# optimizing the expected utility

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:

My approach:

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.

• Let's say you're looking at call options, with $\sigma$ the market implied vol being lower than what you expect the actual volatility $\sigma'$ to be. In theory, if you were to buy the cheap option at $\sigma$ and short a self-financing replicating portfolio (needs to be dynamically rebalanced i.e. delta-hedging using vol $\sigma'$), you would, in fine, make: $\underbrace{V_0(\sigma')}_{\textit{short portfolio replicating option priced at$\sigma'$}} -\underbrace{V_0(\sigma)}_{\text{long option at its market price$\sigma$}} > 0$. All of this is quantifiable in the BS world (...) – Quantuple Aug 8 '17 at 13:15
• (...) see seminal paper 'Which free lunch would you like today sir?'. So under the strong assumption you could hedge perfectly, the best strategy would be to leverage that strategy with an infinite notional. What you could do in practice is estimate the real distribution of the delta-hedging P&L given market-frictions and discrete re-balancing to check whether this trade is indeed viable? I know it's not the approach you took, this is why this appears as a comment rather than an answer. Just my 2 cents. – Quantuple Aug 8 '17 at 13:18
• @Quantuple Thank you for your comment. May I expect some kind of bet, for example a long-butterfly, if I am using this approach in your comment? – quallenjäger Aug 8 '17 at 13:42
• I don't understand your question sorry (well I understand that a butterfly could be interpreted as a bet but I don't see why you ask this?). Have you read the paper? – Quantuple Aug 8 '17 at 13:57
• I don't want to be pedant, but he refers to GBM and he has written the equation of ABM instead... -_-' – james42 Aug 8 '17 at 15:41