American Institute of Mathematical Sciences

September  2012, 17(6): 2017-2046. doi: 10.3934/dcdsb.2012.17.2017

The regularized implied local volatility equations -A new model to recover the volatility of underlying asset from observed market option price

 1 Department of Mathematics, Tongji University, Shanghai 200092 2 Department of mathematics, Tongji University, Shanghai 200092

Received  July 2011 Revised  August 2011 Published  May 2012

In this paper, we propose a new continuous time model to recover the volatility of underlying asset from observed market European option price. The model is a couple of fully nonlinear parabolic partial differential equations (see (34), (36)). As an inverse problem, the model is deduced from a Tikhonov regularization framework. Based on our method, the recovering procedure is stable and accurate. It is justified not only in theoretical proofs, but also in the numerical experiments.
Citation: Lishang Jiang, Baojun Bian. The regularized implied local volatility equations -A new model to recover the volatility of underlying asset from observed market option price. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 2017-2046. doi: 10.3934/dcdsb.2012.17.2017
References:
 [1] Y. Achdou, An inverse problem for a parabolic variational inequality with an integro-differential operator, Siam J. Control Optim., 47 (2008), 733-767. doi: 10.1137/060660692. [2] Y. Achdou and O. Pironneau, Volatility smile by multilevel least square, Int. J. Theor. Appl. Finance, 5 (2002), 619-643. [3] Y. Achdou, G. Indragoby and O. Pironneau, Volatility calibration with American options, Methods and Applications of Analysis, 11 (2004), 533-556. [4] J. Andreasen, Implied modelling: Stable implementation. Hedging and duality, working paper, The Aarhus School of Business, 1996. [5] M. Avellaneda, C. Friedman, R. Holmes and D. Samperi, Calibrating volatility surfaces via entropy, Applied Math. Finance, 4 (1997), 37-64. [6] F. Abergel and R. Tachet, A nonlinear partial integro-differential equations from mathematical finance, Discrete and Continuous Dynamical Systems, 27 (2010), 907-917. doi: 10.3934/dcds.2010.27.907. [7] H. Berestycki, J. Busca and I. Florent, An inverse parabolic problem arising in finance, C. R. Acad. Sci. Paris Sér I Math., 331 (2000), 965-969. [8] H. Berestycki, J. Busca and I. Florent, Asymptotics and calibration of local volatility models, Quantitative Finance, 2 (2002), 61-69. [9] D. Betes, Testing option pricing models, in "Statistical Methods in Finance" (eds. G. S. Maddala and C. R. Rao), Handbook of Statistics, 14, Elsevier Science B.V., (1996), 567-611. [10] F. Black, Fact and fantasy in the use of options, Financial Analysis J., 31 (1975), 36-72. doi: 10.2469/faj.v31.n4.36. [11] J. N. Bodurtha and M. Jermakyan, Non-parametric estimation of an implied volatility surface, Jour. of Computational Finance, 2 (1999), 29-60. [12] I. Bouchouev and V. Isakov, The inverse problem of option pricing, Inverse Problem, 13 (1997), L11-L17. doi: 10.1088/0266-5611/13/5/001. [13] I. Bouchouev and V. Isakov, Uniqueness, Stability and numerical methods for inverse problem that arises in financial markets, Inverse Problem, 15 (1999), R95-R116. doi: 10.1088/0266-5611/15/3/201. [14] D. Breeden and R. Litzenberger, Prices of state-contingent claims implicit