The theoretical estimation accuracy of VaR for portfolios of options is higher when using the quadratic model instead of the linear model. True or false
The correct answer and explanation is:
Correct Answer: True
Explanation:
The theoretical estimation accuracy of Value at Risk (VaR) for portfolios containing options is indeed higher when using the quadratic model compared to the linear model. This is because options and other derivatives exhibit nonlinear price behavior in response to changes in underlying asset prices, and the linear model fails to capture this complexity adequately.
Linear VaR Model:
In the linear model, VaR is estimated assuming a linear relationship between the value of the portfolio and the changes in underlying risk factors (like asset prices or interest rates). This works well for portfolios composed primarily of linear instruments, such as stocks or bonds. However, options have nonlinear payoffs (due to convexity and time decay), so linear models can lead to misestimation of potential losses.
Quadratic VaR Model:
The quadratic model, also known as delta-gamma approximation, incorporates not only the first-order sensitivity (delta) but also the second-order sensitivity (gamma) of the option portfolio. Delta measures how the portfolio’s value changes with small changes in the underlying asset, while gamma accounts for the curvature, or the rate at which delta itself changes.
This model provides a more accurate representation of the potential losses, especially in volatile markets where large movements in the underlying can occur. Because it accounts for the nonlinearity in the option price changes, the quadratic model leads to better theoretical estimation accuracy of VaR for portfolios that include options.
Conclusion:
Therefore, the statement is True. When valuing a portfolio that includes options, the quadratic model provides a more realistic and accurate VaR estimate than the linear model because it captures both linear and nonlinear risk exposures.