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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. Google Scholar |
[3] |
M. Avellaneda, A. Levy and A. Paras, Pricing and hedging derivative securities in markets with uncertain volatilities,, Applied Mathematical Finance, 2 (1995), 73.
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.
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.
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).
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. Google Scholar |
[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.
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. Google Scholar |
[10] |
J. Hull, Options, Futures, and Other Derivatives,, Prentice-Hall, (2005). Google Scholar |
[11] |
N. Jackson, E. Suli and S. Howison, Computation of deterministic volatility surfaces,, Journal of Computational Finance, 2 (1999), 5. Google Scholar |
[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.
doi: 10.1088/1469-7688/3/6/304. |
[13] |
Y. K. Kwok, Mathematical Models of Financial Derivatives,, Springer, (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. Google Scholar |
[15] |
R. Lagnado and S. Osher, Reconciling differences,, Risk, 10 (1997), 79. Google Scholar |
[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.
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.
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. Google Scholar |
[19] |
J. Nocedal and S. Wright, Numerical Optimization Series: Springer Series in Operations Research and Financial Engineering 2nd ed,, Springer, (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.
|
[21] |
G. Wahba, Splines Models for Observational Data. Series in Applied Mathematics,, Vol. 59, (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.
doi: 10.1007/s10957-006-9062-3. |
[23] |
P. Wilmott, Paul Wimott on Quantitave Finance,, Wiley, (2000). Google Scholar |
[24] |
K. Zhang and S. Wang, A computational scheme for uncertain volatility model in option pricing,, Applied Numerical Mathematics, 59 (2009), 1754.
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.
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.
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. Google Scholar |
[3] |
M. Avellaneda, A. Levy and A. Paras, Pricing and hedging derivative securities in markets with uncertain volatilities,, Applied Mathematical Finance, 2 (1995), 73.
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.
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.
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).
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. Google Scholar |
[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.
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. Google Scholar |
[10] |
J. Hull, Options, Futures, and Other Derivatives,, Prentice-Hall, (2005). Google Scholar |
[11] |
N. Jackson, E. Suli and S. Howison, Computation of deterministic volatility surfaces,, Journal of Computational Finance, 2 (1999), 5. Google Scholar |
[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.
doi: 10.1088/1469-7688/3/6/304. |
[13] |
Y. K. Kwok, Mathematical Models of Financial Derivatives,, Springer, (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. Google Scholar |
[15] |
R. Lagnado and S. Osher, Reconciling differences,, Risk, 10 (1997), 79. Google Scholar |
[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.
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.
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. Google Scholar |
[19] |
J. Nocedal and S. Wright, Numerical Optimization Series: Springer Series in Operations Research and Financial Engineering 2nd ed,, Springer, (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.
|
[21] |
G. Wahba, Splines Models for Observational Data. Series in Applied Mathematics,, Vol. 59, (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.
doi: 10.1007/s10957-006-9062-3. |
[23] |
P. Wilmott, Paul Wimott on Quantitave Finance,, Wiley, (2000). Google Scholar |
[24] |
K. Zhang and S. Wang, A computational scheme for uncertain volatility model in option pricing,, Applied Numerical Mathematics, 59 (2009), 1754.
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.
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.
doi: 10.1007/s10614-013-9361-3. |
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