Stable reconstruction of the volatility in a regime-switching local-volatility model

Mourad Bellassoued; Raymond Brummelhuis; Michel Cristofol; Éric Soccorsi

doi:10.3934/mcrf.2019036

Previous Article
MCRF Home
This Issue
Next Article
A direct method based on the Clenshaw-Curtis formula for fractional optimal control problems

March 2020, 10(1): 189-215. doi: 10.3934/mcrf.2019036

Stable reconstruction of the volatility in a regime-switching local-volatility model

Mourad Bellassoued ^1, , Raymond Brummelhuis ^2, , Michel Cristofol ^3, and Éric Soccorsi ^4,,

1.	University of Tunis El Manar, National Engineering School of Tunis, ENIT-LAMSIN, B.P. 37, 1002 Tunis, Tunisia
2.	Laboratoire de Mathématiques de Reims, Université de Reims, EA 4535, France
3.	Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France
4.	Aix-Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France

^* Corresponding author: eric.soccorsi@univ-amu.fr

Received December 2017 Revised April 2019 Published March 2020 Early access August 2019

Fund Project: The last author is partially supported by the Agence Nationale de la Recherche under grant ANR-17- CE40-0029.

Prices of European call options in a regime-switching local-volatility model can be computed by solving a parabolic system which generalizes the classical Black and Scholes equation, giving these prices as functionals of the local-volatilities. We prove Lipschitz stability for the inverse problem of determining the local-volatilities from quoted call option prices for a range of strikes, if the calls are indexed by the different states of the continuous Markov chain which governs the regime switches.

Keywords: Inverse problem, local-volatility, regime-switching model, stability inequality, parabolic Carleman inequality.

Mathematics Subject Classification: Primary: 35R30, 35K10; Secondary: 58J35.

Citation: Mourad Bellassoued, Raymond Brummelhuis, Michel Cristofol, Éric Soccorsi. Stable reconstruction of the volatility in a regime-switching local-volatility model. Mathematical Control and Related Fields, 2020, 10 (1) : 189-215. doi: 10.3934/mcrf.2019036

References:

