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Figure 5: Leverage function (left) and the local volatility surface (right) computed with our proposed method in the synthetic data example.

Figure 5 Leverage function (left) and the local volatility surface (right) computed with our proposed method in the synthetic data example.

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We will now compare the methods described in Sections 3.2 and 3.3 within syn- thetic and real data examples. The following information is common to both cases:
We will now compare the methods described in Sections 3.2 and 3.3 within syn- thetic and real data examples. The following information is common to both cases:
The coarse mesh is the one used in the finite difference methods in our nu- merical examples below. Figure 1: Domestic and foreign interest rates
The coarse mesh is the one used in the finite difference methods in our nu- merical examples below. Figure 1: Domestic and foreign interest rates
Figure 2: The ground truth leverage function (left) and the local volatility surface oj,.(t, x)" (right). Table 2: SV parameters for the synthetic data example
Figure 2: The ground truth leverage function (left) and the local volatility surface oj,.(t, x)" (right). Table 2: SV parameters for the synthetic data example
Table 3: SV parameters calibrated to real data I n this section we present a real data example. We chose FX options on EURUSD on March 18th, 2015. They include the typical 25 liquid option contracts, with 5 maturities (1W, 1M, 3M, 6M, 1Y) and 5 strikes (related to 10 and 25 Call and Put Delta and to ATM) per maturity (see Figure 3). The spot value was 1.0864. nq The parameters of the Heston model are calibrated to this data set and given in n Table 3.
Table 3: SV parameters calibrated to real data I n this section we present a real data example. We chose FX options on EURUSD on March 18th, 2015. They include the typical 25 liquid option contracts, with 5 maturities (1W, 1M, 3M, 6M, 1Y) and 5 strikes (related to 10 and 25 Call and Put Delta and to ATM) per maturity (see Figure 3). The spot value was 1.0864. nq The parameters of the Heston model are calibrated to this data set and given in n Table 3.
Figure 3: EUR-USD local volatility surface and options prices on March 18th, 2015 .3 Numerical Results
Figure 3: EUR-USD local volatility surface and options prices on March 18th, 2015 .3 Numerical Results
Figure 4: Leverage function (left) and the local volatility surface (right) computed with the benchmark method in the synthetic data example. In the figures below we show the recovered leverage function and the local volatil- ity surface using the benchmark method in Section 3.2 and our proposed method shown in Section 3.3.
Figure 4: Leverage function (left) and the local volatility surface (right) computed with the benchmark method in the synthetic data example. In the figures below we show the recovered leverage function and the local volatil- ity surface using the benchmark method in Section 3.2 and our proposed method shown in Section 3.3.
Figure 6: The leverage function in the synthetic data example: the ground truth (with stars), the benchmark method (with squares) and our method (with circles)
Figure 6: The leverage function in the synthetic data example: the ground truth (with stars), the benchmark method (with squares) and our method (with circles)
Figure 7: The local volatility surface in the synthetic data example: the ground truth (with stars), the benchmark method (with squares) and our method (with circles)
Figure 7: The local volatility surface in the synthetic data example: the ground truth (with stars), the benchmark method (with squares) and our method (with circles)
Figure 8: Leverage function (left) and the local volatility surface (right) computed with the benchmark method in the real data example.
Figure 8: Leverage function (left) and the local volatility surface (right) computed with the benchmark method in the real data example.
Figure 9: Leverage function (left) and the local volatility surface (right) computed with our proposed method in the real data example.
Figure 9: Leverage function (left) and the local volatility surface (right) computed with our proposed method in the real data example.
Figure 10: The leverage function in the real data example: the benchmark method (with squares) and our method (with circles)
Figure 10: The leverage function in the real data example: the benchmark method (with squares) and our method (with circles)
Figure 11: The local volatility surface in the real data example: computed from option prices (with stars), the benchmark method (with squares) and our method (with circles)
Figure 11: The local volatility surface in the real data example: computed from option prices (with stars), the benchmark method (with squares) and our method (with circles)
+ a. Once we estimate the leverage function L, we can recover the local volatility urface using the Alternating Direction Implicit (ADI) method for the Fokker- -lanck PDE with the leverage function at both times, t, and t,41, see Section 3.1.1. Comparing with the ground truth of local volatility surface in the synthetic lata example, we can calculate the relative residuals. In Table 4, we present the elative residuals in two intervals of the log-moneyness, which are |[—3,3] and —2,2]. We also report the relative residuals of the real data example. For both xamples, we see that the proposed method generates better results with relative rrors significantly smaller than the benchmark method. We would like to point ut that this failure of convergence of the benchmark method is not related o a boundary issue. Indeed, numerical experiments on smaller log-moneyness ntervals have similar results to the truncated version of the results we have found.
+ a. Once we estimate the leverage function L, we can recover the local volatility urface using the Alternating Direction Implicit (ADI) method for the Fokker- -lanck PDE with the leverage function at both times, t, and t,41, see Section 3.1.1. Comparing with the ground truth of local volatility surface in the synthetic lata example, we can calculate the relative residuals. In Table 4, we present the elative residuals in two intervals of the log-moneyness, which are |[—3,3] and —2,2]. We also report the relative residuals of the real data example. For both xamples, we see that the proposed method generates better results with relative rrors significantly smaller than the benchmark method. We would like to point ut that this failure of convergence of the benchmark method is not related o a boundary issue. Indeed, numerical experiments on smaller log-moneyness ntervals have similar results to the truncated version of the results we have found.
Related topics:
MathematicsComputer ScienceEconomicsEconometricsStochastic VolatilityLocal Volatility
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