Portfolio VaR When Delta and Gamma Vanish

Summary

The document poses a portfolio risk question about estimating value at risk when the portfolio value is a nonlinear cubic function of one normally distributed risk factor. At the stated current factor level, both the first and second derivatives are zero, so a local delta-gamma approximation appears uninformative. The function is also non-monotonic, prompting the question of whether that shape makes standard Taylor-based VaR methods unsuitable.

The author asks about practical alternatives but supplies no answer, calculation, simulation, or empirical evidence. As a result, the text serves mainly as a problem statement: it identifies a limitation of relying on local sensitivities at a point where they vanish, but does not compare full revaluation, simulation, or other VaR approaches. Any estimate would need to account for the factor’s distribution and the portfolio’s nonlinear payoff across possible factor values, rather than infer risk from the local derivatives alone.

Key ideas

• The example maps one risk factor to portfolio value through a nonlinear cubic function.
• At the stated current factor level, both delta and gamma are zero.
• The author questions whether non-monotonicity limits delta-gamma VaR methods.
• No alternative method or VaR estimate is provided in the document.

Tags

# Approximation of portfolio VaR (after mapping) when Delta and Gamma both equal zero


# Approximation of portfolio VaR (after mapping) when Delta and Gamma both equal zero












As titled, I am having trouble estimating the VaR of a portfolio mapped as a function of a single risk factor $S$, in the form :

$$V(S) = S^3 - 30S^2 + 300S + 150$$

with current value $S = 10$.

$S$ is supposed to be normally distributed, with mean $\mu = 10$ and annual volatility $\sigma= 0.3$.

I found that both delta and gamma evaluated at $S = 10$ are zero, then clearly I cannot implement any of Delta-Gamma method or Delta-Gamma-Delta method.

Are these methods not applicable on non-monotonic function?

Then what kinds of other practical methods can I use to estimate the portfolio VaR?

Can someone please briefly explain, please?

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