[1]	D. D. Aingworth, S. R. Das and R. Motwani, A simple approach for pricing equity options with Markov switching state variables, Quant. Fin., 6 (2006), 95-105. doi: 10.1080/14697680500511215.
[2]	V. Albani, A. De Cezaro and J. P. Zubelli, On the choice of the Tikhonov regularization parameter and the discretization level: A discrepancy-based strategy, Inverse Probl. Imaging, 10 (2016), 1-25. doi: 10.3934/ipi.2016.10.1.
[3]	V. Albani, A. De Cezaro and J. P. Zubelli, Convex regularization of local volatility estimation, Int. J. Theor. Appl. Finance, 20 (2017), 1750006, 37 pp. doi: 10.1142/S0219024917500066.
[4]	D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa, 22 (1968), 607-694.
[5]	F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Political Econ., 81 (1973), 637-654. doi: 10.1086/260062.
[6]	N. P. B. Bollen, Valuing options in regime-switching models, J. Derivatives, 6 (1998), 38-49. doi: 10.3905/jod.1998.408011.
[7]	I. Bouchouev and V. Isakov, The inverse problem of option pricing, Inv. Probl., 13 (1997), L11–L17. doi: 10.1088/0266-5611/13/5/001.
[8]	I. Bouchouev and V. Isakov, Uniqueness, stability and numerical methods for the inverse problem that arises in financial markets, Inv. Probl., 15 (1999), R95–R116. doi: 10.1088/0266-5611/15/3/201.
[9]	I. Bouchouev, V. Isakov and N. Valdivia, Recovery of the volatilty coefficient by linearization, Quantitative Finance, 2 (2002), 257-263. doi: 10.1088/1469-7688/2/4/302.
[10]	J. Buffington and R. J. Elliott, American options with regime switching, Int. J. Theor. Appl. Fin., 5 (2002), 497-514. doi: 10.1142/S0219024902001523.
[11]	S. Chin and D. Dufresne, A general formula for option prices in a stochastic volatility model, Appl. Math. Fin., 19 (2012), 313-340. doi: 10.1080/1350486X.2011.624823.
[12]	S. Crépey, Calibration of the local volatility in a trinomial tree using Tikhonov regularization, Inv. Prob, 19 (2003), 91-127. doi: 10.1088/0266-5611/19/1/306.
[13]	S. Crépey, Calibration of the local volatility in a generalized Black-Scholes model using Tikhonov regularization, SIAM J. Math. Anal., 34 (2003), 1183-1206. doi: 10.1137/S0036141001400202.
[14]	M. Cristofol and L. Roques, Simultaneous determination of the drift and diffusion coefficients in stochastic differential equations, Inv. Probl., 33 (2017), 095006, 12 pp. doi: 10.1088/1361-6420/aa7a1c.
[15]	E. B. Davies, Linear Operators and Their Spectra, Cambridge Studies in Advanced Mathematics, 106. Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9780511618864.
[16]	A. De Cezaro, O. Scherzer and J. P. Zubelli, Convex regularization of local volatility models from option prices: Convergence analysis and rates, Nonlinear Analysis, 75 (2012), 2398-2415. doi: 10.1016/j.na.2011.10.037.
[17]	M. V. de Hoop, L. Y. Qiu and O. Scherzer, Local analysis of inverse problems: Hölder stability and iterative reconstruction, Inv. Prob., 28 (2012), 045001, 16 pp. doi: 10.1088/0266-5611/28/4/045001.
[18]	Z. C. Deng, J. N. Yu and L. Yang, An inverse problem of determining the implied volatility in option pricing, J. Math. Anal. Appl., 340 (2008), 16-31. doi: 10.1016/j.jmaa.2007.07.075.
[19]	B. Dupire, Pricing with a smile, Risk, 7 (1994), 18-20.
[20]	H. Egger and H. W. Engl, Tikhonov regularization applied to the inverse problem of option pricing: Convergence analysis and rates, Inv. Probl., 21 (2005), 1027-1045. doi: 10.1088/0266-5611/21/3/014.
[21]	S. D. , Parabolic Systems, North-Holland Publishing Co., Amsterdam-London, Wolters-Noordhoff Publishing, Groningen, 1969.
