Interest Rate Volatility Skew - A Guide For the Preplexed (c)
If you look at the yield vols (BS) on interest rates derivatives, you would see a clear slope across the strikes, with higher vols at lower interest rates and lower vols at higher rates. The longer the expiries and the longer the tails (maturity along the yield curve), the bigger this slope would be. In order to understand the skew, you'd need to think about the nature of the interest rate movement.
First, take the proportional volatility (aka as BS vols or log-normal vols). Lognormal model assumes that the movement in the underlying is proportional, that is if we have 16% annualized BS vols (1% per day), on a rate of 10% we expect to see a daily change of 1% * 10% = 10bp, while on the rate of 1% we would expect the see a change of 1% * 1% = 1bp.
An alternative to the proportional volatility would be normal volatility, were the absolute size of the movement is kept constant no matter what the level of the rate is. In this case, if we see 10bp movement when the rates are at 10%, we'd also see 10bp movement when the rates are at 1%. Clearly, when you translate this movement to log-normal volatility, you would see a major increase in the level of lognormal vols at the rates decrease, from 10bp/10% = 1% to 10bp/1% = 10%.
If you took your time to understand the above exampes, you now understand the reasoning behind the skew. The skew in interest rates is mainly driven by the expectation of the level of lognormal volatilities at different rates level. For the quanty people, that could be translated as "the skew is representative of the underlying random process".
In real life, however, the nature of the movement is somewhere in between the two modes, normal and log-normal. For example, one can imagine that the movement would be such that while at the rate of 10% we'd see a movement of 10bp, while when the rates are at 1% you would see a movement of 5bp.
The street uses a simple model to represent the "intermediate" process called the CEV (Constant Elastisity of Variance). The model states that most of the uneven level in the yield vols across the strikes can be explained as
Volatility At Strike = ATM Volatility * (Forward/Strike)^(1-B)/2
If you plug this thing into a spreadsheet and play around with the value of beta, you can see the different "styles" of skewness at different extremes of beta. If we take beta = 0, our lognormal volatility scales up as the strikes decrease and we are looking at expected normal changes. If we take beta of 1, we see that the lognormal volatilities do not change across strikes, that is we are looking at expected proportional changes.
As a matter of fact, this model holds for other, non-IR markets. If keep changing it around, you can also achieve "super"-lognormal or super-normal skew, where the volatility would super-proportional or inversly proportional, like in inflation or in commodities.
We are still missing an importat part of the volatilities, however. If you try to fit the CEV backbone into vols for shorter-dated expiries, you would find that expiries farther away from the money tend to deviate up from the CEV backbone. If you actuallly substract the CEV backbone from the actual market volatilities, you'd be left with a smile that is driven by gap-risk. There numerous ways of modellng gap risk (stochastic vols approaches, jump-diffusion approaches etc.), but they are not simple and do not really change the nature of the trading.
