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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?
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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 $$ \text{VaR} \approx S p_0 \sigma_d \sqrt{T} \Phi^{-1}(1-\alpha) $$

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