Expected Utility Optimization with Stocks and Call Options Under Volatility Disagreement

Summary

The question sets up a market with one stock and calls priced under Black–Scholes using the market’s volatility estimate. A trader instead believes volatility is different and seeks a portfolio that maximizes exponential expected utility. The proposed return calculation combines prices under the market volatility with payoffs under the trader’s estimate, then evaluates expected utility over portfolio returns.

The central issue is whether the resulting portfolio payoff is Gaussian, which would make the expectation easier to express through its mean and variance. Because call payoffs are nonlinear functions of the stock price, the document’s Gaussian shortcut is not established. It provides no solution or numerical evidence, and leaves open how to compute the utility expectation and optimize the portfolio. The setup therefore highlights the need to model the distribution of option payoffs under the trader’s assumed stock process rather than assume normally distributed portfolio returns.

Key ideas

• The setup distinguishes market volatility used to price options from the trader’s volatility belief about outcomes.
• The proposed objective maximizes exponential expected utility over a portfolio of stock and calls.
• Call option payoffs are nonlinear in the stock price, so portfolio returns cannot be assumed Gaussian from the setup alone.
• The question gives no solution for evaluating the utility expectation or optimizing the portfolio.

Tags

# optimizing the expected utility


# 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.

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