Given that $dS_t=\mu S_tdt+\sigma S_tdW_t$ ,a risk free rate r and defining Value at Risk and Expected Shortfall as $VaR_{t,a}=S_0e^{rt}-x$ where $x$ is the amount such that $P(S_t\leq x)=1-a$ ($a:$confidence level) and $ES_{t,a}=S_0e^{rt}-E(S_t|S_t<x)$ I found
$$VaR_{t,a}=S_0e^{rt} - S_0e^{\sigma\sqrt{t}N^{-1}(1-a)+(\mu-\frac{\sigma^2}{2})t}$$ and $$ES_{t,a}=S_0e^{rt}-\frac{S_0e^{\mu t}N[N^{-1}(1-a)-\sigma \sqrt{t}]}{1-a}$$
I have two questions:
- A popular VaR formula is $S_0\sigma \sqrt{t}N^{-1}(1-a)$. Is this obtained by taking the Taylor expansion and ignoring any power of $t\geq 1$ as well as ignoring the time value of money? ($r=0$)
- Are my Expected Shortfall definition and formula correct? Thanks in advance