Vega

Definition

Vega is the rate of change of an option’s value with respect to a change in implied volatility — dollars gained or lost per one volatility-point move in IV, everything else unchanged. Long options (calls and puts alike) are long vega; short options are short vega. Vega is the exposure that makes options a volatility instrument, not just a directional one.

How it works / structure

• Formula (Black-Scholes-Merton): vega = S·φ(d1)·√T — identical for calls and puts at the same strike/expiry.
• Shape: largest at the money and grows with time to expiry — long-dated options are volatility instruments first, direction instruments second; short-dated options have little vega but much gamma (the structural tradeoff behind strategy-calendar-spread).
• Surface reality: the model assumes one σ; markets trade a surface (opt-volatility-skew, opt-term-structure), so real books measure vega per expiry bucket (term-structure exposure) rather than one aggregate number.
• Position aggregation: net vega by bucket (port-portfolio-greeks); calendar structures are long back-month vega against short front-month vega by design.
• Simulation parameters: repricing under the engine’s IV paths produces vega P&L; stress runs shock the surface (parallel and term-twisted) to expose vega concentration.

When it applies

Any thesis about volatility itself (IV rich vs cheap against realized), event positioning (IV builds into earnings and crushes after — event-earnings), structure selection (calendar vs vertical is largely a vega decision), and risk assessment of any options book (a “directionally hedged” book can still be one big volatility position).

Risk profile & failure modes

• Vol-of-vol: IV can reprice several points in minutes around events; vega exposure realizes faster than direction exposure.
• Correlated stress: IV spikes when markets fall — short vega and long equity are the same trade in stress, defeating diversification accounting that treats them separately.
• Bucket blindness: flat net vega across expiries can hide a steepener/flattener exposure to the term structure.
• Sticky-strike vs sticky-delta: how the surface moves when spot moves is itself a modeling choice; vega P&L attribution differs between conventions.

Evidence & limits

Vega’s mathematics are model-defined (Black-Scholes-Merton; Hull is the standard text treatment). The empirical behavior of IV — mean reversion tendencies, spike asymmetry, premium over realized — is covered with citations in opt-implied-volatility and regime-volatility; this entry makes no empirical volatility claims.

Falsifiable-thesis examples

Illustrations only, not signals:

• “A long 90-day at-the-money straddle on X will gain at least $V per contract if X’s 90-day IV rises 5 points within two weeks, independent of a spot move up to ±1%” — falsified by repricing under those conditions.
• “Y’s front-month IV will fall by more than its back-month IV in the week after earnings (term-structure normalization)” — falsified by the two IV series.

Cross-references

• The quantity it prices: opt-implied-volatility; surface shape: opt-volatility-skew, opt-term-structure
• Companions: greek-theta, greek-gamma
• Strategy consumers: strategy-straddle, strategy-strangle, strategy-calendar-spread, strategy-diagonal-spread
• Aggregation: port-portfolio-greeks