# 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?

• For what underlying are these options? Jun 7 '21 at 16:22
• You need to factor in that volatility is normalised. If you look at this chart where the y axis is the total implied variance, or where you scale the log moneyness to factor in that there is less time (i.e. maybe use delta, or shift the strikes to some time normalised space - i.e. $\frac{\log{\frac{K}{F}}}{\sqrt{t}}$)
– will
Jun 7 '21 at 16:39
• @DaneelOlivaw it's for the SP500 (^GSPC) as the options is SPX Jun 7 '21 at 17:14
• @will not sure I follow your comment. Why would we un-normalise implied vol? Jun 7 '21 at 17:41
• @will, would you care to elaborate on why dividing by sqrt(T) yields such a nice Volatility Smile? Jun 7 '21 at 19:29

## 2 Answers

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 $$\sqrt{T}$$.

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.