# 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?

• Do you have some information about the distribution of the risk factor $S$? Is it observed? In any case, the idea is to simulate this risk factor $S$, say in 1000 scenarios, and recompute $V$ for these scenario values of $S$. This will give you a distribution of PnL for $V$, on which you can just take the 99%-quantile for a VaR 99% let's say. – JejeBelfort May 22 '19 at 7:44
• Not really. To see what is happening in your case, you should plot $V$ and see that its first and second derivatives are indeed flat at $S=0$ (flat inflexion point). Therefore, any small moves of $S$ around $10$ will not change $V$ at the first and second order! Your portfolio is actually delta-gamma hedged at $S = 10$. – JejeBelfort May 22 '19 at 8:00