Option Value at Risk (VaR)

For educational purposes only; not investment advice.

Direct answer

Option Value at Risk (VaR) is a modeled loss threshold for a specified portfolio, horizon, confidence level, data window, and valuation method. If loss is L = -P&L, then:

VaR_alpha = inf { l : P(L <= l) >= alpha }

A one-day 99% VaR of $20,000 means the model places 99% of one-day outcomes at or below a $20,000 loss. It does not cap loss, describe the severity beyond the threshold, or promise exactly one exceedance per 100 trading days.

How option VaR is built

Fix the position snapshot and executable marking convention. Generate joint moves in underlying prices, volatility level and skew by expiry, rates, dividends, correlation, and relevant basis factors. Advance time by the stated horizon, fully reprice every option, add stock and cash P&L, then convert portfolio P&L into losses and take the chosen empirical quantile.

Three approaches have different limitations:

Delta-normal: maps risk factors into an assumed distribution using local sensitivities. It is fast but can miss Gamma, Vega, skew, jumps, and changing sensitivities.
Historical simulation: applies observed joint factor moves and fully reprices. It preserves sampled relationships but cannot contain an event absent from the window.
Monte Carlo: simulates many joint paths and fully reprices. It can model nonlinear interactions, but results depend on distribution, dynamics, calibration, and scenario design.

State whether positions are held constant or dynamically hedged. Do not mechanically multiply a short-horizon VaR by the square root of time for nonlinear, path-dependent, illiquid, or event-exposed options. Backtest realized exceptions, investigate clusters, and compare VaR with expected shortfall and independent stress tests.

Example

A $2,000,000 option portfolio is fully repriced under 10,000 one-day scenarios. After sorting losses from smallest to largest, the selected 99% quantile is $20,000; approximately 100 modeled scenarios are worse. Their average loss is $46,000, and the worst modeled loss is $75,000.

The reported one-day 99% VaR is $20,000, not $46,000 or $75,000. Those additional figures reveal the tail that VaR omits. If an actual day loses $31,000, it is a VaR exception; repeated or clustered exceptions can indicate stale calibration, missing risk factors, structural change, or poor executable-price assumptions.

Risks and limitations

VaR says nothing about loss magnitude beyond its quantile; report expected shortfall and severe scenarios too.
Short options, barriers, near-expiry positions, and concentrated Gamma can change sharply between sampled points.
Historical windows can exclude crashes or overweight one regime; Monte Carlo can impose the wrong tails or correlations.
End-of-day model marks can understate exit losses when spreads widen or market depth disappears.
Exercise, assignment, margin, funding, and settlement cash needs are not automatically included.
Comparing VaR numbers is invalid unless horizon, confidence, portfolio, valuation, data, and weighting conventions match.

Common misconceptions

“99% VaR is the maximum loss.” It is a quantile; losses beyond it may be much larger.

“A higher confidence level makes the model accurate.” It moves deeper into a tail where observations are scarcer and model uncertainty may be greater.

“Backtesting success proves future validity.” It tests past calibration under a particular period and exception definition, not future regimes.