“…In financial literature, a significant number of empirical studies shows that these X j 's are not necessarily independent, for example, X j may follow an ARCH or GARCH process. Under certain technical conditions (see (A1) to (A3) in Tamaki and Taniguchi30), even for dependent process X j , by applying similar argument leading to (6), we can also derive an Edgeworth expansion of the density g ( z ) of \documentclass{article}\usepackage{amsmath}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{amsfonts}\pagestyle{empty}\begin{document}$Z \triangleq \left( 2\pi N f_{X,2} \right)^{-\frac{1}{2}} \sum_{j=1}^N X_j$\end{document}: where f X,k is the k th order cumulant spectral density evaluated at frequency 0: and c X,k ( u 1 , …,u k −1 ) is the k th order joint cumulant, and By substituting this Edgeworth expansion in place of the ‘risk‐neutral density’ in the expectation of discounted terminal contingent claim, say for the European call option, we come up with an approximate option ‘price’ C (see Ref 30 for further details): where for any integer k > 0. Note that G 0 is just the ordinary Black–Scholes formula for a vanilla European call, and the rest of the other terms are asymptotic adjustments for skewness and kurtosis.…”