in option prices, Journal of Business, 51 (1978), 621-651. doi: 10.1086/296025. [15] J. R. Cannon, P. Duchatean and K. Steube, "Identifying a Time Dependent Unknown Coefficient in a Nonlinear Heat Equation," Nonlinear Diffusion Equations & their Equilibrium States, 3, Birkhauser, (1992), 153-169. [16] J. R. Cannon, "The One-Dimensional Heat Equations," Encyclopedia of Mathematics and its Applications, Vol. 23, Addison-Wesley Publishing Company, 1984. [17] S. Crépey, Calibration of local volatility in a trinomial tree using Tikhonov regularization, Inverse Problems, 19 (2003), 91-127. doi: 10.1088/0266-5611/19/1/306. [18] S. Crépey, Calibration of volatility in a generalized Black-Scholes model using Tikhonov regularization, SIAM J. Math. Anal., 34 (2003), 1183-1206. doi: 10.1137/S0036141001400202. [19] E. Derman and I. Kani, Riding on a smile, Risk, 7 (1994), 32-39. [20] E. Derman, I. Kani and J. Zou, The local volatility surface: Unlocking the information in index option prices, Financial Analysis J., 52 (1996), 25-36. doi: 10.2469/faj.v52.n4.2008. [21] B. Dupire, Pricing and hedging with smile, "Mathematics of Derivative Securities" (Cambridge, 1995), Publ. Newton Inst., 15, Cambridge Univ. Press, Cambridge, 1997. [22] B. Dupire, Pricing with a smile, Risk, 7 (1994), 18-20. [23] N. El Karoui, Measuring and hedging financial risks in dynamical world, in "Proceedings of ICM, Vol. III" (Beijing, 2002), Higher Ed. Press, Beijing, (2002), 773-783. [24] T. Hein and B. Hofmann, On the nature of ill-posedness of an inverse problem arising in option pricing, Inverse Problems, 19 (2003), 1319-1338. doi: 10.1088/0266-5611/19/6/006. [25] K.-H. Hoffmann, L. Jiang and M. Niezgódka, Optimal control of phase change processes with terminal state observation, J. Partial Diff. Eqns., 6 (1993), 97-107. [26] L. Jiang and B. Bian, An inverse problem for parabolic equations with non-divergent form, working paper, Tongji University, 2010. [27] L. Jiang and Y. Tao, Identifying the volatility of underlying assets from option prices, Inverse Problems, 17 (2001), 137-155. doi: 10.1088/0266-5611/17/1/311. [28] L. Jiang, Q. Chen, L. Wang and Jin E Zhang, A new well-posed algorithm to recover implied local volatility, Quantitative Finance, 3 (2003), 451-457. [29] L. Jiang, "Mathematical Modelling and Methods of Financial Derivatives," High Education Press, Beijing, 2003. [30] D. Kinderleher and G. Stampacchia, "An Introduction to Variational Inequalities and Their Applications," Academic Press, 1980. [31] R. Lagnado and S. Osher, A technique for calibrating derivative security pricing models: Numerical solution of an inverse problem, J. Computational Finance, 1 (1997), 13-25. [32] J. Macbeth and L. Merville, An empirical estimation of Black-Scholes call option pricing model, Jour. of Finance, 34 (1979), 285-301. [33] S. Mayhew, Implied volatility, Financial Analysis J., 51 (1995), 8-13. [34] K. Shastri and K. Wethyavivorn, The valuation of currency options for alternate stochastic process, Jour. of Financial Research, 10 (1987), 283-293. [35] G. Skiadopoulos, Volatility smile consistent option models: A survey, International Journal of Theoretical and Applied Finance, 4 (2001), 403-437. [36] P. Wilmott, "Derivatives-The Theory of Practice of Financial Engineering," John Wiley & Sons, New York, 1998.

