Simulating Option Returns for Portfolio Optimization
Summary
The document explains why option returns cannot generally be modeled as a fixed fraction of the underlying asset’s returns. Options have nonlinear payoffs, so a more realistic scenario series requires repricing the option under each simulated underlying price, using a Black–Scholes-like model. Portfolio optimization can then use the resulting return scenarios, though the nonlinear objective calls for numerical optimization rather than simple linear algebra.
The discussion also highlights volatility as a simulation input. When historical option prices or implied volatility are unavailable, a volatility forecast such as one from a GARCH model can serve as an estimate, but it may differ from market implied volatility and introduce error. Results also depend on the strategy assumptions, including how and when positions are rolled. Option replication approaches are mentioned as another avenue, but the document provides no worked example or empirical comparison.
Key ideas
- A constant-delta approximation maps option returns to underlying returns but misses nonlinear payoff behavior.
- Repricing options across simulated underlying price scenarios can represent their asymmetric returns.
- Nonlinear option exposures can make portfolio optimization require numerical methods.
- Volatility assumptions affect simulated option prices and may be forecast from underlying returns.
- A volatility forecast can differ from implied volatility, and the simulated strategy must specify its trading and rolling rules.
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# Options return series for portfolio optimization # 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? ## Answer by Brian B (score 1) https://quant.stackexchange.com/a/36891 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. ## Answer by CFW (score 1) https://quant.stackexchange.com/a/36895 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).
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