Time dependent volatility interest rate models

[afterworkguinness]

Hi,
I'm having trouble understanding the arguments against time dependent volatility models to value and hedge fixed income instruments.

Tuckman says:

"The downward-sloping factor structure and term structure of volatility in mean reverting models capture the behavior of interest rate movements better than parallel shifts and a flat term structure of volatility."​

Can you clarify this ? Is a flat term structure of volatility the same as constant volatility? Does mean reversion result in non-parallel shifts ?

Thanks

 

[[email protected]]

Yes, flat volatility term structure is same as constant volatility over different time horizons. However, in reality, interest rates are more volatile in the short term (the volatility term structure is downward sloping) than the long term. A mean reverting model (e.g.; Vasicek Model) conforms to this fact by incorporating how much the short term interest rate deviates from the long term equilibrium rate. The more the deviation in the current period, we will see a larger change in interest rate in the next period (this also depends on the mean reversion parameter) since the rate will be pulled towards the long run equilibrium. Thus, mean reversion will result in non-parallel shifts in interest rates across different maturities as you said.

Models without drift predict a flat term structure of interest rate volatility since they use the same volatility for all maturities. This should not be mixed with the shift in the term structure of interest rates. For example; in this setting, if we'd shock the short term rate by say +1%, interest rates for all maturities would go up by 1%, leading to a parallel shift in the term structure of interest rates (not the volatility, this point is tricky, volatility is constant, interest rates are shifting in a parallel fashion and they are not constant.)

Hope this helps.

 

[afterworkguinness]

Thanks for the detailed reply; makes sense now. One more quick question, what does Tuckman mean by a "factor structure"; from the context I'm assuming he's talking about the term structure of rates. Am I correct ?

 

[[email protected]]

My take is that, he is referring to a, in my own words, structure of mean reversion parameter. That is, some news are long-lived (smaller mean reversion parameter) and some are short-lived (larger parameter) and in that regard it is important how the mean reversion parameter is determined to capture this aspect of the varying nature of news. In Chapter 11 of his book, he refers to the use of multi-factor models that can incorporate multiple mean reversion parameters that can capture that. This part is well beyond what would show up in the FRM exam. You are a very careful reader though.