The Greeks and Option Risk Management › Vega: Sensitivity to Volatility

Vega ($\nu$ or sometimes $\mathcal{V}$) measures the sensitivity of an option's price to changes in implied volatility. While delta captures exposure to price movements that are directly observable in the market, vega captures exposure to volatility, which is the market's expectation of future price variability. This distinction matters because volatility is not directly observable; it must be estimated or implied from option prices themselves.

Mathematically, vega is defined as:

$$\nu = \frac{\partial V}{\partial \sigma}$$

where:

• $\nu$: option vega
• $V$: option value
• $\sigma$: implied volatility

Despite being called a "Greek," vega is not a Greek letter; it is the name of a star. The term has become standard in options markets.

Deriving vega requires differentiating the Black-Scholes formula with respect to volatility $\sigma$. Like theta, volatility appears in multiple places in the formula, including in both $d_1$ and $d_2$ as well as in the scaling of the distribution. After working through the calculus, a remarkably clean result emerges.

For both European calls and puts:

$$\nu = S_0 \sqrt{T} N'(d_1) = S_0 \sqrt{T} \frac{e^{-d_1^2/2}}{\sqrt{2\pi}}$$

where:

• $\nu$: option vega
• $S_0$: underlying price
• $K$: strike price
• $r$: risk-free interest rate
• $\sigma$: volatility
• $T$: time to expiration
• $N'(d_1)$: standard normal PDF
• $d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}$: standardized moneyness factor

Like gamma, vega is identical for calls and puts with the same parameters, which follows from put-call parity (the right side of the parity relationship doesn't depend on volatility). This symmetry makes intuitive sense: both calls and puts benefit equally from increased uncertainty about future prices, because both options benefit from large moves in their favorable direction.

Vega captures exposure to volatility, which is arguably the most important risk factor in options trading. While stock prices are observable in real time, volatility must be estimated or implied from option prices. This inherent uncertainty about the correct volatility makes volatility risk central to options management. We often say that when you trade options, you are really trading volatility, not direction.

The key properties of vega include:

• Always positive for long options: Higher volatility increases the value of both calls and puts because it increases the probability of large moves, which benefit option holders. Whether you hold a call hoping for a big move up or a put hoping for a big move down, more volatility increases your chances of success.

• Maximum at-the-money: ATM options have the highest vega because their value is most sensitive to the probability of ending up in the money, which volatility directly affects. An at-the-money option is balanced between having value and being worthless, so changes in the expected distribution of future prices matter most.

• Increases with time: Unlike gamma (which increases near expiration), vega increases with time to expiration because there's more time for volatility to have an effect. Over a longer horizon, volatility has more opportunity to compound and produce large cumulative price movements.

Our vega implementation calculates the sensitivity to volatility. Note that while input volatility is typically a decimal (e.g., 0.20), we often speak of vega per 1% change in vol, which requires dividing the result by 100: