Vega (Options)

Vega, in options trading, is the sensitivity of an option's price to a change in implied volatility. It is one of the option Greeks and is quoted as the change in the option's value for a one percentage point change in volatility, with the underlying price, time and interest rates held constant. Despite its name, vega is not a letter of the Greek alphabet; some texts use the letter kappa for the same quantity.

Vega is positive for every bought option, whether a call option or a put option, because greater expected movement in the underlying raises the chance of a large payoff while the loss remains limited to the premium. A trader who owns options is said to be long vega and gains when implied volatility rises. A trader who has sold options is short vega.

In the Black-Scholes-Merton model for a European option on an asset paying no dividends, vega is the same for a call and a put with the same terms:

• $S$ is the price of the underlying asset.
• $T$ is the time to expiry in years.
• $N'(d_1)$ is the standard normal density evaluated at the model term $d_1$.

The formula gives the price change for a change of 1.00 in volatility, that is, 100 percentage points, so the result is divided by 100 for the usual quote.

Take a one-year option with a share price of $100, a strike of $100, an interest rate of 5 percent and volatility of 20 percent. Then $d_1 = 0.35$ and $N'(0.35) = 0.3752$. Vega is $100 \times 1 \times 0.3752 = 37.52$, or about 0.375 per percentage point. The call is worth $10.45. If implied volatility rises from 20 to 22 percent, its value increases by roughly $2 \times 0.375 = 0.75$ dollars, to about $11.20. The put with the same strike gains the same amount.

• Time to expiry. Vega grows with the square root of the time remaining. Long-dated options are far more sensitive to volatility than short-dated ones. An option with a week to run has little vega.
• Moneyness. Vega is greatest for an at-the-money option and declines as the strike moves away from the current price in either direction.
• Price level. Vega scales with the price of the underlying, so options on higher-priced assets show larger figures in currency terms.

These patterns contrast with gamma, which is largest for short-dated at-the-money options. A long-dated option is chiefly an exposure to implied volatility, and a short-dated one is chiefly an exposure to actual movement in the underlying. A calendar spread, which sells a near expiry and buys a far one, is long vega for this reason.

Option prices can change substantially with no move in the underlying, purely because the market's assessment of future volatility shifts. Vega measures that exposure. A common illustration is an earnings announcement: implied volatility rises beforehand and falls sharply afterwards, and the holder of a straddle can lose money on the fall in volatility even if the share moves.

Vega cannot be hedged with the underlying asset, which has none. It is hedged with other options. A market maker who has sold options to clients will buy others to bring net vega toward zero, while using the underlying for delta hedging. Traders who want exposure to volatility itself build positions that are neutral to price but long or short vega, the basis of much volatility trading.

A single vega number assumes that implied volatility moves by the same amount for every option. In practice volatility differs by strike and expiry, forming a volatility surface, and the parts move differently. Short-dated volatility is much more variable than long-dated volatility. Risk systems therefore report vega in buckets by expiry and strike, and often weight each expiry by how much its volatility typically moves, so that a point of one-month vega and a point of two-year vega are not treated as equivalent.

Changes in the shape of the surface matter as well. A position can have zero net vega and still lose if the volatility skew steepens or the term structure shifts.

Vega itself changes as market conditions change. Its sensitivity to volatility is called vomma, and its sensitivity to the underlying price is vanna. These second-order measures become important for options far from the money and for large moves.

Vega is a model quantity. The Black-Scholes-Merton model assumes volatility is constant, so a sensitivity to changes in volatility sits awkwardly within it, and models that treat volatility as random give somewhat different figures. Vega is also a local measure, accurate for small changes. In a market shock, implied volatility can jump by many points at once while the underlying falls, and the loss on a short-vega position can be far larger than the vega figure alone suggests. The other first-order Greeks, including theta, should be read alongside it.