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!
