I'm trying to verify a result in the paper of https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1700305 with regards to the tempered stable or CGMY process in option pricing literature, which has Levy measure in ordinary exponential form as


Considering the theorem for the relation between the triplet of ordinary and stochastic exponentials,

If $L$ is a Levy process with characteristic triple $(\mu,\sigma^2,\nu$, then $\mathcal{E}_L(t)=e^{L_1(t)}$ where $L_1$ is a L\'evy process with characteristic triplet $(\mu_1,\sigma_1^2,\nu_1)$, given by

$\nu_1 = \nu\circ f^{-1},\quad f(x)=\log(1+x),$


$\sigma_1 = \sigma$

Conversely, there exists a Levy process $L_2$ with characteristic triplet $(\mu_2,\sigma_2^2,\nu_2)$ such that there exists $e^{L_t}=\mathcal{E}_{L_2}(t)$, where

$\nu_2 = \nu\circ g^{-1},\quad g(x)=e^x-1,$

$\mu_2 = \mu+\frac{1}{2}\sigma^2+\int_{\mathbb{R}-\{0\}}[(e^x-1)\chi_{(-1,1)}(e^x-1)-x\chi_{(-1,1)}(x)]\nu(dx),$

$\sigma_2 = \sigma$

The paper gives the Levy measure under the second scenario (that is, the composite function of the ordinary exponential Levy measure with $log(1+x)$), as

$\nu = \frac{C(1+x)^{-M-1}}{log(1+x)^{Y+1}}\chi_{\{x>0\}}dx+\frac{C(1+x)^{G-1}}{-log(1+x)^{Y+1}}\chi_{\{-1<x<0\}}dx$

I've been looking at this for days and can't for the life of me figure out why the exponential tempering factors are now subtracting 1 under the new representation. Can someone clarify this for me? Thank you!


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