Volatility relationship between underlying and derivative

[Abhinav Agrawal]

The material mentions that the volatility of the derivative is Duration multiplied by the volatility of underlying. E.g Interest rate volatility is 0.5% and duration of gov bond is 8, then volatility of gov bond will be 8*0.5% = 4%.
Any leads as to how this is derived?

 

[ShaktiRathore]

hi,
duration of bond=dP/(P*di) where di is small change in interest rate and dP/P is small percent change in price of bond
=> duration of bond*di=dP/P
=>volatility(duration of bond*di)=volatility(dP/P)
=>volatility(duration of bond)*di+volatility(di)*duration of bond=volatility(dP/P)
=>volatility(di)*duration of bond=volatility(dP/P) assuming duration is constant for small changes in interest rates =>volatility(duration of bond)=0
=>volatility of interest rates*duration of bond=volatility of underlying

thanks