the time-dependent CEV and CIR models, J. Deriv., 28 (2020), 26–50. https://doi.org/10.3905/jod.2020.1.113 doi: 10.3905/jod.2020.1.113
[9]	P. Carr, A. Itkin, D. Muravey, Semi-analytical pricing of barrier options in the time-dependent Heston model, J. Deriv., 30 (2022), 141–171. https://doi.org/10.3905/jod.2022.30.2.141 doi: 10.3905/jod.2022.30.2.141
[10]	H. Chi, P. Beerli, D. W. Evans, M. Mascagni, On the scrambled Sobol sequence, In: International Conference on Computational Science, Berlin: Springer, 2005,775–782. https://doi.org/10.1007/11428862_105
[11]	C. Chiarella, B. Kang, G. H. Meyer, The evaluation of barrier option prices under stochastic volatility, Comput. Math. Appl., 64 (2012). 2034–2048. https://doi.org/10.1016/j.camwa.2012.03.103
[12]	J. Detemple, Y. Kitapbayev, The valuation of corporate securities with finite maturity debt, IMA J. Manag. Math., in press, 2025. https://doi.org/10.1093/imaman/dpaf028
[13]	H. Funahashi, T. Higuchi, An analytical approximation for single barrier options under stochastic volatility models, Ann. Oper. Res., 266 (2018), 129–157. http://doi.org/10.1007/s10479-017-2559-3 doi: 10.1007/s10479-017-2559-3
[14]	H. Geman, M. Yor, Pricing and hedging double-barrier options: A probabilistic approach, Math. Financ., 6 (1996), 365–378. https://doi.org/10.1111/j.1467-9965.1996.tb00122.x doi: 10.1111/j.1467-9965.1996.tb00122.x
[15]	C. Guardasoni, S. Sanfelici, Fast numerical pricing of barrier options under stochastic volatility and jumps, SIAM J. Appl. Math., 76 (2016), 27–57. https://doi.org/10.1137/15100504X doi: 10.1137/15100504X
[16]	C. Guardasoni, M. R. Rodrigo, S. Sanfelici, A Mellin transform approach to barrier option pricing, IMA J. Manag. Math., 31 (2020), 49–67. https://doi.org/10.1093/imaman/dpy016 doi: 10.1093/imaman/dpy016
[17]	T. Guillaume, Analytical valuation of a general form of barrier option with stochastic interest rate and jumps, Rev. Deriv. Res., 28 (2025), 8. https://doi.org/10.1007/s11147-025-09215-6 doi: 10.1007/s11147-025-09215-6
[18]	M. Ha, D. Kim, J. H. Yoon, Pricing of timer volatility-barrier options under Heston's stochastic volatility model, J. Comput. Appl. Math., 457 (2025), 116310. https://doi.org/10.1016/j.cam.2024.116310 doi: 10.1016/j.cam.2024.116310
[19]	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. https://doi.org/10.1093/rfs/6.2.327 doi: 10.1093/rfs/6.2.327
[20]	J. Hull, A. White, The pricing of options on assets with stochastic volatilities, J. Financ., 42 (1987), 281–300. https://doi.org/10.1111/j.1540-6261.1987.tb02568.x doi: 10.1111/j.1540-6261.1987.tb02568.x
[21]	A. Itkin, D. Muravey, Semi-closed form prices of barrier options in the Hull-White model, preprint paper, 2004. https://doi.org/10.48550/arXiv.2004.09591
[22]	A. Itkin, D. Muravey, Semi-analytic pricing of double barrier options with time-dependent barriers and rebates at hit, Front. Math. Financ., 1 (2022), 53–79. https://doi.org/10.3934/fmf.2021002 doi: 10.3934/fmf.2021002
[23]	J. Jeon, J. H. Yoon, C. R. Park, An analytic expansion method for the valuation of double-barrier options under a stochastic volatility model, J. Math. Anal. Appl., 449 (2017), 207–227. https://doi.org/10.1016/j.jmaa.2016.11.061 doi: 10.1016/j.jmaa.2016.11.061
[24]	D. Kim, J. H. Yoon, C. R. Park, Pricing external barrier options under a stochastic volatility model, J. Comput. Appl. Math., 394 (2021), 113555. https://doi.org/10.1016/j.cam.2021.113555 doi: 10.1016/j.cam.2021.113555
[25]	J. L. Kirkby, C. A. Rinella, J. P. Aguilar, N. Rupprecht, Return to the barrier: Option pricing and calibration in foreign exchange markets, J. Risk, 27 (2025), 1–32. http://doi.org/10.21314/JOR.2025.005 doi: 10.21314/JOR.2025.005
[26]	A. Kufner, B. Opic, How to define reasonably weighted Sobolev spaces, Comment. Math. Univ. Carolin., 25 (1984), 537–554.
