# Show equivalent martingale measure Sept 2009 Q6

Discussion in 'CM2' started by Michael Truscott, Mar 18, 2020.

1. Hello!

This thread was originally in the CT8 thread, on which I have a follow-up question (at the bottom), and I was asked to move it here. Hopefully it reads ok!

Any help much appreciated.....I hate this bit of the course. Cheers

Mike

1. e_sitActive Member
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!!

e_sit, Apr 10, 2014 Report

2. John PotterActEd TutorStaff Member
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

John Potter, Apr 10, 2014 Report

3. e_sitActive Member

Thanks John!! I get it now e_sit, Apr 13, 2014 Report

4. Michael Truscott
Hello.....can I ask a follow up question please? I struggle a bit with the probability measure stuff.

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!

2. ### John PotterActEd TutorStaff Member

P-lambda is a probability measure. A probability measure is an equivalent martingale measure if the discounted value of all assets is a martingale under that measure. This is useful because we know that this is also the fair price of that asset.

Good luck!
John