Show equivalent martingale measure Sept 2009 Q6

[e_sit]

In part (ii), we are asked to show that P_lambda is an equivalent martingale measure.

The answer shows this by proving the discounted process: e^(-rt)D_t is a martingale under all scenarios.

Why is showing the discounted price process is a martingale proves that P_lambda is an equivalent martingale?

Shouldn't we try to show that D_t is a martingale instead?

Thanks!!

 

[John Potter]

No, in this question, Dt is the bond price process. In the risk-neutral world, we need the expected return on the bond to equal the return on cash.

E[Dt|Fs] = Ds exp(t-s)r

This is the same as needing the DISCOUNTED bond price process to be a martingale:

exp(-rt)E[Dt|Fs] = Ds exp(-rs)

E[Dt exp(-rt)|Fs] = Ds exp(-rs)

John

 

[e_sit]

No, in this question, Dt is the bond price process. In the risk-neutral world, we need the expected return on the bond to equal the return on cash.

E[Dt|Fs] = Ds exp(t-s)r

This is the same as needing the DISCOUNTED bond price process to be a martingale:

exp(-rt)E[Dt|Fs] = Ds exp(-rs)

E[Dt exp(-rt)|Fs] = Ds exp(-rs)

John

Thanks John!! I get it now

 

[Michael Truscott]

Hello.....can I ask a follow up question please? I struggle a bit with the probability measure stufff.

I think I get what John has written, but don’t understand what this has to do with p-lambda, or what p-lambda really is and so how this answers the question.

Any help greatly appreciated!

Mike