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?

  • $\begingroup$ 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. $\endgroup$ – JejeBelfort May 22 '19 at 7:44
  • $\begingroup$ I am told to assume the risk factor's return follows a normal distribution with annual volatility = 0.3 $\endgroup$ – Cheuk Kwan LAW May 22 '19 at 7:45
  • $\begingroup$ Thanks!!! But is there any features about V(S) itself that makes DG/DGD methods fail? or is it simply because the computed delta and gamma = 0? Also, there's no way to give an approximated VaR without using a computer, right? $\endgroup$ – Cheuk Kwan LAW May 22 '19 at 7:50
  • $\begingroup$ 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 $S = 10$. $\endgroup$ – JejeBelfort May 22 '19 at 8:00
  • $\begingroup$ If this helps, I can formulate a proper answer. Let me know $\endgroup$ – JejeBelfort May 22 '19 at 9:59

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