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Data driven recovery of local volatility surfaces

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.
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  • 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.

    Mathematics Subject Classification: Primary: 45Q05, 97M30, 65R32.


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  • Figure 1.  Data locations for a PBR (Petrobras, an oil company) set in the $(\tau, y)$ domain with our coarsest mesh in the background

    Figure 2.  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 $S_0$

    Figure 3.  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)

    Figure 4.  The estimated spot price converges to the true price

    Figure 5.  Locations of the SPX data in the $(\tau, y)$ domain with our coarsest mesh in the background

    Figure 6.  Reconstructed SPX local volatility surfaces at different maturities obtained with three method variants using scarce data

    Figure 7.  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

    Figure 8.  Reconstructed SPX local volatility surfaces obtained with six method variants. See legends in Figures 6, 7 and 10

    Figure 9.  Reconstructed SPX local volatility surfaces obtained with six method variants for different maturities in the at-the-money ($y=0$) neighbourhood

    Figure 10.  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

    Figure 11.  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

    Figure 12.  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

    Figure 13.  Henry Hub prices: completed data (green line with pentagram), scarce data (blue continuous line), and market (red squares) implied volatilities

    Figure 14.  WTI prices: completed data (green line with pentagram), sparse data (blue continuous line), and market (red squares) implied volatilities

    Table 1.  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 $5.55$ $3.41$ $2.39$ $1.22$ $0.78$ $0.47$ $0.21$ $0.13$
    $S_0$ $0.950$ $0.963$ $0.977$ $0.985$ $0.989$ $0.994$ $0.997$ $0.999$
     | Show Table
    DownLoad: CSV

    Table 2.  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 $a_0$ $0.4^2/2$
     | Show Table
    DownLoad: CSV

    Table 3.  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 $a_0$ $0.14^2/2$
     | Show Table
    DownLoad: CSV

    Table 4.  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
     | Show Table
    DownLoad: CSV

    Table 5.  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
     | Show Table
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    Table 6.  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
     | Show Table
    DownLoad: CSV

    Table 7.  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
     | Show Table
    DownLoad: CSV
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