I'm reading Jarrow and Turnbull (1997) .
They defined p(t,T) as the time t price of a default free zero coupon bond paying a sure dollar at time T where $0\le t \le T$ (in year) . They also defined the discrete time forward rate (for time period (t,T) ) by $$ f(t,T) = - ln\left(\frac{p(t,T+1)}{p(t,T)} \right)$$ However , I suspect the above forward rate is for time period (T,T+1) instead of (t,T) . Here is my interpretation ,
From first look , it's hard to make sense of the above expression : p(t,T) is the present value of the \$ 1 cash flow at time T , we already have enough information to evaluate f(t,T) , there's no need for p(t,T+1) . To more rigorously illustrate my suspicion ,
First 'discrete time' describes the manner in which risk free rate changes , in this case our risk free rate changes discretely and compound continuously .
Second , p(t,T) and p(t,T+1) are discount factors for time period (t,T) , (t,T+1) respectively , directly from wiki , our above expression on the RHS $$- ln\left(\frac{p(t,T+1)}{p(t,T)} \right) = ln[p(t,T)] - ln[p(t,T+1)] $$ $$ = \frac{1}{T+1-T} ( ln[DF(t,T)] - ln[DF(t,T+1)] ) = f(T,T+1)$$ where DF is notation for discount factor , refer to the 'Continuously compounded rate' section of wiki , wiki link : https://en.wikipedia.org/wiki/Forward_rate
is my suspicion correct ? I really want to confirm this because they use this expression throughout the paper !
The notation author used may be confusing , a better notation would be f(t,T,T+1) , that's the current (time t) forward rate for period (T,T+1) .