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Figure 12.10 Observed and estimated 1-month volatilities at 13/01/99

This step follows equations (12.3) and (12.10) and is carried out for all the call-option deltas between 0.05 and 0.95, with an interval equal to 0.05.

Figure 12.10 illustrates the results obtained in this step for the first sample day, with the larger markers representing the three volatility-delta pairs computed from the risk-reversal, the strangle and the at-the-money volatility figures, with the remaining markers being those resulting from the curve in equation (12.10).42 Afterwards, the volatility-delta function is transformed into a volatility-strike (smile) curve, by estimating the strikes that minimise the squared differences to the observed volatilities. With the volatility smile and using the pricing formula in (12.1), call-option prices are obtained.

The observed values in Figure 12.10 are computed in the "Delta-Vol" sheet, while the estimated values are obtained in the "Strike-Call" sheet. In this sheet, the first set of columns (up to column T) contains the squared difference between the observed volatility (in order to the delta) and the estimated volatility (in order to the strike price). The estimated strike prices are in the second set of columns of the same sheet (between columns V and AO). The third set of columns in the same sheet (between AQ and BJ) contains the estimated volatility, which results from equation (12.10), being the call-option delta obtained from inserting in equation (12.3) the estimated strike prices in the second set of columns in the sheet "Strike-Call". Lastly, the sum of the squared residuals is presented in column BL.

It can be seen in Figure 12.10 that the estimated volatilities are very close to the observed figures. This result is usually obtained only after some iterations and trials concerning the starting values, given the non-linear features of the target function. Thus, the choice of those starting values is crucial for the final result.

Afterwards, the RND is estimated, being the parameters of the distributions obtained in order to minimise the squared difference between the estimated and the observed calloption prices, as in equations (12.11) to (12.15).43 Figure 12.11 shows the fitting obtained for the same first sample day concerning call-option prices.

42 The exercise was performed only considering call-option prices, as all relevant formulas (namely (12.5) and (12.6)) were also derived for call options.

43 The estimated RND parameters are presented in the file "Param.". Again, given the non-linear features of the target function in the optimisation problem, the choice of the starting figures is relevant. In the applications presented on the CD-Rom, the estimated values for the RND parameters at the previous date were used as starting values. In order to estimate only the RND parameters, an additional Visual Basic macro is provided ("RND-Estimation").

Strike price

♦ Observed - Estimated

Strike price

♦ Observed - Estimated

Figure 12.11 Observed and estimated 1-month call-option prices at 13/01/99

Figure 12.11 Observed and estimated 1-month call-option prices at 13/01/99

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