Question…. Standard deviation of portfolios with protected puts

Hi all,

A question that I thought someone may have some ideas about.  I’m not sure whether this is covered in the CFA curriculum, so it’s more just a general question….

I’m wondering whether there is a (simple and practical) way to incorporate asymmetric return and volatility characteristics from assets into a portfolio, such that we can perform mean-variance optimisation?  Specifically I am thinking of assets that we have purchased put options over.

I can sort of work out the overall range (and expected probabilities) of investment outcomes through an iterative process, using my own judgement and market expectations, but it’s difficult (and wrong) to oversimplify the results in terms of standard deviation of returns.  Put option is At-The-Market, though imagine a formula should also be able to handle OTM (assume we have delta etc).

If we think about the problem and range of outcomes, it’s logical to think that (with the addition of costs for protection) the distribution of results in the range 0%+ will be lower by the cost of protection (so we could model this as a lower mean return) but because the downside risk (semivariance) is capped we could actually see the mean return increase in markets with high volatility (i.e. the probabilty of market loss x magnitude of loss > cost of protection).  In either event a normal standard deviation doesn’t properly describe the portfolio’s characteristics.

I’m sure there is a way to do it, I’m just not sure how.