[22]	S. D. Eidelman, S. D. Ivasyshen and A. N. Kochubei, Analytic Methods in the Theory of Differential and Pseudo-Differential Equations of Parabolic Type, Operator Theory: Advances and Applications, vol. 152. Birkhäuser Verlag, Basel, 2004. doi: 10.1007/978-3-0348-7844-9.
[23]	R. J. Elliott, L. Aggoun and J. B. Moore, Hidden Markov Models: Estimation and Control, Applications of Mathematics, 29. Springer-Verlag, New York, 1995.
[24]	R. J. Elliott, L. Chan and T. K. Siu, Option pricing and Esscher transform under regime switching, Ann. Fin., 1 (2005), 423-432. doi: 10.1007/s10436-005-0013-z.
[25]	R. J. Elliott, L. Chan and T. K. Siu, Option valuation under a regime-switching constant elasticity of variance process, Appl. Math. Comp., 219 (2013), 4434-4443. doi: 10.1016/j.amc.2012.10.047.
[26]	R. J. Elliott, T. K. Siu and L. Chan, On pricing barrier options with regime switching, J. Comp. Appl. Math., 256 (2014), 196-210. doi: 10.1016/j.cam.2013.07.034.
[27]	L. C. Evans, Partial Differential Equations, Graduate studies in Mathematics, 19. Amer. Math. Soc., Providence, RI, 1998.
[28]	A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N. J. 1964.
[29]	C.-D. Fuh, K. W. R. Ho, I. Hu and R.-H. Wang, Option pricing with Markov switching, J. Data Science, 10 (2012), 483-509.
[30]	A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, 34. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.
[31]	O. Y. Imanuvilov and M. Yamamoto, Lipshitz stability in inverse parabolic problems by Carleman estimate, Inv. Prob., 14 (1998), 1229-1245. doi: 10.1088/0266-5611/14/5/009.
[32]	V. Isakov, Recovery of time dependent volatility coefficient by linearization, Evolution Equations and Control Theory, 3 (2014), 119-134. doi: 10.3934/eect.2014.3.119.
[33]	G. Kresin and V. Maz'ya, Maximum Principles and Sharp Constants for Solutions of Elliptic and Parabolic Systems, Math. Surveys and Monographs vol. 183. Amer. Math. Soc., providence, RI, 2012. doi: 10.1090/surv/183.
[34]	N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, Graduate Studies in Mathematics, 96. Amer. Math. Soc., Providence, RI, 2008. doi: 10.1090/gsm/096.
[35]	J. Le Rousseau and G. Lebeau, On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations, ESAIM: Control, Optimisation and Calculus of Variations, 18 (2012), 712-747. doi: 10.1051/cocv/2011168.
[36]	L. S. Jiang and Y. S. Tao, Identifying the volatility of underlying assets from option prices, Inv. Probl., 17 (2001), 137-155. doi: 10.1088/0266-5611/17/1/311.
[37]	K. Otsuka, On the positivity of the fundamental solutions for parabolic systems, J. Math. Kyoto Univ., 28 (1988), 119-132. doi: 10.1215/kjm/1250520562.
[38]	M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Corrected reprint of the 1967 original, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5.
[39]	P. Stefanov and G. Uhlmann, Boundary rigidity and stability for generic simple metric, J. Amer. Math. Soc., 18 (2005), 975-1003. doi: 10.1090/S0894-0347-05-00494-7.
[40]	X. J. Xi, M. R. Rodrigo and R. S. Mamon, Parameter estimation of a regime-switching model using an inverse Stieltjes moment approach, Stochastic Processes, Finance and Control: A Festschrift in Honor of Robert J. Elliott, World scientific, 1 (2012), 549–569. doi: 10.1142/9789814383318_0022.
[41]	S.-P. Zhu, A. Badran and X. P. Lu, A new exact solution for pricing European options in a two-state regime-switching economy, Computers and Mathematics with Applications, 64 (2012), 2744-2755. doi: 10.1016/j.camwa.2012.08.005.

