# American Institute of Mathematical Sciences

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 and 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), 37-64. doi: 10.1080/135048697334827. [2] H. Berestycki, J. Busca and I. Florent, An inverse parabolic problem arising in finance, C. R. Math. Acad. Sci. Paris, 331 (2000), 965-969. doi: 10.1016/S0764-4442(00)01749-3. [3] H. Berestycki, J. Busca and I. Florent, Asymptotics and calibration of local volatility models, Quant. Finance, 2 (2002), 61-69. doi: 10.1088/1469-7688/2/1/305. [4] I. Bouchouev and V. Isakov, The inverse problem of option pricing, Inverse Problems, 13 (1997), L11-L17. doi: 10.1088/0266-5611/13/5/001. [5] I. Bouchouev and V. Isakov, Uniqueness, stability, and numerical methods for the inverse problem that arises in financial markets, Inverse Problems, 15 (1999), R95-R116. doi: 10.1088/0266-5611/15/3/201. [6] I. Bouchouev, V. Isakov and N. Valdivia, Recovery of volatility coefficient by linearization, Quant. Finance, 2 (2002), 257-263. doi: 10.1088/1469-7688/2/4/302. [7] J. N. Bodurtha and M. Jermakyan, Non-parametric estimation of an implied volatility surface, J. Comput. Finance, 2 (1999), 29-61. [8] F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Political Econ., 81 (1973), 637-654. doi: 10.1086/260062. [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), 16-31. doi: 10.1016/j.jmaa.2007.07.075. [10] B. Dupire, Pricing with a smile, RISK, 7 (1994), 18-20. [11] H. Egger and H. Engl, Tikhonov regularization applied to the inverse problem of option pricing, Inverse Problems, 21 (2003), 1127-1145. doi: 10.1088/0266-5611/21/3/014. [12] H. Egger, T. Hein and B. Hofmann, On decoupling of volatility smile in inverse option pricing, Inverse Problems, 22 (2006), 1247-1259. doi: 10.1088/0266-5611/22/4/008. [13] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964. [14] J. Hull, Options, Futures, and Other Derivatives Prentice-Hall, 1997. [15] V. Isakov, Inverse Source Problems, Providence, R.I., AMS, 1990. [16] V. Isakov, Inverse parabolic problems with the final overdetermination, Comm. Pure Appl., 44 (1991), 185-209. doi: 10.1002/cpa.3160440203. [17] V. Isakov, Inverse Problems for PDE, 2nd edition, Springer-Verlag, New York, 2006. [18] O.A. Ladyzenskaja, V.A. Solonnikov and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type, Academic Press, New York-London, 1969. [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), 13-25. [20] M. A. Lavrentiev and B. A. Shabat, Methods of Theory of Functions of One Complex Variable, Nauka, Moscow, 1965.

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##### References:
 [1] M. Avellaneda, C. Friedman, R. Holmes and L. Samperi, Calibrating volatility surfaces via relative entropy minimization, Appl. Math. Finance, 4 (1997), 37-64. doi: 10.1080/135048697334827. [2] H. Berestycki, J. Busca and I. Florent, An inverse parabolic problem arising in finance, C. R. Math. Acad. Sci. Paris, 331 (2000), 965-969. doi: 10.1016/S0764-4442(00)01749-3. [3] H. Berestycki, J. Busca and I. Florent, Asymptotics and calibration of local volatility models, Quant. Finance, 2 (2002), 61-69. doi: 10.1088/1469-7688/2/1/305. [4] I. Bouchouev and V. Isakov, The inverse problem of option pricing, Inverse Problems, 13 (1997), L11-L17. doi: 10.1088/0266-5611/13/5/001. [5] I. Bouchouev and V. Isakov, Uniqueness, stability, and numerical methods for the inverse problem that arises in financial markets, Inverse Problems, 15 (1999), R95-R116. doi: 10.1088/0266-5611/15/3/201. [6] I. Bouchouev, V. Isakov and N. Valdivia, Recovery of volatility coefficient by linearization, Quant. Finance, 2 (2002), 257-263. doi: 10.1088/1469-7688/2/4/302. [7] J. N. Bodurtha and M. Jermakyan, Non-parametric estimation of an implied volatility surface, J. Comput. Finance, 2 (1999), 29-61. [8] F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Political Econ., 81 (1973), 637-654. doi: 10.1086/260062. [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), 16-31. doi: 10.1016/j.jmaa.2007.07.075. [10] B. Dupire, Pricing with a smile, RISK, 7 (1994), 18-20. [11] H. Egger and H. Engl, Tikhonov regularization applied to the inverse problem of option pricing, Inverse Problems, 21 (2003), 1127-1145. doi: 10.1088/0266-5611/21/3/014. [12] H. Egger, T. Hein and B. Hofmann, On decoupling of volatility smile in inverse option pricing, Inverse Problems, 22 (2006), 1247-1259. doi: 10.1088/0266-5611/22/4/008. [13] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964. [14] J. Hull, Options, Futures, and Other Derivatives Prentice-Hall, 1997. [15] V. Isakov, Inverse Source Problems, Providence, R.I., AMS, 1990. [16] V. Isakov, Inverse parabolic problems with the final overdetermination, Comm. Pure Appl., 44 (1991), 185-209. doi: 10.1002/cpa.3160440203. [17] V. Isakov, Inverse Problems for PDE, 2nd edition, Springer-Verlag, New York, 2006. [18] O.A. Ladyzenskaja, V.A. Solonnikov and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type, Academic Press, New York-London, 1969. [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), 13-25. [20] M. A. Lavrentiev and B. A. Shabat, Methods of Theory of Functions of One Complex Variable, Nauka, Moscow, 1965.
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