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Value at Risk (VaR): Analytical Derivation for Log-Normal Returns

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The Formal Theorem

Assume an asset price St S_t follows Geometric Brownian Motion, such that dSt=μStdt+σStdWt dS_t = \mu S_t dt + \sigma S_t dW_t . Let R=ln⁡(ST/S0) R = \ln(S_T / S_0) be the log-return over horizon T T . The α \alpha -level Value at Risk, defined as the loss threshold such that P(L>VaRα)=1−α P(L > VaR_{\alpha}) = 1 - \alpha , is given by:
VaRα=S0(1−exp⁡((μ−12σ2)T+σTΦ−1(1−α))) VaR_{\alpha} = S_0 \left( 1 - \exp\left( (\mu - \frac{1}{2}\sigma^2)T + \sigma \sqrt{T} \Phi^{-1}(1-\alpha) \right) \right)
where Φ−1 \Phi^{-1} is the quantile function of the standard normal distribution, μ \mu is the drift, and σ \sigma is the volatility.

Analytical Intuition.

Imagine you are standing at the edge of a turbulent ocean, where the asset price St S_t represents a vessel tossed by the waves of market volatility. Unlike simple linear models that assume prices move in straight lines, the log-normal framework recognizes that prices are multiplicative—gains and losses compound over time. We model the log-returns as a Normal distribution N((μ−12σ2)T,σ2T) N((\mu - \frac{1}{2}\sigma^2)T, \sigma^2 T) , capturing the 'drift' towards growth and the 'volatility drag' caused by variance. Value at Risk VaRα VaR_{\alpha} acts as our sentinel at the tail of this distribution. It translates a specific probability α \alpha —our confidence level—into a concrete monetary loss. We are effectively projecting the probability density into the future, identifying the exact threshold where, with 1−α 1-\alpha certainty, the ship will not sink deeper into the red. It is the mathematical boundary of our risk appetite, calculated by tracing the curve of the Gaussian bell back to the worst-case scenarios allowed by our confidence threshold.
CAUTION

Institutional Warning.

Students frequently conflate the normal distribution of log-returns with the distribution of absolute prices. Remember, the price ST S_T is log-normal, meaning it is bounded by zero and right-skewed, whereas the log-returns are symmetric and unbounded.

Academic Inquiries.

01

Why is the term −12σ2 -\frac{1}{2}\sigma^2 included in the drift?

This is the 'convexity adjustment' (or Ito correction). It arises from Ito's Lemma when transforming the dynamics from price St S_t to log-price ln⁡(St) \ln(S_t) , accounting for the skewness inherent in multiplicative processes.

02

How does increasing the time horizon T T affect VaR?

VaR scales primarily with the square root of time T \sqrt{T} due to the diffusive nature of Brownian motion, meaning risk increases significantly as the horizon expands.

Standardized References.

  • Definitive Institutional SourceHull, J. C., Options, Futures, and Other Derivatives

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Institutional Citation

Reference this proof in your academic research or publications.

NICEFA Visual Mathematics. (2026). Value at Risk (VaR): Analytical Derivation for Log-Normal Returns: Visual Proof & Intuition. Retrieved from https://www.nicefa.org/library/advanced-stochastic-processes/value-at-risk--var---analytical-derivation-for-log-normal-returns

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