show all references

References:

[1]	D. D. Aingworth, S. R. Das and R. Motwani, A simple approach for pricing equity options with Markov switching state variables, Quant. Fin., 6 (2006), 95-105. doi: 10.1080/14697680500511215.
[2]	V. Albani, A. De Cezaro and J. P. Zubelli, On the choice of the Tikhonov regularization parameter and the discretization level: A discrepancy-based strategy, Inverse Probl. Imaging, 10 (2016), 1-25. doi: 10.3934/ipi.2016.10.1.
[3]	V. Albani, A. De Cezaro and J. P. Zubelli, Convex regularization of local volatility estimation, Int. J. Theor. Appl. Finance, 20 (2017), 1750006, 37 pp. doi: 10.1142/S0219024917500066.
[4]	D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa, 22 (1968), 607-694.
[5]	F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Political Econ., 81 (1973), 637-654. doi: 10.1086/260062.
[6]	N. P. B. Bollen, Valuing options in regime-switching models, J. Derivatives, 6 (1998), 38-49. doi: 10.3905/jod.1998.408011.
[7]	I. Bouchouev and V. Isakov, The inverse problem of option pricing, Inv. Probl., 13 (1997), L11–L17. doi: 10.1088/0266-5611/13/5/001.
[8]	I. Bouchouev and V. Isakov, Uniqueness, stability and numerical methods for the inverse problem that arises in financial markets, Inv. Probl., 15 (1999), R95–R116. doi: 10.1088/0266-5611/15/3/201.
[9]	I. Bouchouev, V. Isakov and N. Valdivia, Recovery of the volatilty coefficient by linearization, Quantitative Finance, 2 (2002), 257-263. doi: 10.1088/1469-7688/2/4/302.
[10]	J. Buffington and R. J. Elliott, American options with regime switching, Int. J. Theor. Appl. Fin., 5 (2002), 497-514. doi: 10.1142/S0219024902001523.
[11]	S. Chin and D. Dufresne, A general formula for option prices in a stochastic volatility model, Appl. Math. Fin., 19 (2012), 313-340. doi: 10.1080/1350486X.2011.624823.
[12]	S. Crépey, Calibration of the local volatility in a trinomial tree using Tikhonov regularization, Inv. Prob, 19 (2003), 91-127. doi: 10.1088/0266-5611/19/1/306.
[13]	S. Crépey, Calibration of the local volatility in a generalized Black-Scholes model using Tikhonov regularization, SIAM J. Math. Anal., 34 (2003), 1183-1206. doi: 10.1137/S0036141001400202.
[14]	M. Cristofol and L. Roques, Simultaneous determination of the drift and diffusion coefficients in stochastic differential equations, Inv. Probl., 33 (2017), 095006, 12 pp. doi: 10.1088/1361-6420/aa7a1c.
[15]	E. B. Davies, Linear Operators and Their Spectra, Cambridge Studies in Advanced Mathematics, 106. Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9780511618864.
[16]	A. De Cezaro, O. Scherzer and J. P. Zubelli, Convex regularization of local volatility models from option prices: Convergence analysis and rates, Nonlinear Analysis, 75 (2012), 2398-2415. doi: 10.1016/j.na.2011.10.037.
[17]	M. V. de Hoop, L. Y. Qiu and O. Scherzer, Local analysis of inverse problems: Hölder stability and iterative reconstruction, Inv. Prob., 28 (2012), 045001, 16 pp. doi: 10.1088/0266-5611/28/4/045001.
[18]	Z. C. Deng, J. N. Yu and L. Yang, An inverse problem of determining the implied volatility in option pricing, J. Math. Anal. Appl., 340 (2008), 16-31. doi: 10.1016/j.jmaa.2007.07.075.
[19]	B. Dupire, Pricing with a smile, Risk, 7 (1994), 18-20.
[20]	H. Egger and H. W. Engl, Tikhonov regularization applied to the inverse problem of option pricing: Convergence analysis and rates, Inv. Probl., 21 (2005), 1027-1045. doi: 10.1088/0266-5611/21/3/014.
[21]	S. D. , Parabolic Systems, North-Holland Publishing Co., Amsterdam-London, Wolters-Noordhoff Publishing, Groningen, 1969.
