Calculating Value at Risk (VaR) of a Stock position assuming geomtric brownian motion (GBM)

Question

I want to calculate the VaR for a long position (S) in stockprices after one year. Therefore i tried two methods:

1. analytical solution: $VaR = S\cdot p_0\cdot \sigma_d \cdot \Phi^{-1}(1-\alpha)\cdot \sqrt{252}$

2. MC with geometric brownian motion:

   I. model stock price (assuming $\mu = 0$): $p_{t+1} = p_t + p_t\cdot \sigma \cdot dW_t$

   II. perfrom MC-Simulation of multiple price-series

   III. determine $\alpha$-Quantile of price-distribution for t = 252 ( $p^\alpha_{252}$)

   IV. $VaR = S\cdot (p^\alpha_{252} - p_0)$

with

• $S$: Stock - Position
• $\sigma_d$: volatility of daily returns
• $\alpha$: Risk-Quantile
• $p_t$: stock-Price
• $dW$: Wiener process

Now here are my questions:

1. are those methods correct?
2. If yes, i noticed that both methods lead to different results (VaR for GBM is higher). Why is that so? Which method should i use?
3. I was wondering why the analytical VaR-solution is symmetric concerning upside/downside risk although price levels are lognormal-distributed?

Answer 1

Your analytical formula is only an approximation of the GBM VaR for short maturities, hence the difference in numerical results between methods for a 1 year maturity. The correct analytical formula in the GBM case (with no drift) is $$ \text{VaR} = S p_0 \left(\exp\left(-\frac{\sigma_d^2}{2}T + \sigma_d \sqrt{T} \Phi^{-1}(1-\alpha)\right) -1 \right) $$ For short maturities $T$ a Taylor expansion yields $$ \text{VaR} \approx S p_0 \sigma_d \sqrt{T} \Phi^{-1}(1-\alpha) $$