Multiscale asymptotic pricing and free-boundary analysis of vulnerable perpetual American put options under stochastic elasticity of variance

Ji-Hun Yoon; Sangmin Park; Mijin Ha; Ji-Hun Yoon; Sangmin Park; Mijin Ha

doi:10.3934/math.20261241

AIMS Mathematics

2026, Volume 11, Issue 9: 31434-31475. doi: 10.3934/math.20261241

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Multiscale asymptotic pricing and free-boundary analysis of vulnerable perpetual American put options under stochastic elasticity of variance

1.
Department of Mathematics, Pusan National University, Busan 46241, Republic of Korea
2.
Department of Mathematics and Institute of Mathematical Sciences, College of Natural Sciences, Pusan National University, Busan, Republic of Korea

Received: 06 July 2026 Revised: 12 September 2026 Accepted: 21 September 2026 Published: 23 September 2026
MSC : 91G20, 91G80, 35R35, 60H30

This paper develops a pricing framework for vulnerable perpetual American put options under a stochastic elasticity of variance (SEV) model with counterparty default risk and fractional recovery. Counterparty default is modeled as a Cox time with constant intensity, while the elasticity of variance is driven by a fast mean-reverting Ornstein–Uhlenbeck factor. Dynamic programming yields a default-adjusted variational inequality with a free boundary representing the optimal exercise threshold. A multiscale asymptotic expansion provides explicit approximations for the option price and the optimal exercise boundary: The leading-order term incorporates default and recovery effects, while higher-order corrections capture stochastic elasticity, fast mean reversion, asset–elasticity correlation, and the elasticity risk premium. Numerical results show that the corrected approximation closely matches Monte Carlo benchmark prices and substantially improves upon the leading-order approximation. Higher recovery rates increase option values, whereas higher default intensities reduce them. SEV corrections are concentrated in the continuation region, while sensitivity analyses indicate that SEV-specific parameters affect the optimal exercise boundary more strongly than the option price.
Citation: Ji-Hun Yoon, Sangmin Park, Mijin Ha. Multiscale asymptotic pricing and free-boundary analysis of vulnerable perpetual American put options under stochastic elasticity of variance[J]. AIMS Mathematics, 2026, 11(9): 31434-31475. doi: 10.3934/math.20261241

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Abstract

This paper develops a pricing framework for vulnerable perpetual American put options under a stochastic elasticity of variance (SEV) model with counterparty default risk and fractional recovery. Counterparty default is modeled as a Cox time with constant intensity, while the elasticity of variance is driven by a fast mean-reverting Ornstein–Uhlenbeck factor. Dynamic programming yields a default-adjusted variational inequality with a free boundary representing the optimal exercise threshold. A multiscale asymptotic expansion provides explicit approximations for the option price and the optimal exercise boundary: The leading-order term incorporates default and recovery effects, while higher-order corrections capture stochastic elasticity, fast mean reversion, asset–elasticity correlation, and the elasticity risk premium. Numerical results show that the corrected approximation closely matches Monte Carlo benchmark prices and substantially improves upon the leading-order approximation. Higher recovery rates increase option values, whereas higher default intensities reduce them. SEV corrections are concentrated in the continuation region, while sensitivity analyses indicate that SEV-specific parameters affect the optimal exercise boundary more strongly than the option price.

