CT8 April 2018 Q5(i)

[rlsrachaellouisesmith]

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

 

[Steve Hales]

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.