I am working on a paper and am trying to show that the below expectation equals to zero. I suppose computer simulation is the only way to go here? I don't see an easy way to solve this analytically: but I wanted to double check here first.
$$\mathbb{E}\left[\left(\frac{\sigma e^{-0.5\sigma^2t+\sigma W_t}}{1+2e^{-0.5\sigma^2t+\sigma W_t}+e^{-\sigma^2t+2\sigma W_t}}\right)\left(1+\frac{e^{-\sigma^2t+2\sigma W_t}-1}{1+2e^{-0.5\sigma^2t+\sigma W_t}+e^{-\sigma^2t+2\sigma W_t}}\right)\right]$$
Any tips or hints would be greatly appreciated.