risk free pricing of bonds

I couldn't work out what type of replication portfolio you needed to value bonds? So here are some notes from Hull:

The key is black & scholes showed that the derivative price's equation & hence solution, is not reliant on the expected rate of return, this is turn means we can assume any risk aversion (i.e. any expected rate of return) . Hence we can use any probability measure to value derivatives, some just turn out easier than others depending on what we are valuing...

Let f & g be securities prices dependent on one source of uncertainty

(1) df/f = muf*dt +thetaf*dZ
(2) dg/g = mug*dt +thetag*dZ

if you eliminate dZ, by combining (1) & (2), & subtitute the risk free return for the portfolio
(thetaf*f*g+ thetaf*f*g) + it can shown that both f & g have the same market price of risk, this was alluded to in the notes.

More over, choose the probability measure such that the market price of risk is thetag, then if you substitute muf & mug in terms of r, thetag & thetaf

Then d(Lnf –Lng) = d(Ln(f/g)) = 0.5*dt*(thetaf – thetag)^2 + (thetaf – thetag)*dZ

Hence d(f/g) = (thetaf – thetag)*f/g*dZ, i.e. f/g is a martingale under this measure, so

f(0)/g(0) = E [f(T)/g(T)] or f(0)=g(0)* E[f(T)/g(T)]

This our derivative pricing formulae

if you choose g to be cash worth $1, i.e. dg =r*g*dt

then you get f(0)= 1*E[f(T)/exp(integral{r*dt})]

i.e. the short rate appear inside the expectation (this is how notes valued risk free bonds - i.e. let f(T)=1)

if you choose g to be a risk free bond worth $1 at time T, i.e. g(0) =exp(integral{-r*dt}), this is a different probability measure

then you get f(0)= g(0)*E[f(T)/1] = exp(integral{-r*dt}) *E[f(T)]

i.e. the short rate appear outside the expectation (this is how the notes valued options)