Computing Value-at-Risk for an Option Position
You hold a single option (say a vanilla call) on an underlying asset. Explain how you would compute the Value-at-Risk (VaR) of this position over a short horizon. Discuss the simple delta-based approximation, when it breaks down, and how you would improve it.
Hints
- VaR is a quantile of the loss distribution. For a nonlinear instrument, the loss is not a simple linear scaling of the underlying's return.
- First-order: approximate P&L by $\Delta \cdot dS$ to get delta-normal VaR. When does the missing curvature matter?
- Add the gamma term (and vega) for a delta-gamma approximation; for full accuracy, Monte-Carlo simulate scenarios and reprice the option, then read the $\alpha$ quantile.
Worked Solution
How to Think About It: VaR asks: over the chosen horizon and confidence level, what loss will I not exceed except with small probability? For a linear position you would just scale the underlying's return distribution. The complication with an option is that its payoff is nonlinear in the underlying, so a pure linear (delta) approximation understates risk exactly where it matters -- in the tails and around the strike where gamma is large.
Method 1 -- Delta-normal (linear) VaR: Approximate the option's P&L by its first-order sensitivity. If the underlying $S$ has a return with volatility $\sigma_S$ over the horizon, the option value change is approximately $\Delta \cdot dS$, where $\Delta = \partial V / \partial S$. Then the position's P&L standard deviation is about $|\Delta| \cdot S \cdot \sigma_S$, and the
$\text{VaR}_{\alpha} \approx z_{\alpha} \cdot |\Delta| \cdot S \cdot \sigma_S,$
where $z_{\alpha}$ is the standard-normal quantile (e.g.
Method 2 -- Delta-gamma (quadratic) VaR: Add the second-order term. The P&L is approximated by
$dV \approx \Delta \, dS + \tfrac{1}{2} \Gamma \, (dS)^{2} + \Theta \, dt + \mathcal{V} \, d\sigma,$
with $\Gamma = \partial^{2} V / \partial S^{2}$, $\Theta$ the time decay, and vega $\mathcal{V}$ capturing volatility risk. The gamma term makes the P&L distribution skewed and non-normal, so you either propagate moments through a Cornish-Fisher expansion or simulate.
Method 3 -- Full revaluation / Monte Carlo (most accurate): Simulate many scenarios for $(S, \sigma)$ over the horizon, fully reprice the option (Black-Scholes or the relevant model) under each scenario, build the empirical P&L distribution, and read off the loss at the $\alpha$ quantile. This captures the true nonlinearity and any volatility moves, at the cost of compute. Historical simulation (replaying actual past underlying/vol moves through the pricer) is a non-parametric variant.
Practical Considerations: For a deep-in- or deep-out-of-the-money option, delta-normal can be roughly fine because gamma is small; near the strike or near expiry, gamma is large and you must use delta-gamma or full revaluation. Long-option positions have positive gamma (limited downside, so VaR is overstated by the linear approximation); short-option positions have negative gamma (the dangerous case -- linear VaR understates the true tail loss). Do not forget vega: a vol spike can move the option more than the underlying does.
Answer: Start with delta-normal VaR $\approx z_{\alpha} |\Delta| S \sigma_S$; upgrade to delta-gamma when gamma is non-trivial (near the strike / near expiry); for an accurate number simulate or fully reprice the option across scenarios and take the $\alpha$ quantile of the P&L distribution. Always account for vega.
Intuition
The whole point of this question is option convexity. A linear (delta-only) risk measure treats the option like a fixed amount of stock, which is exactly wrong precisely when risk is highest: a short-gamma position bleeds far more than its delta suggests when the underlying moves sharply, so delta-normal VaR dangerously understates the tail. This is why risk desks layer in gamma, vega, and ultimately full revaluation. The practical hierarchy -- delta, then delta-gamma, then full Monte-Carlo -- mirrors the accuracy-versus-speed tradeoff every risk system manages, and knowing when each is good enough (deep ITM/OTM versus at-the-money near expiry) is the mark of someone who has actually run these numbers.