March  2014, 3(1): 119-134. doi: 10.3934/eect.2014.3.119

Recovery of time dependent volatility coefficient by linearization

1. 

Department of Mathematics, Statistics, and Physics, Wichita State University, Wichita, KS 67260-0033, United States

Received  September 2013 Revised  December 2013 Published  February 2014

We study the problem of reconstruction of special time dependent local volatility from market prices of options with different strikes at two expiration times. For a general diffusion process we apply the linearization technique and we conclude that the option price can be obtained as the sum of the Black-Scholes formula and of an operator $W$ which is linear in perturbation of volatility. We further simplify the linearized inverse problem and obtain unique solvability result in basic functional spaces. By using the Laplace transform in time we simplify the kernels of integral operators for $W$ and we obtain uniqueness and stability results for volatility under natural condition of smallness of the spacial interval where one prescribes the (market) data. We propose a numerical algorithm based on our analysis of the linearized problem.
Citation: Victor Isakov. Recovery of time dependent volatility coefficient by linearization. Evolution Equations & Control Theory, 2014, 3 (1) : 119-134. doi: 10.3934/eect.2014.3.119
References:
[1]

M. Avellaneda, C. Friedman, R. Holmes and L. Samperi, Calibrating volatility surfaces via relative entropy minimization,, Appl. Math. Finance, 4 (1997), 37.  doi: 10.1080/135048697334827.  Google Scholar

[2]

H. Berestycki, J. Busca and I. Florent, An inverse parabolic problem arising in finance,, C. R. Math. Acad. Sci. Paris, 331 (2000), 965.  doi: 10.1016/S0764-4442(00)01749-3.  Google Scholar

[3]

H. Berestycki, J. Busca and I. Florent, Asymptotics and calibration of local volatility models,, Quant. Finance, 2 (2002), 61.  doi: 10.1088/1469-7688/2/1/305.  Google Scholar

[4]

I. Bouchouev and V. Isakov, The inverse problem of option pricing,, Inverse Problems, 13 (1997).  doi: 10.1088/0266-5611/13/5/001.  Google Scholar

[5]

I. Bouchouev and V. Isakov, Uniqueness, stability, and numerical methods for the inverse problem that arises in financial markets,, Inverse Problems, 15 (1999).  doi: 10.1088/0266-5611/15/3/201.  Google Scholar

[6]

I. Bouchouev, V. Isakov and N. Valdivia, Recovery of volatility coefficient by linearization,, Quant. Finance, 2 (2002), 257.  doi: 10.1088/1469-7688/2/4/302.  Google Scholar

[7]

J. N. Bodurtha and M. Jermakyan, Non-parametric estimation of an implied volatility surface,, J. Comput. Finance, 2 (1999), 29.   Google Scholar

[8]

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

[9]

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.  doi: 10.1016/j.jmaa.2007.07.075.  Google Scholar

[10]

B. Dupire, Pricing with a smile,, RISK, 7 (1994), 18.   Google Scholar

[11]

H. Egger and H. Engl, Tikhonov regularization applied to the inverse problem of option pricing,, Inverse Problems, 21 (2003), 1127.  doi: 10.1088/0266-5611/21/3/014.  Google Scholar

[12]

H. Egger, T. Hein and B. Hofmann, On decoupling of volatility smile in inverse option pricing,, Inverse Problems, 22 (2006), 1247.  doi: 10.1088/0266-5611/22/4/008.  Google Scholar

[13]

A. Friedman, Partial Differential Equations of Parabolic Type,, Prentice-Hall, (1964).   Google Scholar

[14]

J. Hull, Options, Futures, and Other Derivatives, Prentice-Hall, (1997).   Google Scholar

[15]

V. Isakov, Inverse Source Problems,, Providence, (1990).   Google Scholar

[16]

V. Isakov, Inverse parabolic problems with the final overdetermination,, Comm. Pure Appl., 44 (1991), 185.  doi: 10.1002/cpa.3160440203.  Google Scholar

[17]

V. Isakov, Inverse Problems for PDE,, 2nd edition, (2006).   Google Scholar

[18]

