Traditional mean-variance applies to portfolios of assets where the returns are driven by separate stochastic processes which are usually correlated. Put and call options on an asset, however, are derivative assets whose values are driven by the same stochastic process that drives the underlying asset. This article extends basic Markowitz mean-variance to take account of these kinds of assets in the portfolio selection process.

The layout of this article is as follows: Section 2 is a short recap of basic Markowitz mean-variance. Section 3 discusses the properties of the two probability distributions .RN (S) and .RW (S) which are required for this work. The main result is then derived in section 4, followed by a Markowitz-style discrete approximation to the result in Section 5. Although the risk measure used here is variance, it will be seen that the result is sensitive to the entire probability distribution, not just the second central moment. The results are illustrated in section 6, .rst for log-normal distributions, and then for another couple of distributions including an example taken from the S&P500 index. Section 7 discusses issues beyond the mean-variance result of section 4. To show that an analytic formula is possible when considering an aversion to some higher moments of .RW (S), section 7.1 re-derives the main result when the investor has an aversion to the skewness of .RW (S) as well as the variance. Section 7.2, however, discusses issues raised by a consideration of the probability distribution of the optimal payoff f (S), which raises doubts about the main result of section 4. Nonetheless, the main result seems to have all the right properties, and this is discussed in the conclusion in section 8. The appendix starting on page 18 gives explicit formulas for the results of sections 4 and 5 for the cases of normal and log-normal distributions.