[27]	N. Kunimoto, M. Ikeda, Pricing options with curved boundaries, Math. Financ., 2 (1992), 275–298. http://doi.org/10.1111/j.1467-9965.1992.tb00033.x doi: 10.1111/j.1467-9965.1992.tb00033.x
[28]	M. K. Lee, J. H. Kim, Pricing vanilla, barrier, and lookback options under two-scale stochastic volatility driven by two approximate fractional Brownian motions, AIMS Math., 9 (2024), 25545–25576. http://doi.org/10.3934/math.20241248 doi: 10.3934/math.20241248
[29]	P. Linz, Analytical and numerical methods for Volterra equations, Philadelphia: SIAM, 1985. http://doi.org/10.1137/1.9781611970852
[30]	A. Lipton, Mathematical methods for foreign exchange: A financial engineer's approach, Singapore: World Scientific, 2001. http://doi.org/10.1142/4694
[31]	A. Lipton, A. Sepp, Toward an efficient hybrid method for pricing barrier options on assets with stochastic volatility, Wilmott, 121 (2022), 74–88. http://doi.org/10.54946/wilm.11053 doi: 10.54946/wilm.11053
[32]	C. F. Lo, C. H. Hui, Valuing double-barrier options with time-dependent parameters by Fourier series expansion, IAENG Int. J. Appl. Math., 36 (2007), 1–5.
[33]	G. Loeper, O. Pironneau, A mixed PDE/Monte-Carlo method for stochastic volatility models, CR Acad. Sci. I-Math., 347 (2009), 559–563. http://doi.org/10.1016/j.crma.2009.02.021 doi: 10.1016/j.crma.2009.02.021
[34]	R. C. Merton, Theory of rational option pricing, Bell J. Econ., 4 (1973), 141–183. https://doi.org/10.2307/3003143 doi: 10.2307/3003143
[35]	A. Mijatović, Local time and the pricing of time-dependent barrier options, Financ. Stoch., 14 (2010), 13–48. https://doi.org/10.1007/s00780-008-0077-5 doi: 10.1007/s00780-008-0077-5
[36]	A. Mijatović, M. Pistorius, Continuously monitored barrier options under Markov processes, Math. Financ., 23 (2013), 1–38. https://doi.org/10.1111/j.1467-9965.2011.00486.x doi: 10.1111/j.1467-9965.2011.00486.x
[37]	A. Pelsser, Pricing double barrier options using Laplace transforms, Financ. Stoch., 4 (2000), 95–104. https://doi.org/10.1007/s007800050005 doi: 10.1007/s007800050005
[38]	G. Peskir, A change-of-variable formula with local time on curves, J. Theor. Probab., 18 (2005), 499–535. https://doi.org/10.1007/s10959-005-3517-6 doi: 10.1007/s10959-005-3517-6
[39]	G. Peskir, On the American option problem, Math. Financ., 15 (2005), 169–181. https://doi.org/10.1111/j.0960-1627.2005.00214.x doi: 10.1111/j.0960-1627.2005.00214.x
[40]	G. Peskir, A. Shiryaev, Optimal stopping and free-boundary problems, In: Lectures in Mathematics, ETH Zürich, Birkhäuser Basel, 2006. https://doi.org/10.1007/978-3-7643-7390-0
[41]	QuantLib: a free/open-source library for quantitative finance, Version v1.39, 2025. https://doi.org/10.5281/zenodo.1440997
[42]	S. Shahmorad, M. Mostafazadeh, Existence, uniqueness and regularity of the solution for a system of weakly singular Volterra integral equations of the first kind, J. Integral Equ. Appl., 35 (2023), 105–117. https://doi.org/10.1216/jie.2023.35.105 doi: 10.1216/jie.2023.35.105
[43]	R. Weiss, Product integration for the generalized Abel equation, Math. Comp., 26 (1972), 177–190. https://doi.org/10.1090/s0025-5718-1972-0299001-7 doi: 10.1090/s0025-5718-1972-0299001-7
[44]	G. A. Willard, Calculating prices and sensitivities for path-independent derivatives securities in multifactor models, J. Deriv., 5 (1997), 45–61. https://doi.org/10.3905/jod.1997.407982 doi: 10.3905/jod.1997.407982
[45]	U. Wystup, FX options and structured products, Hoboken: John Wiley & Sons, 2006. https://doi.org/10.1002/9781118673355
[46]	S. Zhang, J. Zhao, Efficient simulation for pricing barrier options with two-factor stochastic volatility and stochastic interest rate, Math. Probl. Eng., (2017), 3912036. https://doi.org/10.1155/2017/3912036
[47]	Y. X. Zhao, J. Y. Bao, Barrier option pricing in regime switching models with rebates, Acta Math. Appl. Sin., Engl., 40 (2024), 849–861. https://doi.org/10.1007/s10255-024-1053-3 doi: 10.1007/s10255-024-1053-3

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)