Derivation of the Local Volatility Formula 50 C2 Finite Difference Techniques

Derivation of the local volatility formula 50 c2

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Derivation of the Local Volatility Formula . . . . . . . . . . . . . . . . 50 C.2 Finite Difference Techniques . . . . . . . . . . . . . . . . . . . . . . . . 51
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List of Figures 2.1 Historical terminal realised volatility using τ -period returns. . . . . . 7 2.2 Historical path-wise realised volatility using τ -period returns. . . . . 8 2.3 EWMA monthly terminal volatility using overlapping and non-overlapping monthly returns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 EWMA monthly path-wise volatility using overlapping and non- overlapping monthly returns. . . . . . . . . . . . . . . . . . . . . . . . 9 2.5 Volatility estimates using non-overlapping daily returns with vary- ing τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.6 GJR-GARCH(1,1) volatility forecasts. . . . . . . . . . . . . . . . . . . . 11 2.7 Comparison between historical and implied volatilities for a range of maturities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.8 Average IVHV Ratios for the JSE Top40 Index from 2010-2018. . . . . 13 3.1 Volatility estimates using Canonical Valuation with increasing GBM sample size while applying different sampling techniques. . . . . . . 18 3.2 Volatility skew using Canonical Valuation while applying different sampling techniques using GBM data. . . . . . . . . . . . . . . . . . . 19 3.3 Profit and loss profile of a three month at-the-money call option us- ing one thousand years of simulated GBM price data with σ = 30% . . 20 3.4 Volatility estimates using Break-Even volatility with increasing GBM sample size while applying different sampling techniques. . . . . . . 21 3.5 Volatility skew using Break-Even volatility while applying different sampling techniques using GBM data. . . . . . . . . . . . . . . . . . . 22 3.6 Volatility estimates using Canonical Valuation with increasing Hes- ton sample size while applying different sampling techniques. . . . . 23 3.7 Volatility skew using Canonical Valuation while applying different sampling techniques using Heston data. . . . . . . . . . . . . . . . . . 24 3.8 Volatility estimates using Break-Even volatility with increasing Hes- ton sample size while applying different sampling techniques. . . . . 25 3.9 Volatility skew using Break-Even volatility while applying different sampling techniques using Heston data. . . . . . . . . . . . . . . . . . 26 3.10 Volatility estimates using Canonical Valuation with increasing Mer- ton sample size while applying different sampling techniques. . . . . 27 3.11 Volatility skew using Canonical Valuation while applying different sampling techniques using Merton data. . . . . . . . . . . . . . . . . . 28 3.12 Volatility estimates using Break-Even volatility with increasing Mer- ton sample size while applying different sampling techniques. . . . . 28
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3.13 Volatility skew using Break-Even volatility while applying different sampling techniques using Merton data. . . . . . . . . . . . . . . . . . 29 4.1 JSE Top40 volatility surfaces on the 20/11/17. . . . . . . . . . . . . . . 33 4.2 Implied volatility estimates using Method 1 for JSE Top40 at-the- money European call. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.3 Implied volatility estimates using Method 2 for JSE Top40 at-the- money European call. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.4 Implied volatility estimates using Method 3 for JSE Top40 at-the- money European call. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 A.1 Volatility estimates using Canonical Valuation with GBM data while applying different sampling techniques with the same Monte Carlo estimates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 A.2 Volatility estimates using Break-Even volatility with GBM data while applying different sampling techniques with the same Monte Carlo estimates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 A.3 Volatility estimates using Canonical Valuation with Heston data while applying different sampling techniques with the same Monte Carlo estimates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 A.4
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