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A family of extragradient methods for solving equilibrium problems
A penalty-based method from reconstructing smooth local volatility surface from American options
1. | China Center for Special Economic Zone Research, Shenzhen University, 3688 Nanhai Ave., Shenzhen, 518060, China |
2. | Department of Mathematics and Statistics, Curtin University, Perth, Western Australia, WA 6845, Australia |
References:
[1] |
Y. Achdou and O. Pironneau, Computational Methods For Option Pricing, Vol. 30. SIAM, 2005.
doi: 10.1137/1.9780898717495. |
[2] |
A. Anderson and J. Andresen, Jump diffusion process: Volatility smile fitting and numerical methods for option pricing, Review of Derivatives Research, 4 (2000), 231-262. |
[3] |
M. Avellaneda, A. Levy and A. Paras, Pricing and hedging derivative securities in markets with uncertain volatilities, Applied Mathematical Finance, 2 (1995), 73-88.
doi: 10.1080/13504869500000005. |
[4] |
F. E. Benth, K. H.Karlsen and K. Reikvam, A semilinear Black and Scholes partial differential equation for valuing American options, Finance and Stochastics, 7 (2003), 277-298.
doi: 10.1007/s007800200091. |
[5] |
F. Black and M. Scholes, The pricing of options and corporate liabilities, The Journal of Political Economy, 81 (1973), 637-654.
doi: 10.1086/260062. |
[6] |
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. |
[7] |
T. F. Coleman, Y. Li and A. Verma, Reconstructing the unknown local volatility function, Journal of Computational Finance, 2 (1999), 77-100. |
[8] |
S. Crepey, Calibration of the local volatility in a generalized Black-Scholes model using Tikhonov regularization, SIAM Journal of Mathematical Analysis, 34 (2003), 1183-1206.
doi: 10.1137/S0036141001400202. |
[9] |
J. Huang and J. S. Pang, A mathematical programming with equilibrium constraints approach to the implied volatility surface of American options, Journal of Computational Finance, 4 (2000), 21-56. |
[10] |
J. Hull, Options, Futures, and Other Derivatives, Prentice-Hall, Englewood Cliffs, 2005. |
[11] |
N. Jackson, E. Suli and S. Howison, Computation of deterministic volatility surfaces, Journal of Computational Finance, 2 (1999), 5-32. |
[12] |
L. Jiang, Q. Chen, L. Wang and J. Zhang, A new well-posed algorithm to recover implied local volatility, Quantitative Finance, 3 (2003), 451-457.
doi: 10.1088/1469-7688/3/6/304. |
[13] |
Y. K. Kwok, Mathematical Models of Financial Derivatives, Springer, Berlin, 2008. |
[14] |
R. Lagnado and S. Osher, A technique for calibrating derivative security pricing models: Numerical solution of the inverse problem, Journal of Computational Finance, 1 (1997), 13-25. |
[15] |
R. Lagnado and S. Osher, Reconciling differences, Risk, 10 (1997), 79-83. |
[16] |
W. Li and S. Wang, Penalty approach to the HJB equation arising in European stock option pricing with proportional transaction costs, Journal of Optimization Theory and Applications, 143 (2009), 279-293.
doi: 10.1007/s10957-009-9559-7. |
[17] |
R. C. Merton, Option pricing when underlying stock return are discontinuous, Journal of financial economics, 3 (1976), 125-144.
doi: 10.1016/0304-405X(76)90022-2. |
[18] |
B. F. Nielsen, O. Skavhaug and A. Tveito, Penalty and front-fixing methods for the numerical solution of American option problems, Journal of Computational Finance, 5 (2002), 69-97. |
[19] |
J. Nocedal and S. Wright, Numerical Optimization Series: Springer Series in Operations Research and Financial Engineering 2nd ed, Springer, Berlin, 2006. |
[20] |
S. Stojanovic, Implied volatility for American options via optimal control and fast numerical solutions of obstacle problems, Differential Equations and Control Theory, 225 (2002), 277-294 . |
[21] |
G. Wahba, Splines Models for Observational Data. Series in Applied Mathematics, Vol. 59, SIAM, Philadelphia, 1990.
doi: 10.1137/1.9781611970128. |
[22] |
S. Wang, X. Q. Yang and K. L. Teo, Power penalty method for a linear complementarity problem arising from American option valuation, Journal of Optimization Theory and Applications, 129 (2006), 227-254.
doi: 10.1007/s10957-006-9062-3. |
[23] |
P. Wilmott, Paul Wimott on Quantitave Finance, Wiley, New York, 2000. |
[24] |
K. Zhang and S. Wang, A computational scheme for uncertain volatility model in option pricing, Applied Numerical Mathematics, 59 (2009), 1754-1767.
doi: 10.1016/j.apnum.2009.01.004. |
[25] |
K. Zhang and S. Wang, Interior penalty approach to american option pricing, Journal of Industrial and Management Optimization, 7 (2011), 435-447.
