# Options return series for portfolio optimization

Is there anyway to construct a return series for an options positions using price data on the underling? I want to introduce options as an asset to the portfolio optimization process. I realize the best way to do this would be with actual historical data, but that's complicated for obvious reasons.

My thinking is it would be easy if you only used delta. You assume a constant delta position, and the option returns just become a fraction of the underlying's returns. But would it be possible to simulate the higher moments?

Because of the asymmetry of option returns (as a function of the underlying) it never works to map options to proportions of the underlying, except in cases where the option position is too small to matter. You really have to run the options through Black-Scholes-like pricing formulas based on each scenario's underlying prices.

You can still do portfolio optimization, even mean-variance portfolio optimization, but you no longer get all that simple linear algebra. Instead it becomes necessary to run some kind of optimization algorithm over the mean-variance objective. (Since the dimensions are high, a conjugate gradient scheme can do well)

It usually makes sense to make volatility part of your simulation as well. I believe the folks at RiskMetrics have published some papers on the whole process.

Yes, however, this requires assumptions and you would introduce a certain degree of error. This is because option prices - as all asset prices - are the result of supply and demand. Without market prices, you are missing an important piece of information, which is captured in the Black Scholes model by volatility (aka implied volatility when inferred from market prices).

In your case, where option prices are not an option ;), you need to find a replacement for implied volatility. You could look into GARCH models to forecast volatility on your underlying price (return) data. Together with the Black Scholes model this could give you an idea on the corresponding option price. Please note, from empirical data we know that this volatility forecast is usually different from the actual implied volatility, hence, you are introducing an error.

Lastly, you have to make some assumptions on your option strategy, e.g. do you roll a protective put every 3 month, etc.?

You could also have a look at option replication strategies, e.g. "delta-replicated put" (see e.g. this paper).