The market consist of one single stock and call options with different strike price based on the given stock.Suppose the market believes the stock follows the following GBM:$$dS_t=\mu S_tdt+\sigma S_tdW_t$$ and a trader believe that the actual volatility should be $\sigma' $. Suppose all price is given according to the black-scholes-framework. I wish to find an optimal strategy by maximizing the exponential utility:
So the return of the stock and call option can be computed by using $\sigma $ for the current price and payoff by using the $\sigma '$ and then substract the payoff from the current price. Then insert in the expected utility:$$E[U(w^TX)]$$
where U is the exponential utility and X is the return. w denots our portfolio.
My question is, is the payoff of the portfolio still gaussian? As I am including call-options. Because if it is gaussian, I can explicit write down the expectation in dependence of the mean and variance. Then the optimization problem is quite easy. If is not, what would be the best way to do it.