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.

show all references

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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