I am studying forward rates, there is a formula in the book that I am trying to understandit states forward rate;f(t,T) = 1/(delT. B(0,T)) x Ern[ exp^{-\int_{0}^{t}r(s)ds} - exp^{-\int_{0}^{T}r(s)ds} ]where delT is (T-t), Ern is risk neutral expectation, B(t,T)=Ern[exp^{-\int_{t}^{T}r(s)ds} ]then it followsf(t,T) = 1/(B(0,T)) x Ern[ exp^{-\int_{0}^{t}r(s)ds} x (1- exp^{-\int_{t}^{T}r(s)ds} )/delT ]and replaces the integral from t to T with f(t,T) = 1/(B(0,T)) x Ern[ exp^{-\int_{0}^{t}r(s)ds} x (1- B(t,T))/delT ]It claims risk neutral expectation are determined working backward in time. I am not sure if this is right. This is like writing B(t,T) = exp^{-\int_{t}^{T}r(s)ds} , not the expectation of the exponential integral Do you have any guesses??Thanks

i am not sure i understand you perfectly.still, i think the formula is fine because of the law of iterated expectation.

Law of iterative expectation states thatErn[B(t,T)] = Ern[ exp^{-\int_{t}^{T}r(s)ds} ]but how can you apply it here, while there is a multiplication by exp^{-\int_{0}^{t}r(s)ds} in front of exp^{-\int_{t}^{T}r(s)ds} ]

because exp^{-\int_{0}^{t}r(s)ds} is independent on exp^{-\int_{t}^{T}r(s)ds}.

I might be able to accept the independency of B(0,t) from B(t,T) but what about the next argument from the same derivationB(t,T) = 1/[1 + R(t,T) delT]where R(t,T) is the LIBOR rate from t to T. Following the derivation that I gave on previous mesgf(t,T) = 1/B(0,T) Ern[ exp^{-\int_{0}^{t}r(s)ds} R(t,T) B(t,T) ]and with same argument of risk neutral expectation are determined working backward in timef(t,T) = 1/B(0,T) Ern[ exp^{-\int_{0}^{t}r(s)ds} R(t,T) exp^{-\int_{t}^{T}r(s)ds}]for this to happen with your argument, you have to claim R(t,T) independent of exp^{-\int_{t}^{T}r(s)ds}] too. How do you prove that?

try to plug in the definition of libor forward rate and you will go back to the original formulation below, if i am not mistaken.

Could you be more specific;what I am trying to get for the forward rate isf(t,T) = Ern[ exp^{-\int_{0}^{T}r(s)ds} R(t,T) ] / Ern[ exp^{-\int_{0}^{T}r(s)ds} ]I believe there is a similar derivation on Quantitative Modeling of Derivative Securities by Avellaneda and Laurance on Section 12.3, but I do not have that book