We have $N_1, N_2, N_3$ normally distributed random variables with $µ_i =E[N_i]$, $σ_{ij}=Cov(N_i,N_j)$. We also have $\tilde{µ}_i=E[N_i|N_2 = x] $, $\tilde{σ}_{ij}=Cov(N_i,N_j|N_2 = x)$ and $v^2 =Var(N_1 - N_3|N_2 = x) = σ ̃_{11} - 2σ ̃_{13} + σ ̃_{33}$, where the tilde refers to the expectation conditioned by $N_2 = x$. All these are known.
We want to find out $E[(e^{N_1} - N_2 e^{N_3})^+|N_2 = x]$.
I started with $N_1 = aN_2 + bZ + c$ and $N_3 = dN_2 + eZ + f$, where Z is standard normal, and a,b,c,d,e,f are constants.
In the article the answer is $e^{µ ̃_1+ 0.5 σ ̃_{11}} ϕ((µ ̃_1- µ ̃_3-logx+ σ ̃_{11}-
σ ̃_{13})/v) - x e^{µ ̃_3+ 0.5 σ ̃_{33}} ϕ((µ ̃_1- µ ̃_3-logx+ σ ̃_{13}- σ ̃_{33})/v)$, where ϕ is the normal cumulative distribution function.
I obtained
$e^{µ ̃_1+ 0.5 σ ̃_{11}} ϕ((µ ̃_1- µ ̃_3-logx)/v) x e^{µ ̃_3+ 0.5 σ ̃_{33}} ϕ((µ ̃_1- µ ̃_3-logx )/v)$.
I edited the post to show that the expectation takes into account only positive values, and to give more information.