2006 Volume 35 Issue 2 Pages 113-128
This research tackles the following two important problems in the valuation of European option. The first problem is to capture the magnitude of the valuation error due to the Edgeworth expansion of the underlying-asset return distribution generated by Merton's jump diffusion model. The second problem is, from the impact of higher cumulants, to examine the effect of CLT (central limit theorem) that appears in the option price when the maturity of the option becomes longer.
Regarding the first problem, we derive the approximated option valuation formula based on the Edgeworth expansion of the distribution generated by MJD model up to fourth order. Then, we compare the approximated formula price with the exact one computed by FFT. Our approach toward the second problem is to focus on the fact that the N-days underlying-return distribution is nothing but the N-times convolution of the one-day underlying-return distribution. Thus, through the numerical experiment, we observe the speed that the option value of the MJD model converges to that of the BS model due to the CLT effect. We also examine the impact of the third and fourth cumulants in the convergence.
The result of our numerical experiment indicates that the precision of our approximated option valuation formula is reasonably well expect the very short maturity option and the impact of the higher cumulants to the option value is not negligible when the maturity of the option is less than 100days even though the CLT effect is actually observed.
