# Basic question/clarification about the LOOP

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:

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.

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.
• 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!). Commented Feb 21, 2023 at 23:38
• 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. Commented Feb 23, 2023 at 12:11