Following the request from last week, let’s discuss this week about IRO volatility.

While the later can encompass many different volatilities: bond vol, future vol, etc… I will focus on 2 for today: cap/floors and swaptions.

## Rates volatility – ATM volatility

### Swaptions

The ATM volatility of swaptions is already 2 dimensions: option volatility and underlying (swap) maturity. It makes things a bit more complex than others when you throw in on top the smile structure. (more about that later)

The interesting bit about swaptions volatility is that you can choose to interpolate the underlying maturity. You can choose to interpolate based on time, but this might need to be corrected if you have an option on an amortizing swap for instance. As such you can choose to interpolate based on BPV where Murex computes the BPV of the reference swap of the vol group.

### Caps/Floors

While cap/floor vols are defined on a selected index (and you link index vol) there is another thing about cap/floor vols: are they forward/forward or for the whole cap (what’s called par)? One thing to understand is that a cap or a floor is a series of options rather than a single one. For instance a cap on EURIBOR 3M means that every 3M you have an option on the EURIBOR 3M. So if you look at a 2Y cap on the EURIBOR 3M, you effectively have 8 options.

So when you choose vol nature forward/forward, Murex expects that you will provide caplet volatility for each pillar of the vol curve. Nature cap means that you provide a volatility that would be the same for each caplet. In the case of our 2Y EURIBOR 3M cap, this means that the 8 options would share the same volatility. Murex can also calibrate the forward/forward volatilities.

Calibrating the fwd/fwd volatilities mean that the 3m fwd/fwd vol is equal to the par vol 3m (as you have only 1 caplet in that case). Then for the 6m pillar, you know the total price of the cap as the 6m par vol could be applied to both caplets to drive the price. But you also have already found the 0m/3m caplet volatility. You can then backsolve the second caplet volatility so that the sum of the premium of each caplet using fwd/fwd vol is the same than the premium using par vol.

This mechanism is very important as in the pricer this will explain why you see 2 volatilities: one on the main pricer screen (par vol) and one (well, multiple) in the flows screen: fwd/fwd volatilities.

## Rates volatility – Volatility nature

Volatility nature has been for a long time lognormal for rates products. Unfortunately the models consuming lognormal volatility have one major flaw: they do not work with negative rates. And given the current rates state, this is quite a problem.

So 2 solutions emerged:

– Shifted lognormal: the idea behind this is to shift all your rates by a certain amount when using the model (ideally you ensure that your lower strikes of your smile are far off the 0% boundary). So for example you work as if your strike at 0% is a strike at 10%. The advantage of that method is that the work to move away from lognormal is light

– Normal volatility: this is actually quite different and there is a fair bit of work to adapt models to accept normal volatilities. Normalized volatility is a volatility that is not at all correlated to interest rates. Lognormal volatility (and vega by extension) actually changes quite significantly if rates are moving by a large amount in one direction. Normal volatility is very stable. It can also be applied to negative rates without any problem. While more work than shifted lognormal, one main advantage for traders is that when you’re hedged on normal vega, your hedge should prove very stable

## Rates volatility – Smile

### Swaptions

You define a smile curve for each underlying swap maturity (I often see a fair bit of linking between maturities). The interpolation is often interesting for swaptions as you can fall between 4 points rather than 2.

### Caps/Floors

The smile is defined for each index, pretty standard. You can (should?) do linking for less traded indices.

## Rates volatility – Smile dynamics

Alright, this is the interesting bit: smile dynamics.

The smile dynamics is how your smile moves when the rates are changing:

– Lognormal

Lognormal dynamics is basically no dynamics at all. Your curve does not change when the rates shift.

– Normal

Normal smile dynamics is that the corresponding lognormal volatilities do change when the rates change (the conversion from normal to lognormal does use the actual rates). So even if your smile is money based, your lognormal volatility can be different for an option at the money

– SABR

SABR is a parametric volatility calibration model. While SABR would deserve a post all for itself, in a nutshell, basically you can assume that the SABR parameters are constant when rates are changing and you can re-calibrate the volatility based on the new rates

More questions, something I need to dig further into? Let me know!