I would like to estimate Value-at-Risk analytically and through delta-gamma aproximation. I don't know if my idea is ok, but i would like to build a portfolio of European option. Suppose that in this moment ($t=0$) investor buying a option with strike price $K=55$, current price of derivative is $S_0=50\$$, $T=1$ (year), $r=5\%$ and $\sigma=30%$. I use Black-Scholes formulas so I can compute $$S_T=S_0 \exp \left( \left(r-\frac{1}{2}\sigma^2 \right)t+ \sigma\sqrt{T}Z \right),$$ where $Z \sim \mathcal{N}(0,1)$. Any ideas what next? In my opinion, I should use Black-Scholes formula to calculate the price of European call option, so the loss is $$L=-\Delta V=-(e^{-rT}C_T -C_0 ).$$ But there is $d_1$ and $d_2$ and I don't know which price and time I should use.
Maybe someone explain it to me?
Here is a link that you can refer to caculate VAR under different methods. While I tried to answer this question, I clarified my doubts. I decided I shall direct you to this site than answer the question directly. Seems like a reasonable start for Delta Normal, Delta Gamma. Nothing complicated. They have given the excel sheet and plough through it to get nitty gritty details on the implementation on a portfolio of stocks and options.
http://quantitative-finance-by-examples.com/Value-at-Risk-Model-II.html
Goodluck