Parametric VaR for Portfolios Combining Stocks and Options

Summary

The document asks how to calculate parametric value at risk for a portfolio containing a stock and an option written on that stock. It proposes applying a covariance-based portfolio formula to stock and option weights, then questions whether that approach is appropriate because the option’s delta does not appear explicitly. It also asks how to extend the calculation to several stock–option pairs.

The issue is the relationship between an option’s nonlinear value and the risk measure’s representation of portfolio changes. The document supplies no derivation or comparison of methods, and its proposed formula is presented as a question rather than a confirmed procedure. It therefore identifies a modeling concern but does not resolve choices such as how to represent option sensitivity, what approximation to use, or how to handle interactions across multiple pairs.

Key ideas

• The question concerns parametric VaR for a portfolio with a stock and an option on that stock.
• The proposed covariance formula prompts concern because option delta is not explicit in it.
• Extending the calculation to multiple stock–option pairs is also raised.
• No derivation, validated method, or empirical VaR comparison is provided.

Tags

# Parametric VaR of a portfolio of a stock and an option on that stock


# Parametric VaR of a portfolio of a stock and an option on that stock












I understand how to calculate the parametric VaR of a stock and an option separately. But I don't understand how one can calculate the VaR of a portfolio of a stock and an option on that stock using the parametric method.

Would the method for multiple stocks be correct here? i.e.,

$ VaR(portfolio) = -\sqrt{w^{T}\Sigma w} \times z \times V, $

where $w$ = weights, $\Sigma$ = covariance matrix for stock and option, $z$ = $2.327$ for $1\%$ $VaR$ and $V$ = portfolio value.

This doesn't seem correct to me as the delta of the option is not involved...

Additionally, how do you extend this to have multiple stock-option pairs in the same portfolio?

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