[22]	S. D. Eidelman, S. D. Ivasyshen and A. N. Kochubei, Analytic Methods in the Theory of Differential and Pseudo-Differential Equations of Parabolic Type, Operator Theory: Advances and Applications, vol. 152. Birkhäuser Verlag, Basel, 2004. doi: 10.1007/978-3-0348-7844-9.
[23]	R. J. Elliott, L. Aggoun and J. B. Moore, Hidden Markov Models: Estimation and Control, Applications of Mathematics, 29. Springer-Verlag, New York, 1995.
[24]	R. J. Elliott, L. Chan and T. K. Siu, Option pricing and Esscher transform under regime switching, Ann. Fin., 1 (2005), 423-432. doi: 10.1007/s10436-005-0013-z.
[25]	R. J. Elliott, L. Chan and T. K. Siu, Option valuation under a regime-switching constant elasticity of variance process, Appl. Math. Comp., 219 (2013), 4434-4443. doi: 10.1016/j.amc.2012.10.047.
[26]	R. J. Elliott, T. K. Siu and L. Chan, On pricing barrier options with regime switching, J. Comp. Appl. Math., 256 (2014), 196-210. doi: 10.1016/j.cam.2013.07.034.
[27]	L. C. Evans, Partial Differential Equations, Graduate studies in Mathematics, 19. Amer. Math. Soc., Providence, RI, 1998.
[28]	A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N. J. 1964.
[29]	C.-D. Fuh, K. W. R. Ho, I. Hu and R.-H. Wang, Option pricing with Markov switching, J. Data Science, 10 (2012), 483-509.
[30]	A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, 34. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.
[31]	O. Y. Imanuvilov and M. Yamamoto, Lipshitz stability in inverse parabolic problems by Carleman estimate, Inv. Prob., 14 (1998), 1229-1245. doi: 10.1088/0266-5611/14/5/009.
[32]	V. Isakov, Recovery of time dependent volatility coefficient by linearization, Evolution Equations and Control Theory, 3 (2014), 119-134. doi: 10.3934/eect.2014.3.119.
[33]	G. Kresin and V. Maz'ya, Maximum Principles and Sharp Constants for Solutions of Elliptic and Parabolic Systems, Math. Surveys and Monographs vol. 183. Amer. Math. Soc., providence, RI, 2012. doi: 10.1090/surv/183.
[34]	N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, Graduate Studies in Mathematics, 96. Amer. Math. Soc., Providence, RI, 2008. doi: 10.1090/gsm/096.
[35]	J. Le Rousseau and G. Lebeau, On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations, ESAIM: Control, Optimisation and Calculus of Variations, 18 (2012), 712-747. doi: 10.1051/cocv/2011168.
[36]	L. S. Jiang and Y. S. Tao, Identifying the volatility of underlying assets from option prices, Inv. Probl., 17 (2001), 137-155. doi: 10.1088/0266-5611/17/1/311.
[37]	K. Otsuka, On the positivity of the fundamental solutions for parabolic systems, J. Math. Kyoto Univ., 28 (1988), 119-132. doi: 10.1215/kjm/1250520562.
[38]	M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Corrected reprint of the 1967 original, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5.
[39]	P. Stefanov and G. Uhlmann, Boundary rigidity and stability for generic simple metric, J. Amer. Math. Soc., 18 (2005), 975-1003. doi: 10.1090/S0894-0347-05-00494-7.
[40]	X. J. Xi, M. R. Rodrigo and R. S. Mamon, Parameter estimation of a regime-switching model using an inverse Stieltjes moment approach, Stochastic Processes, Finance and Control: A Festschrift in Honor of Robert J. Elliott, World scientific, 1 (2012), 549–569. doi: 10.1142/9789814383318_0022.
[41]	S.-P. Zhu, A. Badran and X. P. Lu, A new exact solution for pricing European options in a two-state regime-switching economy, Computers and Mathematics with Applications, 64 (2012), 2744-2755. doi: 10.1016/j.camwa.2012.08.005.

