# Put Volatility Smiles and Implied Volatility

I have been observing the option chains of put options with differing maturities. I have noticed that those puts with a close expiry date have the steepest volatility smiles.

Can someone please explain why this is the case in an intuitive way.

Regards,

The short answer: Your observation is caused by some sort of central limit theorem.

The long answer: The reason for the volatility smile/skew is the non-normality of the assumed return distribution. If the implied volatility of out-of-the-money put options is higher than of at-the-money put options, it implies that the market assumes that the risk-neutral probability of a downside return is higher than implied by a normal distribution. The implied volatility surface can therefore be seen as a convenient way to describe the underlying risk-neutral return distribution.

Let's assume that the true underlying process generates daily log-returns which are left-skewed and leptokurtic, i.e. downside moves are larger than upside moves. If we observe options with a maturity of one day, we will find that the risk-neutral distribution corresponds to the skewness and kurtosis of the underlying return process. However, if we extend the time horizon returns are aggregated and the central limit theorem starts to work. Thus, longer return horizons generate more normal returns. So for example, if we observe the volatility smile of thirty day options we observe the assumed risk-neutral return distribution of the 30 day log-return which is the sum of the next 22 or so log-returns. Therefore, the 30 day return is necessarily less non-normal than a one day return. If we switch back to the corresponding volatility surface, less non-normal returns imply a flatter smile.

In reality this is a common observation of implied volatility smiles and of historical return distributions. If you calculate the skewness and kurtosis of daily, monthly, quarterly and yearly returns you usually find that yearly returns are the least non-normal. The speed of convergence is dependent on the initial non-normality of the process and for highly non-normal distributions convergence can be very slow.

This answer corresponds the answer of Quantuple who named important microstructural and hedging arguments.

This can be due to various effects. I will list you 2 of them off the top of my head:

1. Jumps/Crashes : assume you were to price a put option which expires in a few days from now. Your diffusion model tells you this option should be worth $\$3.25e^{-7}$since it is very out of the money as of today. What price will you quote? Well, in order to be conservative, you'll certainly price in an additional probably of adverse market movement occurring between now and the option expiry (e.g. strong downard jumps or flash crash) that could add value to the put option. This is more marked on put that on call, because one would rarely price in events that would be exceptionally good for him/her in a conservative setting. 2. In practice, there is usually a minimum trade price for any option listed on an exchange. Even if there were none, you would never be able to sell something for$\$3.25e^{-7}$ anyway since the smallest currency unit is 1 cent (even with lots size of 1000 you'd still have a problem.). The effect is almost the same as jumps: it tends to strongly overvalue options that should theoretically be worth almost nothing. To convince yourself, it is illuminating to perform the following experiment: using a classic BS model with a volatility of $20\%$ generate put option prices. Now replace all put prices that are below the minimum trade price by this precise price and imply the Black-Scholes volatilities out of these corrected prices. Tell me, is the smile flat as you would have expected?

Anyway, over-valuating (by pricing-in extreme events that are not accounted for in your models, or simply due to technicalities) short-term OTM put prices that should theoretically be worth almost nothing leads to higher short-term put implied volatilities.