Conditional expectation of exp{xy} given filtration $F_1$

Question

Let $Y_1, Y_2$ be independent en standard normal distributed and $F_1$ a natural filtration. For $0<\lambda <1$, I need to compute the following conditional expectation: $E[e^{\lambda Y_1Y_2}|F_1]$. I don't really know where to start. I tried to write $Y_1Y_2=\frac{1}{2}(Y_1+Y_2)^2-\frac{1}{2}(Y_1^2+Y_2^2)$, so that the exponential would be less hard to work with. However it got me nowwhere.

Answer 1

It is well known that $Ee^{tY_2}=e^{t^{2}/2}$. Hence $E(e^{\lambda Y_1Y_2}|F_1)=e^{\lambda^{2}Y_1^{2}/2}$. [Conditioned on $F_1$, $Y_1$ acts like a constant].