[1]	Wensheng Yin, Jinde Cao, Yong Ren. Inverse optimal control of regime-switching jump diffusions. Mathematical Control and Related Fields, 2021 doi: 10.3934/mcrf.2021034
[2]	Wenhui Shi. An epiperimetric inequality approach to the parabolic Signorini problem. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1813-1846. doi: 10.3934/dcds.2020095
[3]	Yinghui Dong, Kam Chuen Yuen, Guojing Wang. Pricing credit derivatives under a correlated regime-switching hazard processes model. Journal of Industrial and Management Optimization, 2017, 13 (3) : 1395-1415. doi: 10.3934/jimo.2016079
[4]	Chao Xu, Yinghui Dong, Zhaolu Tian, Guojing Wang. Pricing dynamic fund protection under a Regime-switching Jump-diffusion model with stochastic protection level. Journal of Industrial and Management Optimization, 2020, 16 (6) : 2603-2623. doi: 10.3934/jimo.2019072
[5]	Jiaqin Wei, Zhuo Jin, Hailiang Yang. Optimal dividend policy with liability constraint under a hidden Markov regime-switching model. Journal of Industrial and Management Optimization, 2019, 15 (4) : 1965-1993. doi: 10.3934/jimo.2018132
[6]	Meiqiao Ai, Zhimin Zhang, Wenguang Yu. Valuing equity-linked death benefits with a threshold expense structure under a regime-switching Lévy model. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022007
[7]	S. J. Li, Z. M. Fang. On the stability of a dual weak vector variational inequality problem. Journal of Industrial and Management Optimization, 2008, 4 (1) : 155-165. doi: 10.3934/jimo.2008.4.155
[8]	Walter Allegretto, Yanping Lin, Shuqing Ma. On the box method for a non-local parabolic variational inequality. Discrete and Continuous Dynamical Systems - B, 2001, 1 (1) : 71-88. doi: 10.3934/dcdsb.2001.1.71
[9]	Junfeng Yang. Dynamic power price problem: An inverse variational inequality approach. Journal of Industrial and Management Optimization, 2008, 4 (4) : 673-684. doi: 10.3934/jimo.2008.4.673
[10]	Fuke Wu, George Yin, Zhuo Jin. Kolmogorov-type systems with regime-switching jump diffusion perturbations. Discrete and Continuous Dynamical Systems - B, 2016, 21 (7) : 2293-2319. doi: 10.3934/dcdsb.2016048
[11]	Christoforidou Amalia, Christian-Oliver Ewald. A lattice method for option evaluation with regime-switching asset correlation structure. Journal of Industrial and Management Optimization, 2021, 17 (4) : 1729-1752. doi: 10.3934/jimo.2020042
[12]	Jiaqin Wei. Time-inconsistent optimal control problems with regime-switching. Mathematical Control and Related Fields, 2017, 7 (4) : 585-622. doi: 10.3934/mcrf.2017022
[13]	Zhuo Jin, Linyi Qian. Lookback option pricing for regime-switching jump diffusion models. Mathematical Control and Related Fields, 2015, 5 (2) : 237-258. doi: 10.3934/mcrf.2015.5.237
[14]	Engel John C Dela Vega, Robert J Elliott. Conditional coherent risk measures and regime-switching conic pricing. Probability, Uncertainty and Quantitative Risk, 2021, 6 (4) : 267-300. doi: 10.3934/puqr.2021014
[15]	Jun Li, Fubao Xi. Exponential ergodicity for regime-switching diffusion processes in total variation norm. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2021309
[16]	Tak Kuen Siu, Yang Shen. Risk-minimizing pricing and Esscher transform in a general non-Markovian regime-switching jump-diffusion model. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2595-2626. doi: 10.3934/dcdsb.2017100
[17]	T. A. Shaposhnikova, M. N. Zubova. Homogenization problem for a parabolic variational inequality with constraints on subsets situated on the boundary of the domain. Networks and Heterogeneous Media, 2008, 3 (3) : 675-689. doi: 10.3934/nhm.2008.3.675
[18]	Yarui Duan, Pengcheng Wu, Yuying Zhou. Penalty approximation method for a double obstacle quasilinear parabolic variational inequality problem. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022017
[19]	Ian Knowles, Ajay Mahato. The inverse volatility problem for American options. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3473-3489. doi: 10.3934/dcdss.2020235
[20]	Pedro Caro. On an inverse problem in electromagnetism with local data: stability and uniqueness. Inverse Problems and Imaging, 2011, 5 (2) : 297-322. doi: 10.3934/ipi.2011.5.297

American Institute of Mathematical Sciences

Stable reconstruction of the volatility in a regime-switching local-volatility model

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Stable reconstruction of the volatility in a regime-switching local-volatility model

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors