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A direct method based on the Clenshaw-Curtis formula for fractional optimal control problems
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 |
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.
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. |
[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. |
[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. |
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