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A wavelet frame approach for removal of mixed Gaussian and impulse noise on surfaces
Data driven recovery of local volatility surfaces
1. | Dept. of Mathematics, UFSC, Florianopolis, Brazil |
2. | Dept. of Computer Science, University of British Columbia, Canada |
3. | IMPA, Rio de Janeiro, Brazil |
This paper examines issues of data completion and location uncertainty, popular in many practical PDE-based inverse problems, in the context of option calibration via recovery of local volatility surfaces. While real data is usually more accessible for this application than for many others, the data is often given only at a restricted set of locations. We show that attempts to "complete missing data" by approximation or interpolation, proposed and applied in the literature, may produce results that are inferior to treating the data as scarce. Furthermore, model uncertainties may arise which translate to uncertainty in data locations, and we show how a model-based adjustment of the asset price may prove advantageous in such situations. We further compare a carefully calibrated Tikhonov-type regularization approach against a similarly adapted EnKF method, in an attempt to fine-tune the data assimilation process. The EnKF method offers reassurance as a different method for assessing the solution in a problem where information about the true solution is difficult to come by. However, additional advantage in the latter approach turns out to be limited in our context.
References:
[1] |
Y. Achdou and O. Pironneau,
Computational Methods for Option Pricing SIAM, 2005.
doi: 10.1137/1. 9780898717495. |
[2] |
Y. Achdou and O. Pironneau,
Numerical procedure for calibration of volatility with American options, Applied Mathematical Finance, 12 (2007), 201-241.
doi: 10.1080/1350486042000297252. |
[3] |
V. Albani, U. Ascher and J. Zubelli, Local volatility models in commodity markets and online calibration,
J. Computational Finance 2017. Accepted, to appear. |
[4] |
V. Albani and J. P. Zubelli,
Online local volatility calibration by convex regularization, Appl. Anal. Discrete Math., 8 (2014), 243-268.
doi: 10.2298/AADM140811012A. |
[5] |
U. Ascher, H. Huang and K. van den Doel,
Artificial time integration, BIT, 47 (2007), 3-25.
doi: 10.1007/s10543-006-0112-x. |
[6] |
F. Black and M. Scholes,
The pricing of options and corporate liabilities, J. Pol. Econ., 81 (1973), 637-654.
doi: 10.1086/260062. |
[7] |
P. Boyle and D. Thangaraj,
Volatility estimation from observed option prices, Decisions in Economics and Finance, 23 (2000), 31-52.
doi: 10.1007/s102030050004. |
[8] |
D. Calvetti, O. Ernst and E. Somersalo, Dynamic updating of numerical model discrepancy using sequential sampling Inverse Problems 30 (2014), 114019, 19pp.
doi: 10.1088/0266-5611/30/11/114019. |
[9] |
A. De Cezaro, O. Scherzer and J. Zubelli,
Convex regularization of local volatility models from option prices: convergence analysis and rates, Nonlinear Analysis, 75 (2012), 2398-2415.
doi: 10.1016/j.na.2011.10.037. |
[10] |
A. De Cezaro and J. P. Zubelli,
The tangential cone condition for the iterative calibration of local volatility surfaces, IMA Journal of Applied Mathematics, 80 (2015), 212-232.
doi: 10.1093/imamat/hxt037. |
[11] |
B. Dupire,
Pricing with a smile, Risk, 7 (1994), 18-20.
|
[12] |
H. Egger and H. Engl,
Tikhonov regularization applied to the inverse problem of option pricing: convergence analysis and rates, Inverse Problems, 21 (2005), 1027-1045.
doi: 10.1088/0266-5611/21/3/014. |
[13] |
H. W. Engl, M. Hanke and A. Neubauer,
Regularization of Inverse Problems Kluwer, 1996.
doi: 10.1007/978-94-009-1740-8. |
[14] |
J. Gatheral,
The Volatility Surface: A Practitioner's Guide Wiley Finance. John Wiley & Sons, 2006.
doi: 10.1002/9781119202073. |
[15] |
J. Granek and E. Haber, Data mining for real mining: A robust algorithm for prospectivity mapping with uncertainties Proc. SIAM Conference on Data Mining (2015), 9pp.
