(Below is from rates perspective, but applicable to any asset class in general.)

the market price of skew in rates can be captured through the concept of implied and realized blend. The concept in principle is quite simple and intuitive. Usually in rates markets, the underlying (i.e. forward rates) is modeled as a normal process. That is, the absolute change in rates are independent. The other extreme is lognormal, which implies percentage changes (instead of absolute changes) are independent random moves. The actual realized dynamics can be anywhere in between. Let's assume the forward rate dynamics is as given below (sorry for the clumsy picture!)

the market price of skew in rates can be captured through the concept of implied and realized blend. The concept in principle is quite simple and intuitive. Usually in rates markets, the underlying (i.e. forward rates) is modeled as a normal process. That is, the absolute change in rates are independent. The other extreme is lognormal, which implies percentage changes (instead of absolute changes) are independent random moves. The actual realized dynamics can be anywhere in between. Let's assume the forward rate dynamics is as given below (sorry for the clumsy picture!)

Here beta is the blend parameter.

This captures the normal dynamics mentioned above, for a value of beta = 0, and lognormal for beta = 1, and any realistic dynamics for different values of beta.

It is interesting to see what it means in terms of skew pricing. For example if we are talking about skew measured in terms of normal (bps) volatility (as usually done for rates markets), a normal dynamics would mean (without any stochastic volatility assumption) a flat smile. Whereas a beta close to 1 would mean a volatility smile which is higher for payers (higher as yields goes up), and similarly a beta less than 0 means a receiver side skew. This is evident from the model above: e.g. for beta = 1, as yields go up, the ATM implied vol is priced higher, giving rise to a payer side skew.

Now just like implied and realized vol, this beta, or blend can also be thought of having two different sources of estimation. One from the current prices of options at different strikes. This is easy to extract, by taking the price of a (tight) collar around the ATMF and solving for the value of beta which prices the payer and receiver vols correctly (or rather their difference, i.e. the collar price), given the current ATM vol. This can be thought of as the implied blend.

For the realized blend, just like realized volatility, we can take historical ATM vols and regress against the ATMF yields. Note beta here defines the value at which the right-hand side of the model above becomes a random move. So running the above regression and changing the value of beta, we can pick the value for which we get the least R-square (worst fit). This value of beta can be thought of as the realized blend. Just like realized volatility, it will be sensitive to the amount and period of history we look back at.

Now armed with this two measure we can draw some interesting observations. To begin with, the values directly compare the realized vs. implied skew and can be used to find relative value in skew trades. Imagine a scenario where we have seen a high realized beta and a low implied one. This means the markets prices the receiver skews more favorably (implied), although in recent history volatility went up as yields went up (realized). If we assume this past behavior will persist, then it makes sense to buy a payer and sell a receiver (buy a collar or risk-reversal overall) and delta hedge. If yields indeed go up, the portfolio will be long gamma, and the implied vols will be higher than what was priced in. If the yields go down, the portfolio will be short gamma, but implied vols will be lower than priced in.

Some further observations: firstly, the co-movement of blends vs. rates can allow us to conclude about higher orders than skew, in this case about vol of vol. The blend (both implied and realized) will depend on the absolute level of yields. For very low levels of rates, a further rally will mean limited scope for yields to move downwards (by zero bound condition for nominal rates), where as the moves upward is unbound. This means in normal condition the yields and the realized blends should move in the opposite direction (vol stabilizing environment). If they move in same direction, that reflects uncertainty and high vol of vol scenarios (vol blow-up environment). This directionality can be compared with the current vol of vol priced in (say in a strangle or a fly) and use this method to spot relative value in vol of vol.

Here are some charts for the 10y swap point for USD (click to enlarge).

And for EUR (click to enlarge, all data from Bloomberg)

Notice how the market spent most of the recent times constantly re-pricing in USD under increased uncertainties as vol picked up (see here). In fact last October and early this year we have seen highest level of vol of vol for US rates since Lehman crisis. For EUR, initially it was normal. Only after introduction of a negative depo rate and imminent QE, blend caught up in lock-step with rates with a jump in vol as well.

Interesting time ahead.

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