Black Implied Volatility as an Options Quoting Convention

Summary

Black implied volatility is the volatility input that makes the Black pricing formula match an option’s observed market price, given the relevant strike, spot or forward, interest rate, and time to expiry. For options with payoffs resembling a call on a positive underlying, this gives traders a common way to express prices as volatility rather than currency amounts.

The document explains why this convention is useful: it helps compare options across dates and strikes, and implied volatilities may vary more smoothly across strikes and expiries than prices do. That can make interpolation in volatility space preferable when estimating values between quoted contracts, with less risk of introducing arbitrage than interpolating prices directly. These are practical motivations, not a guarantee that volatility quotes are uniform or arbitrage-free. The answer notes that some products, such as credit index options, have more complicated payoff or pricing details, and frames Black volatility primarily as a quoting convention rather than a directly observed forecast of future volatility.

Key ideas

• Black implied volatility is the input that makes a Black model price equal the observed option price.
• It provides a common quoting measure for comparing options with different strikes or observation dates.
• Volatility quotes may be more suitable than raw prices for interpolation across strikes.
• Implied volatilities can still differ across strikes and expiries, and some products need more complex treatment.

Tags

# Black-Equivalent Volatility


# Black-Equivalent Volatility












Can Someone Explain to me what this term means, and how it's used?

## Answer by quant_dev (score 5, accepted)

https://quant.stackexchange.com/a/321

Quite a lot of options on asset $S(t) > 0$ have a payoff at tinme $T$ equal (at least approximately -- it's a bit more complicated in the case of e.g. credit index options)

$ (S(T) - K)^+ $

You can always find a number $\sigma$ such that, when plugged into Black formula together with strike $K$, spot price $S(t)$, interest rate $r$ and time to expiry $T-t$, you will recover the market price of the option $V(t)$. This number is called the Black implied volatility of the option. Basically, it's a quoting convention for the option prices. Traders use it because:

- it makes it easier for them to compare prices of options on different days, with different strikes

- Black vols tend to be similar across strikes and expiries (not always!)

- it is better (in the sense: you're less likely to suffer lots of arbitrage) to interpolate market prices in the $\sigma$ space then directly; that is, if prices for strikes $K_1$ and $K_2$ are quoted, it's better to use some interpolation method on their Black vols $\sigma_1$ and $\sigma_2$ than on their prices $V_1$ and $V_2$

- it fits their intution better (a paramount argument)

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.