O.A. Ladyzenskaja, V.A. Solonnikov and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type,, Academic Press, (1969).   Google Scholar

[19]

R. Lagnado and S. Osher, A technique for calibrating derivation of the security pricing models: Numerical solution of the inverse problem,, J. Comput. Finance, 1 (1997), 13.   Google Scholar

[20]

M. A. Lavrentiev and B. A. Shabat, Methods of Theory of Functions of One Complex Variable,, Nauka, (1965).   Google Scholar

show all references

References:
[1]

M. Avellaneda, C. Friedman, R. Holmes and L. Samperi, Calibrating volatility surfaces via relative entropy minimization,, Appl. Math. Finance, 4 (1997), 37.  doi: 10.1080/135048697334827.  Google Scholar

[2]

H. Berestycki, J. Busca and I. Florent, An inverse parabolic problem arising in finance,, C. R. Math. Acad. Sci. Paris, 331 (2000), 965.  doi: 10.1016/S0764-4442(00)01749-3.  Google Scholar

[3]

H. Berestycki, J. Busca and I. Florent, Asymptotics and calibration of local volatility models,, Quant. Finance, 2 (2002), 61.  doi: 10.1088/1469-7688/2/1/305.  Google Scholar

[4]

I. Bouchouev and V. Isakov, The inverse problem of option pricing,, Inverse Problems, 13 (1997).  doi: 10.1088/0266-5611/13/5/001.  Google Scholar

[5]

I. Bouchouev and V. Isakov, Uniqueness, stability, and numerical methods for the inverse problem that arises in financial markets,, Inverse Problems, 15 (1999).  doi: 10.1088/0266-5611/15/3/201.  Google Scholar

[6]

I. Bouchouev, V. Isakov and N. Valdivia, Recovery of volatility coefficient by linearization,, Quant. Finance, 2 (2002), 257.  doi: 10.1088/1469-7688/2/4/302.  Google Scholar

[7]

J. N. Bodurtha and M. Jermakyan, Non-parametric estimation of an implied volatility surface,, J. Comput. Finance, 2 (1999), 29.   Google Scholar

[8]

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

[9]

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.  doi: 10.1016/j.jmaa.2007.07.075.  Google Scholar

[10]

B. Dupire, Pricing with a smile,, RISK, 7 (1994), 18.   Google Scholar

[11]

H. Egger and H. Engl, Tikhonov regularization applied to the inverse problem of option pricing,, Inverse Problems, 21 (2003), 1127.  doi: 10.1088/0266-5611/21/3/014.  Google Scholar

[12]

H. Egger, T. Hein and B. Hofmann, On decoupling of volatility smile in inverse option pricing,, Inverse Problems, 22 (2006), 1247.  doi: 10.1088/0266-5611/22/4/008.  Google Scholar

[13]

A. Friedman, Partial Differential Equations of Parabolic Type,, Prentice-Hall, (1964).   Google Scholar

[14]

J. Hull, Options, Futures, and Other Derivatives, Prentice-Hall, (1997).   Google Scholar

[15]

V. Isakov, Inverse Source Problems,, Providence, (1990).   Google Scholar

[16]

V. Isakov, Inverse parabolic problems with the final overdetermination,, Comm. Pure Appl., 44 (1991), 185.  doi: 10.1002/cpa.3160440203.  Google Scholar

[17]

V. Isakov, Inverse Problems for PDE,, 2nd edition, (2006).   Google Scholar

[18]

O.A. Ladyzenskaja, V.A. Solonnikov and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type,, Academic Press, (1969).   Google Scholar

[19]

R. Lagnado and S. Osher, A technique for calibrating derivation of the security pricing models: Numerical solution of the inverse problem,, J. Comput. Finance, 1 (1997), 13.   Google Scholar

[20]

M. A. Lavrentiev and B. A. Shabat, Methods of Theory of Functions of One Complex Variable,, Nauka, (1965).   Google Scholar

[1]

Fioralba Cakoni, Rainer Kress. Integral equations for inverse problems in corrosion detection from partial Cauchy data. Inverse Problems & Imaging, 2007, 1 (2) : 229-245. doi: 10.3934/ipi.2007.1.229

