# American Institute of Mathematical Sciences

October 2017, 11(5): 799-823. doi: 10.3934/ipi.2017038

## 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

Received  September 2016 Revised  May 2017 Published  June 2017

Fund Project: VA acknowledges and thanks CNPq through grant 201644/2014-2. UMA and XY acknowledge with thanks a Ciencias Sem Fronteiras (visiting scientist / postdoc) grant from CAPES, Brazil. JPZ thanks the support of CNPq grant 307873 and FAPERJ grant 201.288/2014

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.

Citation: Vinicius Albani, Uri M. Ascher, Xu Yang, Jorge P. Zubelli. Data driven recovery of local volatility surfaces. Inverse Problems & Imaging, 2017, 11 (5) : 799-823. doi: 10.3934/ipi.2017038
##### 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.
Data locations for a PBR (Petrobras, an oil company) set in the $(\tau, y)$ domain with our coarsest mesh in the background
Reconstructed (continuous line) and true (line with circles) local volatility surfaces at the five different maturities. The reconstructed local volatility surface corresponds to the one obtained with the adjustment algorithm of the underlying asset $S_0$
Calibration of the local volatility in 5 iterations. Shown, starting from the upper left, are the 1st, 3rd, and 5th iterations, as well as the ground truth (bottom right)
The estimated spot price converges to the true price
Locations of the SPX data in the $(\tau, y)$ domain with our coarsest mesh in the background
Reconstructed SPX local volatility surfaces at different maturities obtained with three method variants using scarce data
Reconstructed SPX local volatility surfaces at different maturities obtained with Tikhonov-type and EnKF methods using completed data. These results are inferior to the corresponding ones for scarce data, displayed in Figure 6
Reconstructed SPX local volatility surfaces obtained with six method variants. See legends in Figures 6, 7 and 10
Reconstructed SPX local volatility surfaces obtained with six method variants for different maturities in the at-the-money ($y=0$) neighbourhood
Implied (Black-Scholes) volatility corresponding to the local volatility surfaces, obtained with the six method variants (Tikhonov, EnKF and "no $a_0$" applied to real and completed data) and compared to the market one
Reconstructed local volatility for different maturity dates for Henry Hub call option prices, comparing between completed data (green line with pentagram) and scarce data (blue line) results
Reconstructed local volatility for different maturity dates for WTI call option prices, comparing between completed data (green line with pentagram) and scarce data (blue line) results
Henry Hub prices: completed data (green line with pentagram), scarce data (blue continuous line), and market (red squares) implied volatilities
WTI prices: completed data (green line with pentagram), sparse data (blue continuous line), and market (red squares) implied volatilities
Normalized $\ell_2$-distance between the true and the reconstructed local volatility surfaces and the value of $S_0$ at each step of the algorithm for adjusting $S_0$
 Iteration 1 2 3 4 5 6 7 8 Normalized Distance $5.55$ $3.41$ $2.39$ $1.22$ $0.78$ $0.47$ $0.21$ $0.13$ $S_0$ $0.950$ $0.963$ $0.977$ $0.985$ $0.989$ $0.994$ $0.997$ $0.999$
 Iteration 1 2 3 4 5 6 7 8 Normalized Distance $5.55$ $3.41$ $2.39$ $1.22$ $0.78$ $0.47$ $0.21$ $0.13$ $S_0$ $0.950$ $0.963$ $0.977$ $0.985$ $0.989$ $0.994$ $0.997$ $0.999$
Parameters for the example of Figure 3
 $\widehat{S}_0$ initial spot price 2500 $S_{\mathrm{true}}$ optimal spot price 2200 $r$ interest rate 0.25% the maximum maturity 1.8 Minimum $y$ -3.5 Maximum $y$ 3.5 $\Delta \tau$ 0.1 $\Delta y$ 0.1 a priori surface $a_0$ $0.4^2/2$
 $\widehat{S}_0$ initial spot price 2500 $S_{\mathrm{true}}$ optimal spot price 2200 $r$ interest rate 0.25% the maximum maturity 1.8 Minimum $y$ -3.5 Maximum $y$ 3.5 $\Delta \tau$ 0.1 $\Delta y$ 0.1 a priori surface $a_0$ $0.4^2/2$
Parameters for the equity data examples
 $S_0$ initial spot price 2112.7 $S_0$ optimal spot price 2095.6 $r$ interest rate 0.25% the maximum maturity 2.5 Minimum $y$ -4.5 Maximum $y$ 1.5 $\Delta \tau$ 0.05 $\Delta y$ 0.1 initial $a_0$ $0.14^2/2$
 $S_0$ initial spot price 2112.7 $S_0$ optimal spot price 2095.6 $r$ interest rate 0.25% the maximum maturity 2.5 Minimum $y$ -4.5 Maximum $y$ 1.5 $\Delta \tau$ 0.05 $\Delta y$ 0.1 initial $a_0$ $0.14^2/2$
Parameters of the penalty functional (10) or (12) with SPX data
 Parameter $\alpha_0$ $\alpha_1$ $\alpha_2$ $\alpha_3$ Value 4.e+8 1.e+6 or 0 1.e+6 1.e+6
 Parameter $\alpha_0$ $\alpha_1$ $\alpha_2$ $\alpha_3$ Value 4.e+8 1.e+6 or 0 1.e+6 1.e+6
Residuals of the 6 method variants
 Tikhonov-type EnKF Scarce Comp. Scarce (no $a_0$) Comp. (no $a_0$) Scarce Comp. Residual 0.0196 0.0314 0.0247 0.0289 0.0198 0.0294
