doi: 10.3934/mcrf.2019036

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

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  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.

Citation: Mourad Bellassoued, Raymond Brummelhuis, Michel Cristofol, Éric Soccorsi. Stable reconstruction of the volatility in a regime-switching local-volatility model. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2019036
References:
[1]

D. D. AingworthS. 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.  Google Scholar

[2]

V. AlbaniA. 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.  Google Scholar

[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.  Google Scholar

[4]

D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa, 22 (1968), 607-694.   Google Scholar

[5]

F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Political Econ., 81 (1973), 637-654.  doi: 10.1086/260062.  Google Scholar

[6]

N. P. B. Bollen, Valuing options in regime-switching models, J. Derivatives, 6 (1998), 38-49.  doi: 10.3905/jod.1998.408011.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

I. BouchouevV. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15] E. B. Davies, Linear Operators and Their Spectra, Cambridge Studies in Advanced Mathematics, 106. Cambridge University Press, Cambridge, 2007.  doi: 10.1017/CBO9780511618864.  Google Scholar
[16]

A. De CezaroO. 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.  Google Scholar

[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.  Google Scholar

[18]

Z. C. DengJ. 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.  Google Scholar

[19]

B. Dupire, Pricing with a smile, Risk, 7 (1994), 18-20.   Google Scholar

[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.  Google Scholar

[21]

S. D. , Parabolic Systems, North-Holland Publishing Co., Amsterdam-London, Wolters-Noordhoff Publishing, Groningen, 1969.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

R. J. ElliottL. 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.  Google Scholar

[25]

R. J. ElliottL. 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.  Google Scholar

[26]

R. J. ElliottT. 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.  Google Scholar

[27]

L. C. Evans, Partial Differential Equations, Graduate studies in Mathematics, 19. Amer. Math. Soc., Providence, RI, 1998.  Google Scholar

[28]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N. J. 1964.  Google Scholar

[29]

C.-D. FuhK. W. R. HoI. Hu and R.-H. Wang, Option pricing with Markov switching, J. Data Science, 10 (2012), 483-509.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

S.-P. ZhuA. 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.  Google Scholar

show all references

References:
[1]

D. D. AingworthS. 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.  Google Scholar

[2]

V. AlbaniA. 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.  Google Scholar

[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.  Google Scholar

[4]

D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa, 22 (1968), 607-694.   Google Scholar

[5]

F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Political Econ., 81 (1973), 637-654.  doi: 10.1086/260062.  Google Scholar

[6]

N. P. B. Bollen, Valuing options in regime-switching models, J. Derivatives, 6 (1998), 38-49.  doi: 10.3905/jod.1998.408011.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

I. BouchouevV. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15] E. B. Davies, Linear Operators and Their Spectra, Cambridge Studies in Advanced Mathematics, 106. Cambridge University Press, Cambridge, 2007.  doi: 10.1017/CBO9780511618864.  Google Scholar
[16]

A. De CezaroO. 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.  Google Scholar

[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.  Google Scholar

[18]

Z. C. DengJ. 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.  Google Scholar

[19]

B. Dupire, Pricing with a smile, Risk, 7 (1994), 18-20.   Google Scholar

[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.  Google Scholar

[21]

S. D. , Parabolic Systems, North-Holland Publishing Co., Amsterdam-London, Wolters-Noordhoff Publishing, Groningen, 1969.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

R. J. ElliottL. 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.  Google Scholar

[25]

R. J. ElliottL. 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.  Google Scholar

[26]

R. J. ElliottT. 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.  Google Scholar

[27]

L. C. Evans, Partial Differential Equations, Graduate studies in Mathematics, 19. Amer. Math. Soc., Providence, RI, 1998.  Google Scholar

[28]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N. J. 1964.  Google Scholar

[29]

C.-D. FuhK. W. R. HoI. Hu and R.-H. Wang, Option pricing with Markov switching, J. Data Science, 10 (2012), 483-509.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

S.-P. ZhuA. 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.  Google Scholar

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