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This is a very basic question/comment regarding the way that the LOOP is stated in the book "Dan Stefanica - A Primer for the Mathematics of Financial Engineering". The proposition goes as follows:

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What confuses me is that there is no restriction on the number of dividends paid by these portfolios at any time $t < \tau$ prior to maturity.

Indeed:

  • let $V_{1}$ be a coupon-bearing bond and
  • let $V_{2}$ be a zero-coupon bond,

both with same maturity and same payoff/value at maturity. The LOOP can't be true if $V_{1}$ has non-zero coupon payments at $t < \tau$. Should this additional restriction be added or is there something that I'm missing? Thanks in advance!

EDIT:

I’m looking for an elaborate answer on why would the author say that the statement holds only assuming the existence of one $\tau$. I guess that the author is thinking of the V’s as instruments that have only one payoff at a fixed time (like european options or forwards). If that is not the case, (V is an american option, for example) then I guess that the equality of the portfolios should be required on all $\tau$’s between $t$ and maturity.

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You are missing something, you should interpret $\tau$ as every point in time from $t$ to maturity (in the case of your example). Clearly your statement only holds true for $\tau = maturity$. In any point in time such that $t < \tau < maturity$ the portfolio of the zero coupon bond and the coupon bearing bond will be different.

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  • $\begingroup$ It should be interpreted as you say, obviously, but the point is that it's clearly wrong stated, it says explicitly that if there exists some $\tau$ such that both portfolio values agree, then they must agree at any time prior to $\tau$, which is clearly wrong as the example shows. It should say: if for every $\tau > t$ we have equality, then we have equality at time $t$ as well (just as you say!). $\endgroup$
    – user_12345
    Feb 21, 2023 at 23:38
  • $\begingroup$ @Smm it would be good to clarify why you accepted the answer and then unaccepted it. $\endgroup$
    – phdstudent
    Feb 22, 2023 at 20:03
  • $\begingroup$ Sorry, I was looking for a more elaborate answer on why would the author say that the statement holds only assuming the existence of one $\tau$. I guess that the author is thinking of the V’s as instruments that have only one payoff at a fixed time (like european options or forwards). If that is not the case, (V is an american option, for example) then I guess that the equality of the portfolios should be required on all $\tau$’s between $t$ and maturity. $\endgroup$
    – user_12345
    Feb 23, 2023 at 12:11
  • $\begingroup$ It’s surprising that such a simple distinction hasn’t been made, and I wanted to verify why the author would do that. It’s not a matter of interpretation, without additional assumptions the proposition is wrong in general. $\endgroup$
    – user_12345
    Feb 23, 2023 at 12:11

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