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A fully non-linear PDE problem from pricing CDS with counterparty risk
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 |
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] | |
[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] | |
[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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