# 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

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 $$\sqrt{T}$$.