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Math Help - derivative valuation question

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    derivative valuation question

    Question.

    Let V be the value of a derivative written on an asset whose time t=o value is S and whose value in the next period at time t=1 will be Su or Sd, where d<1=r<u and r is the interest rate for the period. Suppose the derivative value in the next period is correspondingly V_u and V_d. Show by constructing an appropriate portfolio that

    (1+r)V=\frac{(V_u-V_d)(1+r)+V_dd-V_ud}{u-d}.

    The background to this question is a chapter in Mathematics for Finance and Valuation entitled "One period, one risky asset and two states" where the risk-neutral probability was introduced and used to value the forward contract and call and put options.

    I cannot get to that equation. My attempt at solution:

    First, I find state probabilities (eg q_u is the probability of "up" state and so on) from the current price of asset S:

    Suq_u+Sdq_d=S<br />
q_u+q_d=1

    Suq_u+Sd(1-q_u)=S eliminate S

    q_uu+d-q_ud=1

    q_u=\frac{1-d}{u-d}

    q_d=1-q_u=\frac{u-1}{u-d}

    Then I assume I can use the same probabilities to value the derivative claim on the asset S (can I?). The value of the derivative in one year's time V(1+r) should be

    V(1+r)=V_uq_u+V_q_d=V_u\frac{1-d}{u-d}+V_d\frac{u-1}{u-d}=\frac{V_u(1-d)+V_d(u-1)}{u-d}=\frac{(V_u-V_d)+V_du-V_ud}{u-d}

    So my answer is different from the instruction by one component: I have V_du and they have V_dd. Any comments?

    Another question, does interest r=1 make sense?... Is that 100%?
    Last edited by Volga; January 8th 2011 at 06:25 AM.
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