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oronimbus
Topic Author
Posts: 4
Joined: January 8th, 2020, 11:52 am

### Implied/Realised Volatility Ratio for Negative Rates?

Hi all, I'm trying to calculate the implied vs realised vol ratio for different swaptions across G5 currencies. This works fine for USD and GBP as rates are positive. However I'm struggling with adapting my algorithm to EUR. For the implied vol data I can directly use the quoted normal ATM bpVol by a broker or composite (e.g. TP, ICAP). For the realised ratio however, I'm calculating the basis point equivalent using the standard deviation * sqrt(T) * ATM_Forward. In terms of calculating the standard deviation I can decide between using daily percentage returns or absolute differences as log-returns obviously won't work.

Now every time rates start to approach zero the I/R-vol ratio starts to explode. Then, finally as rates dip below zero the annualised bpVol goes negative (as I'm multiplying by the ATM forward), which then makes the I/R vol - ratio negative.

My questions are:
a) is my approach wrong and
b) is there any other way to get a bpVol equivalent for realised volatility? The only other solution that came to mind is using shifted Black volatilities (I need to check if I can get those...) and %-realised vol.

Thanks

lePiddu
Posts: 20
Joined: March 17th, 2015, 2:15 pm

### Re: Implied/Realised Volatility Ratio for Negative Rates?

I do believe your approach is wrong: you are treating forward swap rates (the underlying of a swaption priced using BlackScholes/Bachelier formulas) as they were equity.

Swap rates are (sort of) averages of points in a yield curve. Therefore, the "shocks" that generates volatility are applied to the curve as a whole. Or to some low dimensional representations of it (like short rates). As you approach the maturity of a swaption, the points of the yield curve that forms the swap rate are not fixed. Those points start sliding toward time=0. So you have to track the daily variation in the various forward rates (and zcbs of your discount curve) composing your forward swap rate. All these quantities in turn have a (non negligible) dependence from your yield-curve bootstrapping algorithm. So the true realized volatility for a forward swap rate is quite a complex object.

Nonetheless you can pretend swap rates are point-like values like equity. Just remember that it's an approximation.
In that case you must use shifted-Lognormal volatilites and of course, shifted rates. Of course you have to be consistent in what shift magnitude you are using when computing prices and when estimating volatilites.

DavidJN
Posts: 1746
Joined: July 14th, 2002, 3:00 am

### Re: Implied/Realised Volatility Ratio for Negative Rates?

As you approach the maturity of a swaption, the points of the yield curve that forms the swap rate are not fixed. Those points start sliding toward time=0.”

How is aging of bespoke swaps relevant when one is dealing with the implied vols of ATM par swap rates that are quoted as a constant maturity relative to the valuation date?  Every day the 5-year par swap rate is quoted with 5-years relative to the valuation date. There is no aging effect evident in a series of par swap rates.

oronimbus
Topic Author
Posts: 4
Joined: January 8th, 2020, 11:52 am

### Re: Implied/Realised Volatility Ratio for Negative Rates?

Possibly my intentions were not quite clear but I agree with David. Essentially I'm trying to create a chart of Implied vs Realised Volatility Ratios, each value displaying the ratio of ATM implied vol vs realised vol of the ATMF at that point in time. You can see these type of charts a lot in some of the sell side research for example.

lePiddu
Posts: 20
Joined: March 17th, 2015, 2:15 pm

### Re: Implied/Realised Volatility Ratio for Negative Rates?

As you approach the maturity of a swaption, the points of the yield curve that forms the swap rate are not fixed. Those points start sliding toward time=0.”

How is aging of bespoke swaps relevant when one is dealing with the implied vols of ATM par swap rates that are quoted as a constant maturity relative to the valuation date?  Every day the 5-year par swap rate is quoted with 5-years relative to the valuation date. There is no aging effect evident in a series of par swap rates.
What I'm saying is: if you use a time series of swap rates (say your 5y tenor swap rate) to estimate some measure of volatility in order to compare it with an implied volatility (say in a 1y5y swaption) you end up comparing different things.

If I look at today's 1y5y swaption implied volatility (ATM, whatever), say it's 18%. The corresponding realized volatility is going to be "realized" over the next year. And it's going to be realized by the movements of the forward rates embedded in the yield curve. The fixing date of those forward rates is constant of course, but not the time to fixing, which is going down every day. Therefore, the volatility estimated from a series of constant maturity 5y swaps is not going to match (in principle) what was actually "accounted for" in that 18% by the market.

This previous reasoning is forward in time but it's the same if you want to apply it for historical comparison: one year ago quote of 1y5y swaption implied volatility corresponds to the volatility realized by the movements of the portion of the yield curve (i.e. the forwards that makes up the 5y swap rate) that one year ago was one year forward, then one-year-minus-one-day forward, one-year-minus-two-days forward........ until today when if finally fixes the spot 5y swap.

DavidJN
Posts: 1746
Joined: July 14th, 2002, 3:00 am

### Re: Implied/Realised Volatility Ratio for Negative Rates?

If one compares an apple to an orange there should be no surprise that they're not the same.

When talking about swaptions, one is naturally interested in the behaviour of forward-starting par rates corresponding to the swaptions in question. I think that is the point you are making.

Given a complete historical set of spot-starting par swap rates it is not difficult to compute the associated forwards you are interested in and work with those.

bearish
Posts: 5425
Joined: February 3rd, 2011, 2:19 pm

### Re: Implied/Realised Volatility Ratio for Negative Rates?

Wait, you're saying it's fruitless to compare apples and oranges?

lePiddu
Posts: 20
Joined: March 17th, 2015, 2:15 pm

### Re: Implied/Realised Volatility Ratio for Negative Rates?

If one compares an apple to an orange there should be no surprise that they're not the same.

When talking about swaptions, one is naturally interested in the behaviour of forward-starting par rates corresponding to the swaptions in question. I think that is the point you are making.

Given a complete historical set of spot-starting par swap rates it is not difficult to compute the associated forwards you are interested in and work with those.
Sorry for the delay, couple of busy days =)
I may have lost the actual target of OP, but shouldn't these type of comparisons lead to a "rich/cheap" view on options by comparing historical and implied volatility?

I'm actually interested in this story! my question is: what kind of information can I obtain when comparing the estimated volatility of a series of forward swap rates (say 1y5y) with the implied volatility of the corresponding swaption?

lePiddu
Posts: 20
Joined: March 17th, 2015, 2:15 pm

### Re: Implied/Realised Volatility Ratio for Negative Rates?

Wait, you're saying it's fruitless to compare apples and oranges?
Juicy question =)

bbayram
Posts: 3
Joined: October 11th, 2018, 6:02 am

### Re: Implied/Realised Volatility Ratio for Negative Rates?

Wait, you're saying it's fruitless to compare apples and oranges?
Juicy question =)
lePiddu any chance you can DM me ?

lePiddu
Posts: 20
Joined: March 17th, 2015, 2:15 pm

### Re: Implied/Realised Volatility Ratio for Negative Rates?

Wait, you're saying it's fruitless to compare apples and oranges?
Juicy question =)
lePiddu any chance you can DM me ?
Surem, why not? =)

bbayram
Posts: 3
Joined: October 11th, 2018, 6:02 am

### Re: Implied/Realised Volatility Ratio for Negative Rates?

Juicy question =)
lePiddu any chance you can DM me ?
Surem, why not? =)

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