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Choosing a VaR Method for Portfolios with Nonlinear Options

Article Quant Q&A · Author: Etomna
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Summary

The document explains how an out-of-the-money put affects variance-covariance Value at Risk for a portfolio of stocks, bonds, an ETF, and an option. The method begins with market-factor volatilities and correlations, then combines factor sensitivities with the covariance matrix under a normal-distribution assumption. This approach is appropriate when the instruments behave approximately linearly over the chosen confidence interval. If a put remains so far out of the money that it has negligible value throughout that interval, it may be excluded; if it behaves like the underlying, it can be approximated by that underlying.

When the option’s value changes materially and nonlinearly within the risk interval, the document recommends repricing it across simulated scenarios, such as with Monte Carlo, or using historical VaR. It also mentions scenario-based historical shocks and stress scenarios, and suggests considering expected shortfall. These approaches have tradeoffs: simulation requires many option valuations, while Excel can be cumbersome. The normality and linearity assumptions limit the basic covariance method, and the discussion does not specify a complete implementation.

Key ideas

  • Variance-covariance VaR uses factor sensitivities, volatilities, and correlations under a distributional assumption.
  • The linear covariance method is suitable only when instruments are approximately linear over the risk interval.
  • An option with negligible value across that interval may be omitted from the VaR calculation.
  • Material option nonlinearity calls for scenario repricing, such as Monte Carlo, or historical VaR.
  • Historical shocks and stress scenarios offer additional ways to examine portfolio losses.

Tags

Risk managementOptionsPortfolio constructionStatistics
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# Value at risk (portfolio with stocks, bonds, and options)


# Value at risk (portfolio with stocks, bonds, and options)












I am building a VaR model (in Excel using the variance-covariance method) for a portfolio containing stocks, bonds, and an ETF. Additionally, there is a put option (out of the money) that is there to hedge the downside risk of my position in the ETF. The ETF tracks the MSCI index.

My question is, how do I implement the put option in my VaR model?

## Answer by Dimitri Vulis (score 2)

https://quant.stackexchange.com/a/75595

You have a matrix that for all your market factors contains their volatilities and pairwise correlations, probably calculated from historcal market data. You assume that they are normally distributed. You calculate VaR to some confidence interval, often 99%.

As long as all the instruments in your portfolio are linear within your chosen confidence interval, you can use matrix multiplication, which is easy in Excel: the vector of your factor sensitivities times the matrix times the transpose(vector of factor sensitivities) times normsinv(confidence interval) times sqrt(time horizon) and few other things.

In your example, if the option is so far our of the money that it is worthless on your confidence interval, you just assume that it's worthless and effectively exclude it from the VaR calculation. Likewise, if the option were very far in the money that you could assume that you hold the underlying, then you could indeed assume that you hold the underlying. But, conversely, if the option is out of the money now, but would be in the money if the underlying moved, e.g., 2 historical standard deviations, in other words, the value of the option is materially non-linear within your confidence interval, then you can't use matrix multiplication anymore. You have to use a Monte-Carlo simulation (MC). MC is described well, for example, in Value at Risk: The New Benchmark for Managing Financial Risk by Philippe Jorion, section 12 or Market Risk Analysis Volume IV: Value at Risk Models by Carol Alexander, section IV.1.9.3. For MC, you will need to estimate the change in the value of option under thousands of risk scenarions. This is likely to be computationally intensive, so you may want to look into shortcuts where you reprice the option under a few scenarios, and linearly iterpolate between them. Or, instead of a MC and covariance matrix, you can use "Historical VaR'.

While it's possible to implement MC or Historical VaR in Excel, it's really painful, and I urge you to use another tool more suitable for such calculations.

I also urge you to look at expected shortfall (ES), rather than VaR.

Another approach you might try instead of MC is not to assume that a small number of parameters summarizes the dynamics of your historical market data, not to use any covrance matrix, but rather to calculate how the value of your portfolio would change if the market data changed like it did in your historial data; and use the worst cases for VaR or ES.

You should also consider the impact of additional historical or hypothetical market stress scenarios.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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