Implied Volatility

Inverting the Formula

Black-Scholes takes five inputs — S, K, T, r, and σ — and produces an option price. But in real markets, you observe prices, not volatilities. Every day, options trade at market prices that are set by supply and demand. The question becomes: what value of σ makes the Black-Scholes formula match the observed market price?

That σ is called implied volatility (IV). It is the volatility "implied" by the market price. If an option is trading at $5.80 and Black-Scholes with σ = 0.20 produces $5.80, then the implied vol is 20%.

This inversion cannot be done analytically — there is no closed-form formula for σ given C. Instead, we use a numerical root-finding method. We want to find the zero of:

f(σ) = BS(S, K, T, r, σ) − C_market = 0

The two most common methods are bisection (reliable, slow) and Newton-Raphson (fast, occasionally brittle). Professionals use Newton-Raphson because it converges in 4–6 iterations for typical option inputs.

Newton-Raphson: The Fast Way

Newton-Raphson iteratively improves an estimate of the root using the derivative of the function:

σ_n+1 = σ_n − f(σ_n) / f′(σ_n)

Here f(σ) = BS_price(σ) − C_market, and f′(σ) = ∂BS/∂σ = vega (divided by 100, in the same units). Specifically:

σ_n+1 = σ_n − (BS(σ_n) − C_market) / Vega(σ_n)

where Vega is the raw vega (S · n(d₁) · √T), not the per-1% version we computed in Lesson 7. Starting from σ₀ = 0.20 (20%) and iterating 5–10 times typically gives convergence to 6+ decimal places for standard options.

The method can fail when vega is near zero (deep OTM or very short-dated options), so good implementations add a fallback to bisection when vega is too small.

IV as the Market's Price of Uncertainty

Implied vol is not a forecast of realized volatility — it is the market's price for uncertainty. Several things follow from this:

• IV ≠ realized vol: On average across history, IV has been about 1–3 percentage points above subsequently realized volatility. This gap is the variance risk premium — compensation for the risk of providing vol insurance. It's why systematic option selling has historically been profitable on average.
• IV can be wrong in both directions: In quiet markets, IV undershoots realized vol (options are cheap). Before major events (Fed decisions, earnings), IV overshoots — the premium for the known uncertainty.
• IV is forward-looking: Realized vol looks backward (what has happened). IV looks forward (what the market fears may happen). They can diverge dramatically around events.

The Volatility Surface

If Black-Scholes were literally true, every option on the same underlying and expiry would have the same IV. In practice, IVs vary across both strike and time to expiry, forming a two-dimensional volatility surface.

Term structure: IV typically increases with time to expiry (the market fears more as time horizon lengthens), but can invert during crises (short-dated IV spikes more than long-dated IV when fear is acute).

Skew / smile: For equity index options (SPX, SPY), OTM puts consistently trade at higher IV than ATM options, which trade at higher IV than OTM calls. The shape is called the volatility skew. It reflects:

1. Demand for downside protection: Investors buy OTM puts as portfolio insurance. High demand → high price → high IV.
2. The leverage effect: When stocks fall, equity volatility rises (empirically well-documented). OTM puts protect against this correlated scenario, so they're especially valuable.
3. Crash risk premium: Markets can crash much faster than they can rise (asymmetric tails), so OTM puts command an asymmetric premium.

For FX options, you often see a true symmetric smile — OTM puts and calls both trade above ATM — because currencies can move sharply in both directions.

Practical IV Numbers

Some intuition-building reference points:

• SPY 30-day ATM IV: normally 12–18% in calm markets, 25–40% during stress
• Individual stocks: often 25–60% for large-caps, 60–150%+ for small-caps and biotechs
• VIX (SPX 30-day IV index): 11–15 = extreme calm, 20–25 = normal, 30–40 = stressed, 40+ = crisis

High IV means options are expensive. Low IV means options are cheap. Options traders say they are "buying vol" (going long options) or "selling vol" (going short options) to describe their exposure to IV changes, independent of any directional view.

IV and the Greeks

Because IV is the key input that traders control, the Greeks take on a vol-centric interpretation:

• Vega: Direct IV exposure. If you buy a straddle and IV rises, vega profits regardless of where the stock went.
• Gamma: The "realized vol" Greek. If actual daily moves are larger than implied, gamma scalping profits exceed theta cost — you win. This is why people say gamma is "long realized vol."
• Theta: The cost of owning the IV insurance. You're paying the variance risk premium in real time.

The core question every options trader asks is: Is IV cheap or expensive relative to what the stock will actually do? If IV = 25% and you think the stock will realize 35% vol, buy options. If IV = 25% and you think the stock will barely move (5% realized vol), sell options. Everything else is details.