Derivation of the local volatility formula 50 c2

Derivation of the Local Volatility Formula . . . . . . . . . . . . . . . . 50
C.2 Finite Difference Techniques . . . . . . . . . . . . . . . . . . . . . . . . 51

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

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