Implied Volatility from zero
1/5What is implied volatility?
Black-Scholes takes five inputs and spits out a price. Implied volatility does the reverse: given a market price, what volatility makes the model match?
Four of the five BS inputs are directly observable — spot, strike, time to expiry, and the risk-free rate. Volatility is the odd one out. Nobody can look it up. So the market reveals its volatility estimate through the price it sets on an option.
The formula is the same. The direction is different. Instead of plugging in σ to get a price, you plug in the price to get σ.
The inversion
There is no closed-form inverse of Black-Scholes. You solve for IV numerically — bisection or Newton’s method, iterating until BS(σ) matches the market price within tolerance.
The blue curve below shows the BS call price as a function of σ. The orange dashed line is the market price. Where they intersect is the implied volatility.
This always works because BS price is strictly increasing in σ — higher vol always means a higher option price. So for any market price between the intrinsic value and the spot price, there is exactly one σ that fits.
Drag the market price up. The intersection moves right — higher market price implies higher vol. Drag it down near zero and the IV approaches zero too. The mapping is monotonic.
Why IV matters
IV is the market’s consensus estimate of future uncertainty. It encodes information that historical data alone cannot capture — upcoming events, shifting sentiment, supply-demand for hedges.
Historical volatility (HV) measures what the asset actually did over some lookback window. Implied volatility (IV) measures what the options market expects going forward.
When IV is above HV, the options market is pricing in more risk than recently observed. Traders call these options “expensive.” When IV is below HV, options are “cheap” relative to recent moves.
ETH 30-day HV is 45%. But IV on 30-day ATM options is 70%. The market expects significantly more turbulence than recent history suggests — maybe a merge, a regulatory event, or a macro catalyst. If nothing happens and realized vol stays at 45%, option sellers collect the 25-point premium.
The vol smile and skew
If Black-Scholes were literally true, every strike on the same expiry would have the same IV. They don’t. Plot IV against strike and you get a curve — the volatility smile.
The skew (tilt) reflects directional fear. In equity and crypto markets, downside protection is in higher demand, so OTM puts trade at higher IV than OTM calls. The curve tilts left.
The kurtosis (curvature) reflects tail fear — the market’s belief that extreme moves in either direction are more likely than a normal distribution predicts. More curvature means both wings are expensive.
Drag the skew slider left to increase downside fear — watch the left wing lift. Increase kurtosis and both wings rise, forming the classic smile shape.
In practice, crypto vol surfaces show steep negative skew (crash protection is expensive) and significant kurtosis (the market prices fat tails).
Reading IV in practice
An IV number by itself is meaningless until you translate it to an expected price range. 80% IV on ETH sounds abstract. ±5% daily move is concrete.
IV is annualized. To convert to a shorter horizon, multiply by the square root of the time fraction. For a daily move using trading days:
Set ETH at $3,500 with 80% IV. The calculator shows a daily move of about ±$175 and a 30-day range of roughly ±$800. That is what 80% IV means — not that ETH will move 80% this year, but that the market assigns a 68% probability the annual move stays within ±80%.
Where to go next:
Vega — how option prices change when IV moves
Black-Scholes — the model IV inverts
Option valuation — connecting IV to extrinsic value