Implied Volatility (IV)

Implied volatility (IV) is the level of volatility that, when entered into an option pricing model, makes the model's price equal to the option's market price. It is not observed directly and is not a forecast issued by anyone: it is backed out of the option premium that buyers and sellers have agreed. It is quoted as an annualised percentage and is read as the market's pricing of how much the underlying asset may move over the life of the option.

IV is forward-looking, which distinguishes it from realized volatility, the movement that has already occurred. Options traders often quote and compare options by implied volatility instead of by price, because it puts contracts with different strikes and expiries on a common scale. The best-known index of implied volatility is the Cboe Volatility Index (VIX), which is calculated from the prices of options on the S&P 500.

A pricing model such as the Black-Scholes-Merton model takes five main inputs: the price of the underlying, the strike price, the time to expiry, the interest rate and the volatility. The first four are known. Implied volatility is the value of the fifth that satisfies:

• $C_{\text{market}}$ is the option's price in the market.
• $S$ is the price of the underlying and $K$ is the strike price.
• $T$ is the time to expiry in years and $r$ is the risk-free interest rate.
• $\sigma_{\text{imp}}$ is the implied volatility.

The equation cannot be rearranged to give volatility directly, so it is solved by trial and improvement, using numerical methods such as Newton-Raphson or bisection. Because an option's price rises with volatility, there is only one solution. For an option whose strike is close to the forward price, a common shortcut is $C \approx 0.4 \times S \times \sigma \times \sqrt{T}$. A one-month call priced at 2.29 on a stock at 100 implies $\sigma \approx 2.29 / (0.4 \times 100 \times 0.2887) = 0.198$, or about 20 percent.

Implied volatility is a one-standard-deviation annual move. It is scaled to shorter horizons by the square root of time:

Here $d$ is the number of calendar days. For a stock at 100 with an implied volatility of 20 percent, the one-standard-deviation move over 30 days is $100 \times 0.20 \times \sqrt{30/365} = 5.73$. Option prices are therefore consistent with the stock finishing between about 94.3 and 105.7 roughly two-thirds of the time. A rule of thumb used on trading desks divides annual volatility by 16, close to the square root of 252 trading days, to get a daily figure: 20 percent a year is about 1.25 percent a day.

The number says nothing about direction. It is also a risk-neutral quantity that includes a premium for bearing risk, not a pure forecast.

If the model's assumptions held exactly, every option on the same underlying would have the same implied volatility. In practice it differs by strike and by expiry. Plotted against strike it forms a volatility smile or a volatility skew. In equity index markets, puts with low strikes have carried higher implied volatility than at-the-money options since the crash of 1987. Plotted against both strike and expiry it forms the volatility surface, which dealers use to price and hedge. The pattern reflects features that the basic model leaves out, including fat tails, price jumps and demand for downside protection.

• Relative value. Traders compare IV with their own forecast of future volatility. Buying options when IV looks low and selling when it looks high is the basis of volatility arbitrage.
• Ranking. Measures such as IV rank place current IV within its range over the past year.
• Risk. The sensitivity of an option's price to a change in IV is its vega. A position can lose money from a fall in IV even when the underlying moves the right way, as often happens after an earnings release, when IV drops sharply.
• Risk premium. On average, index IV has exceeded the volatility later realized. The gap is the volatility risk premium, which option sellers collect in return for bearing the risk of sudden large moves.

Implied volatility depends on the model used to extract it. A different treatment of dividends, interest rates or early exercise gives a different number, so figures from two sources may not match. For options that are far from the money or thinly traded, a wide bid-ask spread makes IV imprecise, since a small change in price implies a large change in volatility. And because it is derived from prices, IV reflects supply and demand for options as much as any consensus view of future movement. It can stay well above or below subsequent realized volatility for long periods.

• Cboe Global Markets