References

[1]	H. Johnson, R. Stulz, The pricing of options with default risk, J. Financ., 42 (1987), 267–280. https://doi.org/10.1111/j.1540-6261.1987.tb02567.x doi: 10.1111/j.1540-6261.1987.tb02567.x
[2]	J. Hull, A. White, The impact of default risk on the prices of options and other derivative securities, J. Bank. Financ., 19 (1995), 299–322. https://doi.org/10.1016/0378-4266(94)00050-D doi: 10.1016/0378-4266(94)00050-D
[3]	P. Klein, Pricing Black–Scholes options with correlated credit risk, J. Bank. Financ., 20 (1996), 1211–1229. https://doi.org/10.1016/0378-4266(95)00052-6 doi: 10.1016/0378-4266(95)00052-6
[4]	P. Klein, M. Inglis, Pricing vulnerable European options when the option's payoff can increase the risk of financial distress, J. Bank. Financ., 25 (2001), 993–1012. https://doi.org/10.1016/S0378-4266(00)00109-6 doi: 10.1016/S0378-4266(00)00109-6
[5]	R. C. Merton, On the pricing of corporate debt: The risk structure of interest rates, J. Financ., 29 (1974), 449–470. https://doi.org/10.2307/2978814 doi: 10.2307/2978814
[6]	F. Black, J. C. Cox, Valuing corporate securities: Some effects of bond indenture provisions, J. Financ., 31 (1976), 351–367. https://doi.org/10.1111/j.1540-6261.1976.tb01891.x doi: 10.1111/j.1540-6261.1976.tb01891.x
[7]	R. A. Jarrow, S. M. Turnbull, Pricing derivatives on financial securities subject to credit risk, J. Financ., 50 (1995), 53–85. https://doi.org/10.1111/j.1540-6261.1995.tb05167.x doi: 10.1111/j.1540-6261.1995.tb05167.x
[8]	D. Lando, On Cox processes and credit risky securities, Rev. Deriv. Res., 2 (1998), 99–120. https://doi.org/10.1007/BF01531332 doi: 10.1007/BF01531332
[9]	D. Duffie, K. J. Singleton, Modeling term structures of defaultable bonds, Rev. Financ. Stud., 12 (1999), 687–720. https://doi.org/10.1093/rfs/12.4.687 doi: 10.1093/rfs/12.4.687
[10]	T. R. Bielecki, M. Rutkowski, Credit risk: Modeling, valuation and hedging, Springer, 2002. https://doi.org/10.1007/978-3-662-04821-4
[11]	F. Black, M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ., 81 (1973), 637–654. https://doi.org/10.1086/260062 doi: 10.1086/260062
[12]	R. C. Merton, Theory of rational option pricing, Bell J. Econ. Manage. Sci., 4 (1973), 141–183. https://doi.org/10.1142/5860 doi: 10.1142/5860
[13]	J. C. Cox, S. A. Ross, The valuation of options for alternative stochastic processes, J. Financ. Econ., 3 (1976), 145–166. https://doi.org/10.1016/0304-405X(76)90023-4 doi: 10.1016/0304-405X(76)90023-4
[14]	M. Schroder, Computing the constant elasticity of variance option pricing formula, J. Financ., 44 (1989), 211–219. https://doi.org/10.1111/j.1540-6261.1989.tb02414.x doi: 10.1111/j.1540-6261.1989.tb02414.x
[15]	J. H. Kim, J. H. Yoon, J. Lee, S. Y. Choi, On the stochastic elasticity of variance diffusions, Econ. Model., 51 (2015), 263–268. https://doi.org/10.1016/j.econmod.2015.08.011 doi: 10.1016/j.econmod.2015.08.011
[16]	J. Cao, J. H. Kim, S. W. Kim, W. Zhang, Rough stochastic elasticity of variance and option pricing, Financ. Res. Lett., 37 (2020), 101381. https://doi.org/10.1016/j.frl.2019.101381 doi: 10.1016/j.frl.2019.101381
[17]	J. P. Fouque, G. Papanicolaou, R. Sircar, K. Solna, Multiscale stochastic volatility asymptotics, Multiscale Model. Simul., 2 (2003), 22–42. https://doi.org/10.1137/030600291 doi: 10.1137/030600291
[18]	J. P. Fouque, G. Papanicolaou, R. Sircar, K. Solna, Multiscale stochastic volatility for equity, interest rate, and credit derivatives, Cambridge University Press, 2011. https://doi.org/10.1017/CBO9781139020534
