Fixed Income Mathmetics

LIBOR rate and Bond price
L(t,S,T) = -(p(t,T) - p(t,S))/((T-S) p(t,T))
* LIBOR is just discretely compounded forward rate

Continuously compounded forward rate and Bond price
R(t,S,T) = -(log(p(t,T)) - log(p(t,S)))/(T-S)


Instantaneous forward rate and Bond price
f(t,T) = -d(log(p(t,T)))/dT

Instantaneous short rate
r(t) = f(t,t)

Bond price and Instantaneous forward rate
p(t,T) = exp[-\int^T_t f(t,s) ds]
Note: this is just the inversion of the f vs. p formula above

Bond price and Instantaneous short rate
p(t,T) = E^Q[exp[-\int^T_t r(s) ds]]
Note: this comes from valuing a 1 dollar payoff under the Q-measure