# Characteristic function under risk neutral measure

I am trying to derive a characteristic function (in Levy-Khintchine form) of a compound Poisson process $X_T$ under a risk neutral measure $\mathbb{Q}$, using the Esscher transfrom to change the original measure. The jumps of the compound Poisson are normally distributed with $\sigma=1$, $\mu=0$, and we set the jump intensity $\lambda=1$.

I am having trouble getting it to work "right", i.e. that $E_{\mathbb{Q}}[e^{X_T}]=1$ under the martingale condition.

Using Cont/Tankov -Financial Modelling with Jump Processes (section 9.5) we find that the Esscher trasnformed characteristic exponent is $$\psi_Q(u) = iu\int_{-1}^1x(e^{\theta x}-1)F(dx)+\int_{\mathbb{R}}(e^{iux}-1)e^{\theta x}F(dx)$$ where $F$ is the normal distribution function. Now, under the martingale condition we should have $\psi_Q(-i)=0$. However, using* $$\text{mgf}(\theta +1)=\text{mgf}(\theta)$$ to find the risk neutral Esscher parameter (mgf is the moment-gnereating function of $X_T$), I get $\theta = -1/2$, and with this we have $$\psi_Q(-i) = \int_{-1}^1x(e^{-x/2}-1)F(dx) +0$$ which is non-zero.

Is it possible that I have misunderstood something important here? I have checked my math quite thoroughly at nearly every step.

*This equation is taken from the paper Gerber and Shiu - Option pricing by Esscher transform

$$\int_{\mathbb{R}}(e^{iux}-1)e^{\theta x}F(dx)$$ it should be
$$\int_{\mathbb{R}}(e^{iux}-1-x_{|x| \leq1})e^{\theta x}F(dx)$$