doi: 10.1137/1.9781611974010.17. |
[16] |
E. Haber, U. Ascher and D. Oldenburg,
Inversion of 3D electromagnetic data in frequency and time domain using an inexact all-at-once approach, Geophysics, 69 (2004), 1216-1228.
doi: 10.1190/1.1801938. |
[17] |
B. Hofmann and R. Krämer,
On maximum entropy regularization for a specific inverse problem of option pricing, J. Inverse Ill-Posed Problems, 13 (2005), 41-63.
doi: 10.1515/1569394053583739. |
[18] |
B. Hofmann, B. Kaltenbacher, C. Pöschl and O. Scherzer,
A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators, Inverse Problems, 23 (2007), 987-1010.
doi: 10.1088/0266-5611/23/3/009. |
[19] |
H. Huang and U. Ascher, Fast denoising of surface meshes with intrinsic texture Inverse Problems 24 (2008), 034003, 18pp.
doi: 10.1088/0266-5611/24/3/034003. |
[20] |
M. Iglesias, K. Law and A. Stuart, Ensemble Kalman methods for inverse problems Inverse Problems 29 (2013), 045001, 20pp.
doi: 10.1088/0266-5611/29/4/045001. |
[21] |
R. Jarrow, Y. Kchia and P. Protter,
How to detect an asset bubble, SIAM J. Financial Mathematics, 2 (2011), 839-865.
doi: 10.1137/10079673X. |
[22] |
C. Johns and J. Mandel,
A two-stage ensemble Kalman filter for smooth data assimilation, Environmental and Ecological Statistics, 15 (2008), 101-110.
doi: 10.1007/s10651-007-0033-0. |
[23] |
N. Kahale,
Smile interpolation and calibration of the local volatility model, Risk Magazine, 1 (2005), 637-654.
|
[24] |
R. Korn and E. Korn,
Option Price and Portfolio Optimization: Modern Methods of Mathematical Finance volume 31 of Graduate Studies in Mathematics, AMS, 2001.
doi: 10.1007/978-3-322-83210-8. |
[25] |
R. Kumar, C. da Silva, O. Aklain, A. Aravkin, H. Mansour, B. Recht and F. Herrmann,
Efficient matrix completion for seismic data reconstruction, Geophysics, 80 (2015), 97-114.
doi: 10.1190/geo2014-0369.1. |
[26] |
G. Nakamura and R. Potthast,
Inverse Problems. An Introduction to the Theory and Methods of Inverse Problems and Data Assimilation IOP Publishing, 2015. |
[27] |
S. Reich and C. Cotter,
Probabilistic Forecasting and Bayesian Data Assimilation Cambridge, 2015.
doi: 10.1017/CBO9781107706804. |
[28] |
F. Roosta-Khorasani, K. van den Doel and U. Ascher,
Data completion and stochastic algorithms for PDE inversion problems with many measurements, ETNA, 42 (2014), 177-196.
|
[29] |
C. Vogel,
Computational Methods for Inverse Problem SIAM, Philadelphia, 2002.
doi: 10.1137/1.9780898717570. |
show all references
References:
[1] |
Y. Achdou and O. Pironneau,
Computational Methods for Option Pricing SIAM, 2005.
doi: 10.1137/1. 9780898717495. |
[2] |
Y. Achdou and O. Pironneau,
Numerical procedure for calibration of volatility with American options, Applied Mathematical Finance, 12 (2007), 201-241.
doi: 10.1080/1350486042000297252. |
[3] |
V. Albani, U. Ascher and J. Zubelli, Local volatility models in commodity markets and online calibration,
J. Computational Finance 2017. Accepted, to appear. |
[4] |
V. Albani and J. P. Zubelli,
Online local volatility calibration by convex regularization, Appl. Anal. Discrete Math., 8 (2014), 243-268.
doi: 10.2298/AADM140811012A. |
[5] |
U. Ascher, H. Huang and K. van den Doel,
Artificial time integration, BIT, 47 (2007), 3-25.
doi: 10.1007/s10543-006-0112-x. |
[6] |
F. Black and M. Scholes,
The pricing of options and corporate liabilities, J. Pol. Econ., 81 (1973), 637-654.
doi: 10.1086/260062. |
[7] |
P. Boyle and D. Thangaraj,
Volatility estimation from observed option prices, Decisions in Economics and Finance, 23 (2000), 31-52.
