Why At-the-Money Implied Volatility Need Not Fall with Time

Summary

The document addresses a common confusion between an option's declining time value and its implied volatility as expiry approaches. The answer explains that, near the money, option value scales approximately with annualized volatility multiplied by the square root of time remaining. As a result, the option's value can decline as time passes even when the implied volatility input stays constant.

Implied volatility is not itself the option premium or a quantity that must decrease at a fixed rate with time to expiry. It may rise or fall according to market conditions and supply and demand for options. The questioner's comparison with realized volatility therefore does not establish an arbitrage: a falling premium caused by shorter remaining time is distinct from a change in the annualized volatility implied by that premium. The explanation is concise and uses the at-the-money relationship as intuition. It does not develop the pricing formula, discuss moneyness effects away from at the money, or specify a model for how implied volatility evolves, so it should be read as a basic conceptual clarification rather than a full term-structure analysis.

Key ideas

• At-the-money option value depends on both annualized implied volatility and the square root of time remaining.
• An option premium can decline as expiry approaches even when implied volatility is unchanged.
• Implied volatility may move over time in response to market conditions and option supply and demand.
• A falling option price from shorter time to expiry does not by itself imply an arbitrage against realized volatility.

Tags

# What is implied volatility relationship with time?


# What is implied volatility relationship with time?












My take on the question is that, with all else being equal, as time to expiry approaches, the option price decreases (due to theta). And since implied vol is the price of the option in volatility points, the implied vol should also decrease as TTE approaches.

But this doesn't make sense to me. Implied vol can be thought of as the market's expectation of the realize vol over the life of the option, in annualized vol. If all else remains equal, and the underlying continues moving at the same realize vol as usual, how could the implied vol decrease? Wouldn't that present an arbitrage opportunity, as implied vol keeps falling as TTE approaches, but the realize vol remains the same (since we assume ceteris paribus), thus creating a large disparity between implied vol and realized vol?

## Answer by dm63 (score 1)

https://quant.stackexchange.com/a/80390

It is well known that the value of At the money options is proportional to $\sigma\sqrt{T-t}$ where $\sigma$ is the implied annualized volatility. So as time evolves, the value falls naturally due to the square root factor. The volatility factor may or may not move with time, depending on market conditions and supply and demand for options.

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