# Price volatility and yield volatility

This question is a bit confused, but please bear with me. Now and then I see people use the terminology "price volatility" and "yield volatility" in connection with bond options. I understand the concept of implied volatility for equity options, and I know that option prices are often quoted in terms of their implied volatility.

Is this something similar? Take a Bond options for example. Given a market price in e.g. EUR it is possible (analogues to equity options using Black-Scholes) to find an implied volatility (using Black-76). I guess this is the so called price (bond) volatility.

But when it comes to the so called "yield volatility". I cannot understand how these are implied. I have spent some time on Google trying to find a solid source where I can read more about this, but I can't find it.

Thanks in advance for any assistance!

• FYI option prices are not quoted in terms of their volatility, they are quoted in price, either a "live price" or a price with a spot ref at which you would agree to trade the delta at the same time. People may ask for the vol at the same time though if they can't be bothered to input the quotes into their own pricing tools. – will Oct 6 '18 at 11:45

## 2 Answers

The price volatility of a bond option is the implied volatility using a Black type model, so it is exactly analogous to an equity option, using the bond price instead of the equity price.

Because long dated bonds have naturally more price volatility than short dated bonds due to the extra duration , using price volatility is not very helpful when comparing the prices of options on different bonds. Hence traders also look at yield volatility, which is the implied lognormal volatility if you price the bond option on a lognormal binomial tree using the bond yield as the underlying variable. One can also calculate the normalized yield volatility in this manner by assuming a normal instead of lognormal distribution for the yield. This calculation can also be done analytically instead of on a tree, by integrating the option payoff as a function of the bond yield , against the assumed pdf of the yield.

They are not referring to any implied volatility but actual volatility, i.e. statistical standard deviation. The price volatility is the annualized standard deviation of bond price changes and the yield volatility the annualized standard deviation of bond yield changes. These quantities are usually estimated using a historical estimator. If you have n observations of a quantity X with a sample mean of $\bar{x}$ then its standard deviation is estimated as: $$\hat{\sigma}_x = \left( \frac{1}{n-1} \sum_{i=1}^{n} (x_i-\bar{x})^2\right)^{\frac{1}{2}}$$ Most frequently, daily closing price/yield observations are used in which case you have to multiply with $\sqrt{N}$ where N are the number of days per year (depending on the day count convention).

• This doesn't look correct. For price vol then one should take log P(t)/P(t-1) and for yield, the descrete change as you have written. Moreover, in markets where swaptions are actively traded, then you do indeed have market implied yield and market implied price volatility. – Yugmorf Oct 4 '18 at 2:59
• @yugmorf the above is correct for normal vol, and they state pretty clearly that they're talking about the standard deviation... – will Oct 6 '18 at 11:46
• I think the question is quite clear in that they're are talking about 'implied volatility', and this was well adressed by @dm63's answer. – Yugmorf Oct 11 '18 at 2:07