Exact One-Year Value at Risk Under Geometric Brownian Motion
Summary
The document compares two ways to calculate the value at risk of a long stock position over one year, assuming geometric Brownian motion with no drift. It contrasts a volatility-based approximation, which scales daily volatility by the square root of time, with Monte Carlo simulation that estimates a lower-tail stock price and converts it into a position loss or gain.
The explanation gives the exact GBM quantile formula and shows that the simpler volatility formula is its short-horizon Taylor approximation. Because the approximation omits nonlinear effects in the lognormal price distribution, it can differ from the full calculation over a year. The document therefore explains why the two approaches need not agree and provides a closed-form benchmark against which a simulation can be compared. Its scope is limited to the stated no-drift GBM assumptions; it does not discuss parameter estimation, alternative return distributions, or practical calibration of volatility and the risk quantile.
Key ideas
- The square-root-of-time volatility formula approximates GBM value at risk for short horizons.
- The exact no-drift GBM calculation uses a lognormal price quantile.
- Monte Carlo estimates the terminal price distribution and derives the position loss from its lower tail.
- The approximation and exact formula can diverge over a one-year horizon.
Tags
Full text
# Calculating Value at Risk (VaR) of a Stock position assuming geomtric brownian motion (GBM)
# Calculating Value at Risk (VaR) of a Stock position assuming geomtric brownian motion (GBM)
I want to calculate the VaR for a long position (S) in stockprices after one year. Therefore i tried two methods:
- analytical solution: $VaR = S\cdot p_0\cdot \sigma_d \cdot \Phi^{-1}(1-\alpha)\cdot \sqrt{252}$
- 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:
- are those methods correct?
- 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?
- I was wondering why the analytical VaR-solution is symmetric concerning upside/downside risk although price levels are lognormal-distributed?
## Answer by Antoine Conze (score 1, accepted)
https://quant.stackexchange.com/a/38284
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 $$ \text{VaR} \approx S p_0 \sigma_d \sqrt{T} \Phi^{-1}(1-\alpha) $$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.