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References:
 [1] Y. Achdou, An inverse problem for a parabolic variational inequality with an integro-differential operator, Siam J. Control Optim., 47 (2008), 733-767. doi: 10.1137/060660692. [2] Y. Achdou and O. Pironneau, Volatility smile by multilevel least square, Int. J. Theor. Appl. Finance, 5 (2002), 619-643. [3] Y. Achdou, G. Indragoby and O. Pironneau, Volatility calibration with American options, Methods and Applications of Analysis, 11 (2004), 533-556. [4] J. Andreasen, Implied modelling: Stable implementation. Hedging and duality, working paper, The Aarhus School of Business, 1996. [5] M. Avellaneda, C. Friedman, R. Holmes and D. Samperi, Calibrating volatility surfaces via entropy, Applied Math. Finance, 4 (1997), 37-64. [6] F. Abergel and R. Tachet, A nonlinear partial integro-differential equations from mathematical finance, Discrete and Continuous Dynamical Systems, 27 (2010), 907-917. doi: 10.3934/dcds.2010.27.907. [7] H. Berestycki, J. Busca and I. Florent, An inverse parabolic problem arising in finance, C. R. Acad. Sci. Paris Sér I Math., 331 (2000), 965-969. [8] H. Berestycki, J. Busca and I. Florent, Asymptotics and calibration of local volatility models, Quantitative Finance, 2 (2002), 61-69. [9] D. Betes, Testing option pricing models, in "Statistical Methods in Finance" (eds. G. S. Maddala and C. R. Rao), Handbook of Statistics, 14, Elsevier Science B.V., (1996), 567-611. [10] F. Black, Fact and fantasy in the use of options, Financial Analysis J., 31 (1975), 36-72. doi: 10.2469/faj.v31.n4.36. [11] J. N. Bodurtha and M. Jermakyan, Non-parametric estimation of an implied volatility surface, Jour. of Computational Finance, 2 (1999), 29-60. [12] I. Bouchouev and V. Isakov, The inverse problem of option pricing, Inverse Problem, 13 (1997), L11-L17. doi: 10.1088/0266-5611/13/5/001. [13] I. Bouchouev and V. Isakov, Uniqueness, Stability and numerical methods for inverse problem that arises in financial markets, Inverse Problem, 15 (1999), R95-R116. doi: 10.1088/0266-5611/15/3/201. [14] D. Breeden and R. Litzenberger, Prices of state-contingent claims implicit in option prices, Journal of Business, 51 (1978), 621-651. doi: 10.1086/296025. [15] J. R. Cannon, P. Duchatean and K. Steube, "Identifying a Time Dependent Unknown Coefficient in a Nonlinear Heat Equation," Nonlinear Diffusion Equations & their Equilibrium States, 3, Birkhauser, (1992), 153-169. [16] J. R. Cannon, "The One-Dimensional Heat Equations," Encyclopedia of Mathematics and its Applications, Vol. 23, Addison-Wesley Publishing Company, 1984. [17] S. Crépey, Calibration of local volatility in a trinomial tree using Tikhonov regularization, Inverse Problems, 19 (2003), 91-127. doi: 10.1088/0266-5611/19/1/306. [18] S. Crépey, Calibration of volatility in a generalized Black-Scholes model using Tikhonov regularization, SIAM J. Math. Anal., 34 (2003), 1183-1206. doi: 10.1137/S0036141001400202. [19] E. Derman and I. Kani, Riding on a smile, Risk, 7 (1994), 32-39. [20] E. Derman, I. Kani and J. Zou, The local volatility surface: Unlocking the information in index option prices, Financial Analysis J., 52 (1996), 25-36. doi: 10.2469/faj.v52.n4.2008. [21] B. Dupire, Pricing and hedging with smile, "Mathematics of Derivative Securities" (Cambridge, 1995), Publ. Newton Inst., 15, Cambridge Univ. Press, Cambridge, 1997. [22] B. Dupire, Pricing with a smile, Risk, 7 (1994), 18-20. [23] N. El Karoui, Measuring and hedging financial risks in dynamical world, in "Proceedings of ICM, Vol. III" (Beijing, 2002), Higher Ed. Press, Beijing, (2002), 773-783. [24] T. Hein and B. Hofmann, On the nature of ill-posedness of an inverse problem arising in option pricing, Inverse Problems, 19 (2003), 1319-1338. doi: 10.1088/0266-5611/19/6/006. [25] K.-H. Hoffmann, L. Jiang and M. Niezgódka, Optimal control of phase change processes with terminal state observation, J. Partial Diff. Eqns., 6 (1993), 97-107. [26] L. Jiang and B. Bian, An inverse problem for parabolic equations with non-divergent form, working paper, Tongji University, 2010. [27] L. Jiang and Y. Tao, Identifying the volatility of underlying assets from option prices, Inverse Problems, 17 (2001), 137-155. doi: 10.1088/0266-5611/17/1/311. [28] L. Jiang, Q. Chen, L. Wang and Jin E Zhang, A new well-posed algorithm to recover implied local volatility, Quantitative Finance, 3 (2003), 451-457. [29] L. Jiang, "Mathematical Modelling and Methods of Financial Derivatives," High Education Press, Beijing, 2003. [30] D. Kinderleher and G. Stampacchia, "An Introduction to Variational Inequalities and Their Applications," Academic Press, 1980. [31] R. Lagnado and S. Osher, A technique for calibrating derivative security pricing models: Numerical solution of an inverse problem, J. Computational Finance, 1 (1997), 13-25. [32] J. Macbeth and L. Merville, An empirical estimation of Black-Scholes call option pricing model, Jour. of Finance, 34 (1979), 285-301. [33] S. Mayhew, Implied volatility, Financial Analysis J., 51 (1995), 8-13. [34] K. Shastri and K. Wethyavivorn, The valuation of currency options for alternate stochastic process, Jour. of Financial Research, 10 (1987), 283-293. [35] G. Skiadopoulos, Volatility smile consistent option models: A survey, International Journal of Theoretical and Applied Finance, 4 (2001), 403-437. [36] P. Wilmott, "Derivatives-The Theory of Practice of Financial Engineering," John Wiley & Sons, New York, 1998.
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