Hi all,there is a number formulas which exploit Fourier transforms in order to price derivatives.I'm interested into an application of Lewis (2000) reported in the paper below in Eq. (3.10) and (3.13):http://finance.wharton.upenn.edu/~ishal/papers/cj.pdfIn such a model [$]\varrho_X^Q[$] is the laplace transform of discounted log prices under Q, with both log prices and interest rate affine in the state-variables X (which in turn belong to the jump diffusion affine class).I want to compute the price by direct integration (ie no fft or frft) but I just don't understand the notation in the equation mentioned above.If I have correctly understood z is supposed to be complex and the integrand also. I don't understand how should I perform the integration. I don't understand what does it exactly mean to take the integral from [$]i z_i -\infty[$] to [$]i z_i +\infty[$] where [$]z_i[$] is the imaginary part of z which is the integration variable.I guess my question is somewhat stupid. But assuming I have a model for [$]\varrho_X^Q[$], I don't understand how to write the integral in my Mathematica code to compute the option price!Thank you in advance.

In Mathematica, say you want to integrate some f[z] along this contour you mention with [$]z_i = 1/2[$].Then, the integral is strictlyNIntegrate[f[x + I/2],{x,-Infinity,0,Infinity}]but in practice, you will often have a cutoff and, because the result is real, use2 NIntegrate[Re[f[x + I/2]],{x, 0, xmax}]Of course, if you want [$]z_i = c[$], then f[x + I c], etc.

Thank you, Alan.My code works with:NIntegrate[ f[ i x], {x, i c-Infinity, i c + Infinity}]where c is a constant > 1 and infinity is substituted by a number large enough.