doi: 10.3934/jimo.2011.7.435. |
[26] |
K. Zhang, K. L. Teo and M. Swartz, A robust numerical scheme for pricing American options under regime switching based on penalty method, Computational Economics, 43 (2014), 463-483.
doi: 10.1007/s10614-013-9361-3. |
show all references
References:
[1] |
Y. Achdou and O. Pironneau, Computational Methods For Option Pricing, Vol. 30. SIAM, 2005.
doi: 10.1137/1.9780898717495. |
[2] |
A. Anderson and J. Andresen, Jump diffusion process: Volatility smile fitting and numerical methods for option pricing, Review of Derivatives Research, 4 (2000), 231-262. |
[3] |
M. Avellaneda, A. Levy and A. Paras, Pricing and hedging derivative securities in markets with uncertain volatilities, Applied Mathematical Finance, 2 (1995), 73-88.
doi: 10.1080/13504869500000005. |
[4] |
F. E. Benth, K. H.Karlsen and K. Reikvam, A semilinear Black and Scholes partial differential equation for valuing American options, Finance and Stochastics, 7 (2003), 277-298.
doi: 10.1007/s007800200091. |
[5] |
F. Black and M. Scholes, The pricing of options and corporate liabilities, The Journal of Political Economy, 81 (1973), 637-654.
doi: 10.1086/260062. |
[6] |
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. |
[7] |
T. F. Coleman, Y. Li and A. Verma, Reconstructing the unknown local volatility function, Journal of Computational Finance, 2 (1999), 77-100. |
[8] |
S. Crepey, Calibration of the local volatility in a generalized Black-Scholes model using Tikhonov regularization, SIAM Journal of Mathematical Analysis, 34 (2003), 1183-1206.
doi: 10.1137/S0036141001400202. |
[9] |
J. Huang and J. S. Pang, A mathematical programming with equilibrium constraints approach to the implied volatility surface of American options, Journal of Computational Finance, 4 (2000), 21-56. |
[10] |
J. Hull, Options, Futures, and Other Derivatives, Prentice-Hall, Englewood Cliffs, 2005. |
[11] |
N. Jackson, E. Suli and S. Howison, Computation of deterministic volatility surfaces, Journal of Computational Finance, 2 (1999), 5-32. |
[12] |
L. Jiang, Q. Chen, L. Wang and J. Zhang, A new well-posed algorithm to recover implied local volatility, Quantitative Finance, 3 (2003), 451-457.
doi: 10.1088/1469-7688/3/6/304. |
[13] |
Y. K. Kwok, Mathematical Models of Financial Derivatives, Springer, Berlin, 2008. |
[14] |
R. Lagnado and S. Osher, A technique for calibrating derivative security pricing models: Numerical solution of the inverse problem, Journal of Computational Finance, 1 (1997), 13-25. |
[15] |
R. Lagnado and S. Osher, Reconciling differences, Risk, 10 (1997), 79-83. |
[16] |
W. Li and S. Wang, Penalty approach to the HJB equation arising in European stock option pricing with proportional transaction costs, Journal of Optimization Theory and Applications, 143 (2009), 279-293.
doi: 10.1007/s10957-009-9559-7. |
[17] |
R. C. Merton, Option pricing when underlying stock return are discontinuous, Journal of financial economics, 3 (1976), 125-144.
doi: 10.1016/0304-405X(76)90022-2. |
[18] |
B. F. Nielsen, O. Skavhaug and A. Tveito, Penalty and front-fixing methods for the numerical solution of American option problems, Journal of Computational Finance, 5 (2002), 69-97. |
[19] |
J. Nocedal and S. Wright, Numerical Optimization Series: Springer Series in Operations Research and Financial Engineering 2nd ed, Springer, Berlin, 2006. |
[20] |
S. Stojanovic, Implied volatility for American options via optimal control and fast numerical solutions of obstacle problems, Differential Equations and Control Theory, 225 (2002), 277-294 . |
[21] |
G. Wahba, Splines Models for Observational Data. Series in Applied Mathematics, Vol. 59, SIAM, Philadelphia, 1990.
doi: 10.1137/1.9781611970128. |
[22] |
S. Wang, X. Q. Yang and K. L. Teo, Power penalty method for a linear complementarity problem arising from American option valuation, Journal of Optimization Theory and Applications, 129 (2006), 227-254.
doi: 10.1007/s10957-006-9062-3. |
[23] |
P. Wilmott, Paul Wimott on Quantitave Finance, Wiley, New York, 2000. |
[24] |
K. Zhang and S. Wang, A computational scheme for uncertain volatility model in option pricing, Applied Numerical Mathematics, 59 (2009), 1754-1767.
doi: 10.1016/j.apnum.2009.01.004. |
[25] |
K. Zhang and S. Wang, Interior penalty approach to american option pricing, Journal of Industrial and Management Optimization, 7 (2011), 435-447.
doi: 10.3934/jimo.2011.7.435. |
[26] |
K. Zhang, K. L. Teo and M. Swartz, A robust numerical scheme for pricing American options under regime switching based on penalty method, Computational Economics, 43 (2014), 463-483.
doi: 10.1007/s10614-013-9361-3. |
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