Limits of VIX and Black–Scholes for Historical Option Pricing
Summary
The document considers how to estimate historical S&P 500 option prices for backtests, especially for options far in or out of the money. It asks whether public volatility measures such as VIX, SKEW, and VVIX can improve on using a single volatility input in the Black–Scholes model. The responses explain that a single volatility assumption is poorly suited to representing the full implied volatility smile, so deep in- and out-of-the-money prices may be inaccurate.
One response cautions that VIX is not a direct substitute for an option-specific implied volatility, since VIX itself is influenced by derivatives trading. It suggests using historical option prices to calculate implied volatilities and build a volatility curve. The other response notes that smile corrections require a model beyond standard Black–Scholes assumptions. The discussion gives no tested pricing method or comparison of accuracy, and it does not explain how to construct a curve from SKEW or VVIX. Its practical recommendation depends on access to historical option data.
Key ideas
- A single volatility input may not represent prices across different option strikes.
- VIX is shaped by derivatives markets and is not a complete measure of each option’s implied volatility.
- Historical option prices can be used to calculate implied volatilities and estimate a volatility curve.
- Modeling the smile requires assumptions beyond standard Black–Scholes.
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Full text
# How to calculate the most realistic historical option prices with additional publicly available parameters # How to calculate the most realistic historical option prices with additional publicly available parameters This is a follow up question of this one. My aim is to create the most realistic historical option prices possible with publicly available data. I want to do this for backtesting purposes. The following paper gives a good master plan how to backtest option strategies with the standard Black Scholes formula. If you use the publicly available implied volatility indices (like VIX) for the vol parameter the results are pretty good for ATM options: How Students Can Backtest Madoff’s Claims by Michael J. Stutzer (2009) Problems arise when you want to backtest strategies with (deep) ITM or OTM options. My question is: How can you produce even better historical option prices (for S&P 500 index options) with corrections for the smile with other publicly available data like SKEW and VVIX? ## Answer by Piroinno (score 5, accepted) https://quant.stackexchange.com/a/3213 I suggest you avoid using the VIX for implied vols. Why? One has to consider that the VIX is not simply solely dependant on the dynamics on the S&P 500 anymore because the VIX can be traded via options, etc. Thus many more parameters affect the trajectory of the VIX. The VIX has to equal the ATM option vol because this is where arbitrage assumption manifest; the price of a derivative on expiry has to be equal the spot price on the same date. Since you are back-testing, why can't you get the historical option prices and then calculate the implied vols (generate your volatility curve using inv) for your calculation? ## Answer by tagoma (score -1) https://quant.stackexchange.com/a/3211 Volatility is one of the inputs in the original Black-Scholes formula. If the VIX index tells you much on the volatility to plug into your pricer for the estimation of the value of the options on the S&P500. Good for you! Now, one has to recall that Black-Scholes world assumes log-normality of the underlying as well as constant volatility. From that, it is clear you have to resort on an extended version of the original Black-Scholes framework for the corrections you wish. Now, one has to recall that Black-Scholes world assumes log-normality of the underlying as well as constant volatility. From that, it is clear you have to resort on an extended version of the original Black-Scholes framework for the corrections you wish.
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