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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

Lishang Jiang 1, and  Baojun Bian 2,

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.
Keywords: Implied local volatility., Inverse problem, Variational inequality, Parabolic equation.
Mathematics Subject Classification: 35K85, 35R30, 49J20, 49J40, 49K20, 49K4.
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 & 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.  Google Scholar

[2]

Y. Achdou and O. Pironneau, Volatility smile by multilevel least square, Int. J. Theor. Appl. Finance, 5 (2002), 619-643.  Google Scholar

[3]

Y. Achdou, G. Indragoby and O. Pironneau, Volatility calibration with American options, Methods and Applications of Analysis, 11 (2004), 533-556.  Google Scholar

[4]

J. Andreasen, Implied modelling: Stable implementation. Hedging and duality, working paper, The Aarhus School of Business, 1996. Google Scholar

[5]

M. Avellaneda, C. Friedman, R. Holmes and D. Samperi, Calibrating volatility surfaces via entropy, Applied Math. Finance, 4 (1997), 37-64. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

H. Berestycki, J. Busca and I. Florent, Asymptotics and calibration of local volatility models, Quantitative Finance, 2 (2002), 61-69.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. N. Bodurtha and M. Jermakyan, Non-parametric estimation of an implied volatility surface, Jour. of Computational Finance, 2 (1999), 29-60. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

J. R. Cannon, "The One-Dimensional Heat Equations," Encyclopedia of Mathematics and its Applications, Vol. 23, Addison-Wesley Publishing Company, 1984.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

E. Derman and I. Kani, Riding on a smile, Risk, 7 (1994), 32-39. Google Scholar

[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.  Google Scholar

[21]

B. Dupire, Pricing and hedging with smile, "Mathematics of Derivative Securities" (Cambridge, 1995), Publ. Newton Inst., 15, Cambridge Univ. Press, Cambridge, 1997.  Google Scholar

[22]

B. Dupire, Pricing with a smile, Risk, 7 (1994), 18-20. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

L. Jiang and B. Bian, An inverse problem for parabolic equations with non-divergent form, working paper, Tongji University, 2010. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

L. Jiang, "Mathematical Modelling and Methods of Financial Derivatives," High Education Press, Beijing, 2003. Google Scholar

[30]

D. Kinderleher and G. Stampacchia, "An Introduction to Variational Inequalities and Their Applications," Academic Press, 1980.  Google Scholar

[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. Google Scholar

[32]

J. Macbeth and L. Merville, An empirical estimation of Black-Scholes call option pricing model, Jour. of Finance, 34 (1979), 285-301. Google Scholar

[33]

S. Mayhew, Implied volatility, Financial Analysis J., 51 (1995), 8-13. Google Scholar

[34]

K. Shastri and K. Wethyavivorn, The valuation of currency options for alternate stochastic process, Jour. of Financial Research, 10 (1987), 283-293. Google Scholar

[35]

G. Skiadopoulos, Volatility smile consistent option models: A survey, International Journal of Theoretical and Applied Finance, 4 (2001), 403-437.  Google Scholar

[36]

P. Wilmott, "Derivatives-The Theory of Practice of Financial Engineering," John Wiley & Sons, New York, 1998. Google Scholar

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.  Google Scholar

[2]

Y. Achdou and O. Pironneau, Volatility smile by multilevel least square, Int. J. Theor. Appl. Finance, 5 (2002), 619-643.  Google Scholar

[3]

Y. Achdou, G. Indragoby and O. Pironneau, Volatility calibration with American options, Methods and Applications of Analysis, 11 (2004), 533-556.  Google Scholar

[4]

J. Andreasen, Implied modelling: Stable implementation. Hedging and duality, working paper, The Aarhus School of Business, 1996. Google Scholar

[5]

M. Avellaneda, C. Friedman, R. Holmes and D. Samperi, Calibrating volatility surfaces via entropy, Applied Math. Finance, 4 (1997), 37-64. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

H. Berestycki, J. Busca and I. Florent, Asymptotics and calibration of local volatility models, Quantitative Finance, 2 (2002), 61-69.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. N. Bodurtha and M. Jermakyan, Non-parametric estimation of an implied volatility surface, Jour. of Computational Finance, 2 (1999), 29-60. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

J. R. Cannon, "The One-Dimensional Heat Equations," Encyclopedia of Mathematics and its Applications, Vol. 23, Addison-Wesley Publishing Company, 1984.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

E. Derman and I. Kani, Riding on a smile, Risk, 7 (1994), 32-39. Google Scholar

[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.  Google Scholar

[21]

B. Dupire, Pricing and hedging with smile, "Mathematics of Derivative Securities" (Cambridge, 1995), Publ. Newton Inst., 15, Cambridge Univ. Press, Cambridge, 1997.  Google Scholar

[22]

B. Dupire, Pricing with a smile, Risk, 7 (1994), 18-20. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

L. Jiang and B. Bian, An inverse problem for parabolic equations with non-divergent form, working paper, Tongji University, 2010. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

L. Jiang, "Mathematical Modelling and Methods of Financial Derivatives," High Education Press, Beijing, 2003. Google Scholar

[30]

D. Kinderleher and G. Stampacchia, "An Introduction to Variational Inequalities and Their Applications," Academic Press, 1980.  Google Scholar

[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. Google Scholar

[32]

J. Macbeth and L. Merville, An empirical estimation of Black-Scholes call option pricing model, Jour. of Finance, 34 (1979), 285-301. Google Scholar

[33]

S. Mayhew, Implied volatility, Financial Analysis J., 51 (1995), 8-13. Google Scholar

[34]

K. Shastri and K. Wethyavivorn, The valuation of currency options for alternate stochastic process, Jour. of Financial Research, 10 (1987), 283-293. Google Scholar

[35]

G. Skiadopoulos, Volatility smile consistent option models: A survey, International Journal of Theoretical and Applied Finance, 4 (2001), 403-437.  Google Scholar

[36]

P. Wilmott, "Derivatives-The Theory of Practice of Financial Engineering," John Wiley & Sons, New York, 1998. Google Scholar

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