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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. :)



    1. [​IMG]
      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?


      e_sit, Apr 10, 2014 Report
      #1 Like Reply

    2. [​IMG]
      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 Potter, Apr 10, 2014 Report
      #2 Like Reply

    3. [​IMG]
      e_sitActive Member

      Thanks John!! I get it now :)

      e_sit, Apr 13, 2014 Report
      #3 Like Reply

    4. [​IMG]
      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 Potter

    John Potter ActEd Tutor Staff 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!

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