# Basic Replication of European Call Option

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!