We develop a new method to optimize portfolios of options in a market where European calls and puts are available with many exercise prices for each of several potentially correlated underlying assets. We identify the combination of asset-specific option payoffs that maximizes the Sharpe ratio of the overall portfolio: such payoffs form the unique solution to a system of integral equations, which reduces to a linear matrix equation under discrete representations of the underlying probabilities. Even when risk-neutral volatilities are all higher than physical volatilities, it can be optimal to sell options on some assets while buying options on other assets, for which the positive hedging demand outweighs negative demand stemming from asset-specific returns.