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Option VaR Requires Modeling Market Implied-Volatility Inputs

Article Quant Q&A · Author: Oscar
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Summary

The document addresses how to revalue an options portfolio in a Monte Carlo VaR simulation when the underlying is simulated with a historical stochastic-volatility model such as GARCH. Its central distinction is that historical volatility of the underlying and implied volatility used to price options are different quantities. Implied volatility is a model input that encodes the option’s market premium under a pricing convention; it need not track realized or GARCH volatility directly.

For revaluation, the relevant inputs are market observables and model inputs, including implied volatility and, where applicable, other features such as risk reversal. A robust historical simulation would need time series for these inputs and their dynamics, rather than automatically substituting terminal GARCH volatility or scaling current implied volatility by a GARCH ratio. The response does not prescribe a specific volatility-surface evolution or correlation model, so it leaves those implementation choices unresolved.

Key ideas

  • Historical volatility and option implied volatility represent different quantities and need not move together.
  • Implied volatility is a pricing input that conveys option-premium information under a chosen model.
  • Option revaluation should reflect the simulated behavior of market observables and relevant option-model inputs.
  • Other inputs, such as risk reversal, may matter when the pricing model represents more than a simple volatility level.
  • The document does not specify a complete implied-volatility or correlation simulation method.

Tags

OptionsVolatilityRisk managementBacktesting
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# Answer by Dimitri Vulis (score 2)


# How do you handle implied volatility performing a VaR Monte-Carlo simulation using a stochastic volatility process calibrated on the underlying












Say you have a portfolio consisting of options each having a market implied volatility. If you now use some stochastic volatility model like GARCH to calibrate the real world volatility of the underlying, and then perform simulations of the correlated stock processes using the GARCH stochastic volatility. What volatility would you use when revaluing the options at the end of the simulation? Some possibilities I imagine are:

1.The same IV used at the start to get the market prices of the options.

2.The stochastic GARCH volatility at the end of the period.

3.The Implied volatility multiplied by $\frac{\sigma_{2}^{GARCH}}{\sigma_{1}^{GARCH}}$, where $\sigma_{1}^{GARCH}$ and $\sigma_{2}^{GARCH}$ is the volatility at the start and end period respectively.

Does any of these options make sense? Is there a "right" answer?

As a bonus question, what correlation would you use for the simulation? Can you use simply the standard EWM correlation as you would with constant volatility, or is there an equivalent to GARCH volatility for correlation?

## Answer by Dimitri Vulis (score 2)

https://quant.stackexchange.com/a/53008

The historical volatilities of the market factors is not the same as the implied volatilty used to price the options. The "implied volatility" is just one of the model inputs. It does not need to be similar to the historical volatility of the underlying.

The mark to market of an option is the premium that one would have to pay in the market for this option. Sometimes you can just observe this premium in the market. Other times, there is a widely recognized, but simplistic model that explains the option premium as the function of (underlying, risk-free interest rate, implied vol). People can back out the implied vol from option premium, and quote the implied vol, which conveys exactly the same information as the option premium if the underlying and interest rates are observable. A more sophisticated model might have some structure for the underlyings and interest rates and also perhaps additional model inputs. For example, if you choose not to assume that the underlying is normally distributed, then you could have some way of quantifying this assumptions, such as risk reversal. Then, hopefully, you have historical time series for all these market observables and model inputs, and can use their historical volatilties.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.