doi: 10.1007/s102030050004. |
[8] |
D. Calvetti, O. Ernst and E. Somersalo, Dynamic updating of numerical model discrepancy using sequential sampling Inverse Problems 30 (2014), 114019, 19pp.
doi: 10.1088/0266-5611/30/11/114019. |
[9] |
A. De Cezaro, O. Scherzer and J. Zubelli,
Convex regularization of local volatility models from option prices: convergence analysis and rates, Nonlinear Analysis, 75 (2012), 2398-2415.
doi: 10.1016/j.na.2011.10.037. |
[10] |
A. De Cezaro and J. P. Zubelli,
The tangential cone condition for the iterative calibration of local volatility surfaces, IMA Journal of Applied Mathematics, 80 (2015), 212-232.
doi: 10.1093/imamat/hxt037. |
[11] |
B. Dupire,
Pricing with a smile, Risk, 7 (1994), 18-20.
|
[12] |
H. Egger and H. Engl,
Tikhonov regularization applied to the inverse problem of option pricing: convergence analysis and rates, Inverse Problems, 21 (2005), 1027-1045.
doi: 10.1088/0266-5611/21/3/014. |
[13] |
H. W. Engl, M. Hanke and A. Neubauer,
Regularization of Inverse Problems Kluwer, 1996.
doi: 10.1007/978-94-009-1740-8. |
[14] |
J. Gatheral,
The Volatility Surface: A Practitioner's Guide Wiley Finance. John Wiley & Sons, 2006.
doi: 10.1002/9781119202073. |
[15] |
J. Granek and E. Haber, Data mining for real mining: A robust algorithm for prospectivity mapping with uncertainties Proc. SIAM Conference on Data Mining (2015), 9pp.
doi: 10.1137/1.9781611974010.17. |
[16] |
E. Haber, U. Ascher and D. Oldenburg,
Inversion of 3D electromagnetic data in frequency and time domain using an inexact all-at-once approach, Geophysics, 69 (2004), 1216-1228.
doi: 10.1190/1.1801938. |
[17] |
B. Hofmann and R. Krämer,
On maximum entropy regularization for a specific inverse problem of option pricing, J. Inverse Ill-Posed Problems, 13 (2005), 41-63.
doi: 10.1515/1569394053583739. |
[18] |
B. Hofmann, B. Kaltenbacher, C. Pöschl and O. Scherzer,
A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators, Inverse Problems, 23 (2007), 987-1010.
doi: 10.1088/0266-5611/23/3/009. |
[19] |
H. Huang and U. Ascher, Fast denoising of surface meshes with intrinsic texture Inverse Problems 24 (2008), 034003, 18pp.
doi: 10.1088/0266-5611/24/3/034003. |
[20] |
M. Iglesias, K. Law and A. Stuart, Ensemble Kalman methods for inverse problems Inverse Problems 29 (2013), 045001, 20pp.
doi: 10.1088/0266-5611/29/4/045001. |
[21] |
R. Jarrow, Y. Kchia and P. Protter,
How to detect an asset bubble, SIAM J. Financial Mathematics, 2 (2011), 839-865.
doi: 10.1137/10079673X. |
[22] |
C. Johns and J. Mandel,
A two-stage ensemble Kalman filter for smooth data assimilation, Environmental and Ecological Statistics, 15 (2008), 101-110.
doi: 10.1007/s10651-007-0033-0. |
[23] |
N. Kahale,
Smile interpolation and calibration of the local volatility model, Risk Magazine, 1 (2005), 637-654.
|
[24] |
R. Korn and E. Korn,
Option Price and Portfolio Optimization: Modern Methods of Mathematical Finance volume 31 of Graduate Studies in Mathematics, AMS, 2001.
doi: 10.1007/978-3-322-83210-8. |
[25] |
R. Kumar, C. da Silva, O. Aklain, A. Aravkin, H. Mansour, B. Recht and F. Herrmann,
Efficient matrix completion for seismic data reconstruction, Geophysics, 80 (2015), 97-114.
doi: 10.1190/geo2014-0369.1. |
[26] |
G. Nakamura and R. Potthast,
Inverse Problems. An Introduction to the Theory and Methods of Inverse Problems and Data Assimilation IOP Publishing, 2015. |
[27] |
S. Reich and C. Cotter,
Probabilistic Forecasting and Bayesian Data Assimilation Cambridge, 2015.
