I've been taught the following European call valuation formula under jump-diffusion model: \begin{equation} price = E[e^{-rT}max(S_T-K,0)] =\sum_{j = 0}^\infty e^{-rT}P_j(\lambda)E[max(S_T-K,0)|J=j] \end{equation} where $J$ is the number of jumps, probability term $P$ is the poisson process: \begin{equation} e^{-\lambda T} \frac{(\lambda T)^j}{j!} \end{equation} And I've been given directly the final form of the valuation formula: \begin{equation} price = \sum_{j = 0}^\infty P_j(\lambda^{'})BS(S_0,r_j,\sigma_j^2) \end{equation} where: \begin{equation} \lambda^{'}=\lambda(1+k) \end{equation} $k$ is the average jump size, and: \begin{equation} r_j = r-\lambda k+j\ln(1+k) / T, \sigma_j^2=\sigma^2+j\sigma_s^2/T \end{equation} $\sigma^2 $ is original Black-Scholes volitility, $\sigma_s^2$ is the jump variance.

By expanding the Black-Scholes equation, I've managed to derive $r_j$ and $\sigma_j^2$, but I'm confused about where $\lambda^{'}$ comes from. Can someone help derive this term?

In case of anything unclear, the formula can also be found here and here (in the second link, $m=1+k$)


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