[19]	P. Klein, J. Yang, Vulnerable American options, Manage. Financ., 36 (2010), 414–430. https://doi.org/10.1108/03074351011039436 doi: 10.1108/03074351011039436
[20]	J. H. Yoon, J. H. Kim, S. Y. Choi, Multiscale analysis of a perpetual American option with the stochastic elasticity of variance, Appl. Math. Lett., 26 (2013), 670–675. https://doi.org/10.1016/j.aml.2012.11.015 doi: 10.1016/j.aml.2012.11.015
[21]	H. P. McKean, Appendix: A free boundary problem for the heat equation arising from a problem in mathematical economics, Ind. Manage. Rev., 6 (1965), 32–39. https://cir.nii.ac.jp/crid/1573105974041404416
[22]	I. J. Kim, The analytic valuation of American options, Rev. Financ. Stud., 3 (1990), 547–572. https://doi.org/10.1093/rfs/3.4.547 doi: 10.1093/rfs/3.4.547
[23]	S. D. Jacka, Optimal stopping and the American put, Math. Financ., 1 (1991), 1–14. https://doi.org/10.1111/j.1467-9965.1991.tb00007.x doi: 10.1111/j.1467-9965.1991.tb00007.x
[24]	G. Peskir, A. Shiryaev, Optimal stopping and free-boundary problems, Birkhäuser, 2006. https://doi.org/10.1007/978-3-7643-7390-0_4
[25]	J. H. Yoon, J. H. Kim, A closed-form analytic correction to the Black–Scholes–Merton price for perpetual American options, Appl. Math. Lett., 26 (2013), 1146–1150. https://doi.org/10.1016/j.aml.2013.06.012 doi: 10.1016/j.aml.2013.06.012
[26]	M. K. Lee, S. J. Yang, J. H. Kim, Pricing vulnerable options with constant elasticity of variance versus stochastic elasticity of variance, Econ. Comput. Econ. Cybern. Stud. Res., 51 (2017), 233–247.
[27]	T. S. Zaevski, O. Kounchev, M. Savov, Two frameworks for pricing defaultable derivatives, Chaos Soliton. Fract., 123 (2019), 309–319. https://doi.org/10.1016/j.chaos.2019.04.025 doi: 10.1016/j.chaos.2019.04.025
[28]	J. Detemple, W. Tian, The valuation of American options for a class of diffusion processes, Manage. Sci., 48 (2002), 917–937. https://doi.org/10.1287/mnsc.48.7.917.2815 doi: 10.1287/mnsc.48.7.917.2815
[29]	C. Cai, T. De Angelis, J. Palczewski, The American put with finite-time maturity and stochastic interest rate, Math. Financ., 32 (2022), 1170–1213. https://doi.org/10.1111/mafi.12361 doi: 10.1111/mafi.12361
[30]	T. De Angelis, G. Peskir, Global $C^1$ regularity of the value function in optimal stopping problems, Ann. Appl. Probab., 30 (2020), 1007–1031. https://doi.org/10.1214/19-AAP1517 doi: 10.1214/19-AAP1517
[31]	A. G. Ramm, A simple proof of the Fredholm alternative and a characterization of the Fredholm operators, Amer. Math. Mon., 108 (2001), 855–860. https://doi.org/10.1080/00029890.2001.11919820 doi: 10.1080/00029890.2001.11919820
[32]	M. Ha, J. H. Yoon, Asymptotic analysis of vulnerable perpetual American put options under fast mean-reverting stochastic volatility, Working paper, 2026.
[33]	S. P. Zhu, J. Zhang, Using Laplace transform to price American puts, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 19 (2012), 447–469.
[34]	F. A. Longstaff, E. S. Schwartz, Valuing American options by simulation: A simple least-squares approach, Rev. Financ. Stud., 14 (2001), 113–147. https://doi.org/10.1093/rfs/14.1.113 doi: 10.1093/rfs/14.1.113
[35]	A. Alfonsi, B. Jourdain, Exact volatility calibration based on a Dupire-type call–put duality for perpetual American options, Nonlinear Differ. Equ. Appl., 16 (2009), 523–554. https://doi.org/10.1007/s00030-009-0027-8 doi: 10.1007/s00030-009-0027-8

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