doi: 10.1017/CBO9781107706804. |
[28] |
F. Roosta-Khorasani, K. van den Doel and U. Ascher,
Data completion and stochastic algorithms for PDE inversion problems with many measurements, ETNA, 42 (2014), 177-196.
|
[29] |
C. Vogel,
Computational Methods for Inverse Problem SIAM, Philadelphia, 2002.
doi: 10.1137/1.9780898717570. |












Iteration | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
Normalized Distance | ||||||||
Iteration | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
Normalized Distance | ||||||||
2500 | |
2200 | |
0.25% | |
the maximum maturity | 1.8 |
Minimum |
-3.5 |
Maximum |
3.5 |
0.1 | |
0.1 | |
a priori surface |
2500 | |
2200 | |
0.25% | |
the maximum maturity | 1.8 |
Minimum |
-3.5 |
Maximum |
3.5 |
0.1 | |
0.1 | |
a priori surface |
2112.7 | |
2095.6 | |
0.25% | |
the maximum maturity | 2.5 |
Minimum |
-4.5 |
Maximum |
1.5 |
0.05 | |
0.1 | |
initial |
2112.7 | |
2095.6 | |
0.25% | |
the maximum maturity | 2.5 |
Minimum |
-4.5 |
Maximum |
1.5 |
0.05 | |
0.1 | |
initial |
Parameter | ||||
Value | 4.e+8 | 1.e+6 or 0 | 1.e+6 | 1.e+6 |
Parameter | ||||
Value | 4.e+8 | 1.e+6 or 0 | 1.e+6 | 1.e+6 |
Tikhonov-type | EnKF | |||||
Scarce | Comp. | Scarce (no |
Comp. (no |
Scarce | Comp. | |
Residual | 0.0196 | 0.0314 | 0.0247 | 0.0289 | 0.0198 | 0.0294 |
Tikhonov-type | EnKF | |||||
Scarce | Comp. | Scarce (no |
Comp. (no |
Scarce | Comp. | |
Residual | 0.0196 | 0.0314 | 0.0247 | 0.0289 | 0.0198 | 0.0294 |
Tikhonov-type | EnKF | |||||
Scarce | Comp. | Scarce (no |
Comp. (no |
Scarce | Comp. | |
RMSE | 0.0195 | 0.0321 | 0.0290 | 0.0325 | 0.0255 | 0.0324 |
RWMSE | 0.0175 | 0.0241 | 0.0252 | 0.0242 | 0.0241 | 0.0242 |
RR | 0.1407 | 0.1987 | 0.2292 | 0.2186 | 0.1766 | 0.2186 |
Tikhonov-type | EnKF | |||||
Scarce | Comp. | Scarce (no |
Comp. (no |
Scarce | Comp. | |
RMSE | 0.0195 | 0.0321 | 0.0290 | 0.0325 | 0.0255 | 0.0324 |
RWMSE | 0.0175 | 0.0241 | 0.0252 | 0.0242 | 0.0241 | 0.0242 |
RR | 0.1407 | 0.1987 | 0.2292 | 0.2186 | 0.1766 | 0.2186 |
WTI | Henry Hub | |||
Comp. Data | Sparse Data | Comp. Data | Sparse Data | |
Running Time (sec.) | ||||
1.0e4 | 1.0e3 | 1.0e3 | 1.0e3 | |
4.5 | 1.0 | 1.3 | 1.0 | |
Price Residual | 2.16e-2 | 3.21e-3 | 3.47e-2 | 2.14e-2 |
Implied Vol. Residual | 1.26e-1 | 2.66e-2 | 9.61e-2 | 5.98e-2 |
WTI | Henry Hub | |||
Comp. Data | Sparse Data | Comp. Data | Sparse Data | |
Running Time (sec.) | ||||
1.0e4 | 1.0e3 | 1.0e3 | 1.0e3 | |
4.5 | 1.0 | 1.3 | 1.0 | |
Price Residual | 2.16e-2 | 3.21e-3 | 3.47e-2 | 2.14e-2 |
Implied Vol. Residual | 1.26e-1 | 2.66e-2 | 9.61e-2 | 5.98e-2 |
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