forward rate vol to zero coupon vol

palsky · October 25th, 2002, 2:48 pm

Is there a simple way to compute the zero coupon volatility if you know the forward rate volatility ?And reciprocally ?Thanks for your help.

Monk · October 25th, 2002, 4:06 pm

zero rate is an average of forward rate.Zero(t,T) = 1/(T-t) * Integral fwd(t,u)du from u=t to TSo if you know the dynamics of fwd(t,*), then you can theoretically calculate the dynamics of zero rate.You may find it in some papers on HJM.Hope it helps.Monk

palsky · October 30th, 2002, 9:07 am

Question was not clear.Suppose I know the volatility of the x years zero rate (this rate has a log-normal distribution)What is the volatility of the constant maturity x years zero coupon ?Relation is not very linear ZC = (1+zr)^(-x)First approach is to write ZC (close to) 1-x*zr, but then ?

jonath024 · October 30th, 2002, 4:59 pm

palsky · November 4th, 2002, 10:25 am

Thanks, JonThis does mean that the zero coupon vol is equal to the zero rate vol multiplied by the modified duration?Vol is always positive, so I suppose you mean only that rates and prices always move in opposite direction.

jonath024 · November 4th, 2002, 2:59 pm