Volatility skew/smile for long term options is flatter compared to short term options, could someone help to explain why is that the case? Thanks
One possible reason could be jumps. Over the longer maturity, there could be more jumps so the jumps average out in a way; whereas over the short term, a jump can make a bigger difference and hence the risk of jump increases demand.
This reasoning is used to justify Stochastic volatility with jumps models in some books.
The skew/smile of long term options is flatter than short term options, the reason for this can be explained in several ways.
The Vega of a shorter-dated option is smaller than a longer-dated option. Vega is the dollar value of a 1% change in implied volatility.
i.e., 30d ATM option, $65 strike, .31 ivol = VEGA .07
30d 25 delta option,31% ivol = VEGA .055
180d ATM option $65 strike 31% ivol =VEGA .18;
180d 25 delta 31% ivol = VEGA.135
Remember, the implied vol smile exists to price in skewness in the underlying assets price returns. Skewness is observed in PRICE CHANGES of the underlying asset, which is then converted to volatility.
So we need to go back to the price in order to bake in expected skewness (observed in the ivol of the OTM options).
Let's presume the skewness of the underlying asset shows a long left-sided tail (larger down moves vs up moves) of roughly 6 cents.
To bake that 6 cents into our OTM option we'd price the 30-day, 25 delta option at ~32% Ivol (1% higher than ATM option)
To bake in that same expected skewness of 6 cents into the 180-day, 25 delta option, the ivol only needs to be increased by .4% or 31.4% Ivol.
Yanyi Yuan's answer is making the same point. The difference in the Square root of time:
i.e., 30 days = SQRT(30/365) = .289; 180 days = SQRT(180/365) = .702 The ratio between the SQRT's of time = .285 / .702 = 40%.
In other words, using the SQRT of time, for the price value of skew to be the same, the implied vol of 180-day option only needs to be increased by .4% for each 1% ivol increase in a 30-day option.