Hi, I am not sure why we are showing that EP[exp(-rt)Bt|Fs]=exp(-rs)Bs? We have been asked to show P is an equivalent martingale so I was looking to show EP[Bt|Fs]=Bs. I appreciate these are equivalent statements, but how do we know to include the PV of Bt? Also why do we have to look at the 3 scenarios default before time s default after time s but before time t not defaulted by time t Thank you Rachael
The two statements you've made aren't actually equivalent. Under the equivalent martingale measure, risky assets are expected to grow at the risk-free rate, ie: EP[Bt|Fs]=exp(r*(t-s))*Bs. If they're expected to grow at the risk-free rate, then discounting at the risk-free rate means that they aren't expected to go anywhere, ie EP[exp(-rt)Bt|Fs]=exp(-rs)Bs. This is just a rearranging of the first statement. So it's the discounted asset prices that are martingales under the equivalent martingale measure (aka the risk-neutral measure). The martingale condition EP[exp(-rt)Bt|Fs]=exp(-rs)Bs need to hold for all values of s<t, so all possible outcomes from the bond need to be considered.
