Why Implied Volatility Skew Scales with Time to Expiry
Summary
The document asks why plotted call-option implied-volatility smiles appear steeper for shorter expiries and why scaling log-moneyness by the square root of time can make smiles look more comparable. One response points to work on smiles in stochastic-volatility models, where a general-dynamics result links skew to terminal-distribution skewness divided by the square root of time. This offers a model-based explanation for expiry-related changes in the observed slope.
A second response describes square-root-of-time scaling as a way to compare moves over different horizons. Under a normal-return approximation, standard deviation grows with the square root of time, so a strike’s distance from spot can be expressed relative to the move expected over that expiry. This is an intuition for rescaling, not a universal derivation of the shape of every implied-volatility surface. The discussion gives no fitted data or proof for all models, and the scaling should be interpreted in the context of the underlying dynamics and distribution assumptions.
Key ideas
- The observed implied-volatility smile can appear steeper at shorter expiries.
- A cited stochastic-volatility result relates skew to terminal-distribution skewness divided by the square root of time.
- Square-root-of-time scaling helps compare expected price moves across different expiries under normal-return assumptions.
- Rescaling log-moneyness by expiry can make smiles more comparable in standardized move units.
- The explanation is model-dependent and does not establish a universal rule for every volatility surface.
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# Black-Scholes: Volatility Smile "sharpens" with time to expiry
# Black-Scholes: Volatility Smile "sharpens" with time to expiry
I have tried to calculate IV and log-moneyness (=log(S/K)) for different times to expiry (M = less than 1 month, Q = less than 1 quarter, S = less than 1/2 of an year, Y = less than 1 year, Y (+) = more than 1 year). Doing this I've plotted the IV-smile for Call-options:
Notice that the IV-smile seems to "sharpen" when options get close to expiry. It other words: Small changes in log-moneyness implies large changes in IV when the option is closer to expiry - but why is that?
Following @will's suggestion of dividing log-moneyness by sqrt(T) results in a very nice Volatility Smile. Would someone care to explain why this is the case?
## Answer by MainCom (score 1)
https://quant.stackexchange.com/a/65464
You may take a look at the paper 'The smile in stochastic volatility models' by Bergomi and Guyon. In appendix B of the paper, they derive that if we assume some general dynamics, then the skew is proportional to the skewness of the terminal distribution of the underlying divided by $\sqrt{T}$.
## Answer by kdragger (score 1)
https://quant.stackexchange.com/a/65475
The sqrt(T) is an annualization factor. Conceptually it is the equivalent of comparing a 1 month interest rate to a one year interest rate. If you do not convert into an APY or annualized percentage yield, then you would get low interest rates for 1 month because there is very little time. For example, a 1 month interest rate of 1% would become ~12% (for the purposes of this example, obv not compounded).
The same is in volatility space. Given a volatility of N%, you would expect a range of outcomes in one month that might be, say 0.25underlying wide. The same volatility over 12 months would be far wider. When dealing with a normal distribution, that would be 0.25underlying/sqrt(T).
This then holds for the strikes. If a stock was 100 with volatility of 16%, you would expect a move of about 1% per day as a one standard deviation move (using 256 days per year so that sqrt(T) is 16). The math is vol*underlying/sqrt(T). In one year, you would expect 16% standard deviation. So a 1 std deviation option for one day would be 1 away from the current price and 16 away for a one year option. To appropriately compare the implied volatility, you would want to compare the 1 to the 16 as they are equivalently far away in movement terms.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.