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I am looking at the very basics of replicating an option with a portfolio of risky and risk free assets. As such we can define a portfolio of $x$ no. of shares, $y$ bonds & $z$ options at time $(T)$ as;

\begin{equation} V(T) = xS(T) + yA(T) +zC(T) \end{equation}

I understand that due to the No-Arbitrage Principle that there is:

No portfolio that includes a position $z$ in call options and has initial value $V(0) = 0$ such that $V(T) \geq 0$ with probability 1 and $V(T) > 0$ with non-zero probability.

In the textbook we are then presented the value of an option to be;

$C(0) = x S(0) + yA(0)$

Or else arbitrage could occur.

I have completed exercises on this and understand the principle/procedures in real life that would lead to this opportunity, however the proof of this equality then takes the form of supposing:

$C(0) > xS(0) + yA(0)$

  • We issue and sell one option for $C(0)$
  • Take a long position in the equivalent portfolio $(x,y)$, i.e. buying shares and borrowing cash for replication of a call option

We thus have a positive balance of $C(0) -xS(0) - yA(0) > 0$ and invest this excess amount risk free.

The point I take objection with then follows in the assertion that the resulting portfolio then has an initial value of $V(0) = 0$, as if we were to sub this into our first equation, we would obviously end up with a non zero positive value due to our excess balance given by the inequality. I would also assume we can't just look at the $(x,y)$ portion of the portfolio following the definition given in the first quote block?

I understand I am probably missing a very basic concept in trying to understand this but it is annoying me enough that I wish to understand before progressing. If anyone can clear this up for me it would be greatly appreciated.

Cheers!

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There is no excess balance. As you stated, call option replication includes borrowing to fund the stock purchase. If you borrow and lend at the same interest rate, there is no profit.

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