 Tikhonov-type EnKF Scarce Comp. Scarce (no $a_0$) Comp. (no $a_0$) Scarce Comp. Residual 0.0196 0.0314 0.0247 0.0289 0.0198 0.0294
Measures of data misfit of the 6 models
 Tikhonov-type EnKF Scarce Comp. Scarce (no $a_0$) Comp. (no $a_0$) 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 $a_0$) Comp. (no $a_0$) 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
Parameters obtained in the local volatility calibration with Henry Hub and WTI call prices using sparse data and completed data
 WTI Henry Hub Comp. Data Sparse Data Comp. Data Sparse Data Running Time (sec.) $1.40\times10^{3}$ $3.07\times10^{2}$ $1.41\times10^{3}$ $1.02\times10^{3}$ $\alpha_0$ 1.0e4 1.0e3 1.0e3 1.0e3 $\alpha_1=\alpha_2=\alpha_3$ 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.) $1.40\times10^{3}$ $3.07\times10^{2}$ $1.41\times10^{3}$ $1.02\times10^{3}$ $\alpha_0$ 1.0e4 1.0e3 1.0e3 1.0e3 $\alpha_1=\alpha_2=\alpha_3$ 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
 [1] Junyoung Jang, Kihoon Jang, Hee-Dae Kwon, Jeehyun Lee. Feedback control of an HBV model based on ensemble kalman filter and differential evolution. Mathematical Biosciences & Engineering, 2018, 15 (3) : 667-691. doi: 10.3934/mbe.2018030 [2] Pedro Caro. On an inverse problem in electromagnetism with local data: stability and uniqueness. Inverse Problems & Imaging, 2011, 5 (2) : 297-322. doi: 10.3934/ipi.2011.5.297 [3] Victor Isakov. On uniqueness in the inverse conductivity problem with local data. Inverse Problems & Imaging, 2007, 1 (1) : 95-105. doi: 10.3934/ipi.2007.1.95 [4] Qinghua Ma, Zuoliang Xu, Liping Wang. Recovery of the local volatility function using regularization and a gradient projection method. Journal of Industrial & Management Optimization, 2015, 11 (2) : 421-437. doi: 10.3934/jimo.2015.11.421 [5] Alexander Bibov, Heikki Haario, Antti Solonen. Stabilized BFGS approximate Kalman filter. Inverse Problems & Imaging, 2015, 9 (4) : 1003-1024. doi: 10.3934/ipi.2015.9.1003 [6] Sebastian Reich, Seoleun Shin. On the consistency of ensemble transform filter formulations. Journal of Computational Dynamics, 2014, 1 (1) : 177-189. doi: 10.3934/jcd.2014.1.177 [7] Russell Johnson, Carmen Núñez. The Kalman-Bucy filter revisited. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4139-4153. doi: 10.3934/dcds.2014.34.4139 [8] Gabriel Peyré, Sébastien Bougleux, Laurent Cohen. Non-local regularization of inverse problems. Inverse Problems & Imaging, 2011, 5 (2) : 511-530. doi: 10.3934/ipi.2011.5.511 [9] Luca Rondi. On the regularization of the inverse conductivity problem with discontinuous conductivities. Inverse Problems & Imaging, 2008, 2 (3) : 397-409. doi: 10.3934/ipi.2008.2.397 [10] Laurent Devineau, Paul Bonnefoy, Alexandre Boumezoued. Fast calibration of the Libor market model with stochastic volatility and displaced diffusion. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-31. doi: 10.3934/jimo.2019025 [11] Z. G. Feng, Kok Lay Teo, N. U. Ahmed, Yulin Zhao, W. Y. Yan. Optimal fusion of sensor data for Kalman filtering. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 483-503. doi: 10.3934/dcds.2006.14.483 [12] Seung-Yeal Ha, Jaeseung Lee, Zhuchun Li. Emergence of local synchronization in an ensemble of heterogeneous Kuramoto oscillators. Networks & Heterogeneous Media, 2017, 12 (1) : 1-24. doi: 10.3934/nhm.2017001 [13] Francis J. Chung. Partial data for the Neumann-Dirichlet magnetic Schrödinger inverse problem. Inverse Problems & Imaging, 2014, 8 (4) : 959-989. doi: 10.3934/ipi.2014.8.959 [14] Valter Pohjola. An inverse problem for the magnetic Schrödinger operator on a half space with partial data. Inverse Problems & Imaging, 2014, 8 (4) : 1169-1189. doi: 10.3934/ipi.2014.8.1169 [15] Li Liang. Increasing stability for the inverse problem of the Schrödinger equation with the partial Cauchy data. Inverse Problems & Imaging, 2015, 9 (2) : 469-478. doi: 10.3934/ipi.2015.9.469 [16] Jussi Korpela, Matti Lassas, Lauri Oksanen. Discrete regularization and convergence of the inverse problem for 1+1 dimensional wave equation. Inverse Problems & Imaging, 2019, 13 (3) : 575-596. doi: 10.3934/ipi.2019027 [17] Qiyu Jin, Ion Grama, Quansheng Liu. Convergence theorems for the Non-Local Means filter. Inverse Problems & Imaging, 2018, 12 (4) : 853-881. doi: 10.3934/ipi.2018036 [18] 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 [19] Qihong Chen. Recovery of local volatility for financial assets with mean-reverting price processes. Mathematical Control & Related Fields, 2018, 8 (3&4) : 625-635. doi: 10.3934/mcrf.2018026 [20] Sombuddha Bhattacharyya. An inverse problem for the magnetic Schrödinger operator on Riemannian manifolds from partial boundary data. Inverse Problems & Imaging, 2018, 12 (3) : 801-830. doi: 10.3934/ipi.2018034

2017 Impact Factor: 1.465