[2]

Jaan Janno, Kairi Kasemets. A positivity principle for parabolic integro-differential equations and inverse problems with final overdetermination. Inverse Problems & Imaging, 2009, 3 (1) : 17-41. doi: 10.3934/ipi.2009.3.17

[3]

Davide Guidetti. Convergence to a stationary state of solutions to inverse problems of parabolic type. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 711-722. doi: 10.3934/dcdss.2013.6.711

[4]

Davide Guidetti. Some inverse problems of identification for integrodifferential parabolic systems with a boundary memory term. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 749-756. doi: 10.3934/dcdss.2015.8.749

[5]

Colin Guillarmou, Antônio Sá Barreto. Inverse problems for Einstein manifolds. Inverse Problems & Imaging, 2009, 3 (1) : 1-15. doi: 10.3934/ipi.2009.3.1

[6]

Sergei Avdonin, Pavel Kurasov. Inverse problems for quantum trees. Inverse Problems & Imaging, 2008, 2 (1) : 1-21. doi: 10.3934/ipi.2008.2.1

[7]

Maciej Zworski. A remark on inverse problems for resonances. Inverse Problems & Imaging, 2007, 1 (1) : 225-227. doi: 10.3934/ipi.2007.1.225

[8]

Guanghui Hu, Peijun Li, Xiaodong Liu, Yue Zhao. Inverse source problems in electrodynamics. Inverse Problems & Imaging, 2018, 12 (6) : 1411-1428. doi: 10.3934/ipi.2018059

[9]

Michael Herty, Giuseppe Visconti. Kinetic methods for inverse problems. Kinetic & Related Models, 2019, 12 (5) : 1109-1130. doi: 10.3934/krm.2019042

[10]

Mehdi Badra, Fabien Caubet, Jérémi Dardé. Stability estimates for Navier-Stokes equations and application to inverse problems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2379-2407. doi: 10.3934/dcdsb.2016052

[11]

Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Inverse problems for evolution equations with time dependent operator-coefficients. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 737-744. doi: 10.3934/dcdss.2016025

[12]

Janne M.J. Huttunen, J. P. Kaipio. Approximation errors in nonstationary inverse problems. Inverse Problems & Imaging, 2007, 1 (1) : 77-93. doi: 10.3934/ipi.2007.1.77

[13]

Masoumeh Dashti, Stephen Harris, Andrew Stuart. Besov priors for Bayesian inverse problems. Inverse Problems & Imaging, 2012, 6 (2) : 183-200. doi: 10.3934/ipi.2012.6.183

[14]

Xiaosheng Li, Gunther Uhlmann. Inverse problems with partial data in a slab. Inverse Problems & Imaging, 2010, 4 (3) : 449-462. doi: 10.3934/ipi.2010.4.449

[15]

Sergei A. Avdonin, Sergei A. Ivanov, Jun-Min Wang. Inverse problems for the heat equation with memory. Inverse Problems & Imaging, 2019, 13 (1) : 31-38. doi: 10.3934/ipi.2019002

[16]

Mohsen Tadi. A computational method for an inverse problem in a parabolic system. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 205-218. doi: 10.3934/dcdsb.2009.12.205

[17]

Sergiy Zhuk. Inverse problems for linear ill-posed differential-algebraic equations with uncertain parameters. Conference Publications, 2011, 2011 (Special) : 1467-1476. doi: 10.3934/proc.2011.2011.1467

[18]

Mohammed Al Horani, Angelo Favini. Inverse problems for singular differential-operator equations with higher order polar singularities. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2159-2168. doi: 10.3934/dcdsb.2014.19.2159

[19]

Johannes Elschner, Guanghui Hu. Uniqueness in inverse transmission scattering problems for multilayered obstacles. Inverse Problems & Imaging, 2011, 5 (4) : 793-813. doi: 10.3934/ipi.2011.5.793

[20]

François Monard, Guillaume Bal. Inverse diffusion problems with redundant internal information. Inverse Problems & Imaging, 2012, 6 (2) : 289-313. doi: 10.3934/ipi.2012.6.289

2018 Impact Factor: 1.048

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]