Vega: Volatility Sensitivity
How option prices respond to changes in implied vol · 15 min
The Volatility Greek
Vega (ν) measures how much an option price changes for a one percentage-point change in implied volatility. It is in some ways the most important Greek for professional options traders, because professional options trading is largely about trading volatility — not direction.
ν = ∂V / ∂σ
An ATM one-year call with ν = 0.40 gains $0.40 in value for every 1% rise in implied vol. If IV moves from 20% to 25%, the call gains approximately $2.00 from vega alone, even if the stock doesn't move.
Note: despite the Greek letter, "vega" is not actually a Greek letter — it was invented by traders. In academic literature it sometimes appears as κ (kappa) or λ (lambda). The name stuck because it sounds Greek-ish and starts with 'v' for volatility.
The Formula
ν = S · n(d₁) · √T / 100
where n(d₁) is the standard normal PDF. The formula is identical for calls and puts — a call and put with the same strike and expiry have the same vega. This again follows from put-call parity: C − P = S − PV(K), and differentiating both sides with respect to σ gives ∂C/∂σ = ∂P/∂σ.
The division by 100 converts from "per unit vol" to "per percentage point." Some books define vega without the /100, which means the answer is 100x larger — always check which convention is being used.
Vega Is Largest for ATM Long-Dated Options
Two factors drive vega:
- Moneyness: Vega peaks at the ATM strike and falls off for ITM and OTM options. Deep OTM options are nearly binary (they'll either be worth a lot or nothing), so an incremental vol change doesn't matter much. Deep ITM options will likely be exercised regardless.
- Time to expiry: Vega scales with √T. A 1-year ATM option has roughly √(12) ≈ 3.46× the vega of a 1-month ATM option. Longer-dated options have more time for vol to matter, making them much more sensitive to vol changes.
This means long-dated options are "vol products" more than "directional products." A 2-year LEAPS option might have vega of $0.80 — a 5 vol-point move shifts its price by $4, far more than most delta-driven moves over the same period.
Implied Volatility: The Market's Forecast
Black-Scholes takes σ as an input. In reality, σ is not directly observable. What we observe is the market price of an option. Traders invert the formula: given the market price, what σ makes Black-Scholes match it? That σ is called implied volatility (IV).
Implied vol is not a prediction of actual future volatility — it's the market's consensus price for uncertainty. Historically, IV tends to exceed realized volatility by 1–3 percentage points on average. This gap — called the variance risk premium — is why option selling (collecting IV premium) has historically been profitable on average, at the cost of occasional large losses when volatility spikes.
The Volatility Surface: Where Black-Scholes Breaks Down
If Black-Scholes were literally true, every option on the same stock and expiry would have the same implied vol. In practice, they don't. Plotting IV against strike gives the famous volatility smile (or skew).
Equity skew: For equity index options (SPX, SPY), OTM puts almost always trade at higher IV than ATM options, which trade at higher IV than OTM calls. The shape is asymmetric — a skew rather than a smile. This reflects:
- Crash risk demand: Investors pay a premium for put protection (portfolio insurance). High demand for OTM puts drives their IV up.
- Leverage effect: When stocks fall, companies become more levered (debt stays constant, equity falls), increasing volatility. Crashes historically come with vol spikes.
- Stochastic volatility: Real volatility is not constant — it varies, and it's negatively correlated with stock returns.
FX smile: Currency options often show a true smile — both OTM puts and calls trade at elevated IV relative to ATM. This reflects the possibility of large moves in either direction (currencies can appreciate or depreciate sharply).
VIX: The Fear Index
The CBOE Volatility Index (VIX) is perhaps the most important single number in financial markets. It measures the market's expectation for S&P 500 volatility over the next 30 days, derived from a portfolio of SPX options across many strikes.
VIX is expressed as an annualized volatility percentage. A VIX of 20 means the market expects roughly 20% annualized volatility, which translates to daily moves of about 20%/√252 ≈ 1.26% per day.
Historical VIX context:
- 2017: VIX averaged 11.1 — the calmest year in recorded history. Options were historically cheap.
- 2008 (Lehman crisis): VIX hit 89.5 in October 2008. Options became extraordinarily expensive.
- March 2020 (COVID crash): VIX peaked at 85.5 on March 18, 2020 — the second-highest reading ever.
- Normal range: 12–25 in calm markets, 25–40 in stressed markets, 40+ in crises.
VIX is often called the "fear index" because it spikes when investors are buying protective puts aggressively. A rising VIX means the options market is pricing in more uncertainty — regardless of whether the stock market itself is up or down.
Vega in Portfolio Management
A large options portfolio has a net vega — the sensitivity of its total value to a parallel shift in the vol surface. A portfolio that is long vega gains when IV rises (benefits from fear/uncertainty). A portfolio that is short vega — like a fund that systematically sells options for premium — gains when IV falls and loses when it spikes.
The 2018 "Volmageddon" event illustrates the risk. Several funds were short VIX (short vega) using instruments tied to VIX futures. On February 5, 2018, VIX doubled in a single day — an unprecedented move. One product (XIV) lost 96% of its value overnight and was liquidated. Understanding vega, and the risk of sudden vol spikes, is not academic — it is survival.