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## There is nothing normal about LTGA volatility

Mon, 11 Feb 2013 10:20:38 GMT

## Sandy Sharpe, David Roseburgh, Paul MacCarney

Following on from our first blog which discussed the yield curve methodologies applied in the LTGA, here we highlight a few of our thoughts on the volatility data that been issued by EIOPA which can be used by participants for interest rate calibrations.

The LTG assessment provides participants with example interest rate volatility surfaces which can be used along with the provided yield curves to recalibrate the stochastic nominal interest rate models. These  are then used to generate interest rates scenarios for the valuation of (amongst other things) embedded interest rate guarantees. The level of interest rate volatility is a first order driver of the value of these guarantees – the higher the volatility, the higher the value of the embedded option. For each date a surface is provided to be used for all economies being modelled.

Our first observation is with respect to the nature of the volatilities provided by EIOPA.  Implied volatilities quoted by the market are not measures of realised or even expected volatility – they are parameters which, under certain modelling assumptions about the distribution of the underlying interest rates, reproduce the price of the swaption as quoted by the market.

# Defining volatility

There are two common, differing, sets of assumptions about the rate distribution used in quoting volatilities – in one rates are log-normally distributed, and in the other they have a normal distribution, and the way the simplified volatility is quoted in each case differs markedly.

Under a log normal assumption, volatility is proportional to the level of the interest rate, and implied volatilities are quoted as this relative volatility. This is often called the Black volatility, or lognormal volatility. In this case, a quoted volatility of 20 can be loosely interpreted as meaning that the standard deviation of relative changes in interest rates is 20%.

Under a normal assumption for rates, volatility is not related to the level of the rate, and volatilities are quoted on an absolute basis. This volatility measure is often called the absolute, normal or basis point volatility. Here, a quoted volatility of 20 can be interpreted as implying the standard deviation of absolute changes in interest rates is 20 basis points, or 0.2%.

Now, these measures are equivalent, and mostly exist as quoting conventions. A swaption has a single unambiguous price, which can be converted to a Black vol or a normal vol according to the modeller’s preference.  Conversely, as long as the modeller knows which convention is used, it’s a fairly trivial task to convert the volatility to a price, which can then be used to calibrate a model.

# Understanding  the data

In the insurance sector, when modellers talk about interest rate volatility in the context of market- consistent valuation, they are almost always referring to Black swaption implied volatilities— and most would expect the volatilities supplied by EIOPA to be quoted using this convention.

Unfortunately, this does not appear to be the case.  Exhibit 1 shows the EIOPA swaption volatility for the EUR 10 yr tenor swaps and also market quoted normal volatility for the same tenor. The near match would suggest the IV provided by EIOPA is derived using the normality assumption rather than the commonly used lognormal convention.

Exhibit 1: EIOPA volatilities for options on 10 year EUR swaps.

As we discuss above, this isn’t an issue fundamentally — as long as the user understands which convention is used, the data is fine.

# Not understanding the data

The problem though is that EIOPA do not state which convention is used – the data is simply labelled “volatility”. This could lead a number of modellers to incorrectly interpret the data as Black volatilities according to standard market practice. This could have a number of consequences, the most obvious (and important) of which is that guarantee costs calculated using incorrectly interpreted data will be materially wrong.

Exhibit 2 shows how important this distinction is — if the normal volatility is misinterpreted as Black implied vol, users might overestimate interest rate volatility by as much as 300%.

Exhibit 2: EIOPA and  Black implied volatilities for options on 10 year swaps.

The effect this has on price depends on the yield curve, but for the EIOPA EUR yield curve at end December 2011, the 5 year into 10 swaption price, assuming the EIOPA vols are Black volatilities, is around 1500 basis points. The correct answer is closer to 470 basis points. An insurance liability with similar features to a European swaption– a guaranteed annuity option, for example – would suffer similar mispricing.

This potential for inaccuracy has the potential to severely obfuscate the effects of changes to the yield curve – which in turn negates the purpose of the LTGA.

As a final point it’s important to note that the LTGA specifications provided by EIOPA are not wrong, but they are open to the vagaries of financial ambiguity – and modellers would do well to approach with their full wits about them.