TITLE: No-Arbitrage Principle
QUESTION [9 upvotes]: I hope you do not mind me asking a financial question in this section.
I am having trouble understanding the concept of the no-arbitrage principle for a particular example in my notes:
Suppose a share has price 100 at time $t=0$. Suppose further that at time $t=1$ its price will rise to 150 or fall to 50 with unkown probability.
Suppose the effective rate of interest is $r$ p.a.
Suppose $C$ denotes the price at time $t=0$ of a call option for 1 share for the exercise price of 125 at the exercise time $t=1$.
Determine C so there is no arbitrage.
Solution:
I understand that if we buy 1 share and -4 (i.e. sell 4) call options at $t=0$, then the value of our portfolio at time $t=1$ is 50 regardless of whether the share value rises or falls.
The example then proceeds to say, suppoer an investor has capital $z > \text{max}\{100,4C\}$ consider the 2 possible decisions:
Investment decision A: Buy one share, invest the rest 
Investment decision B: Buy 4 options, invest the rest.
At time $t=0$, the value of his portfolio is $z$ in both cases.
The example then calculates the value of the portfolio at time $t=1$ for both A and B and states that these should be equal if we wish to not have arbitrage.
This is the part I do not understand - why should these values be the same so that we don't have arbitrage? It would seem to have something to do with the sentence I have written below 'Solution:', but I cannot seem to grasp it.
Thanks for your time and help.

REPLY [4 votes]: There is, I think, a more intuitive way to approach this problem.
If you buy the call option, its value in period 1 will be either \$25 (if the share price goes up) or \$0 (if the share price goes down).
Therefore --- here's the no-arbitrage principle --- the price of the call option has to be equal to the price of ANY portfolio that has the same payoffs in the same circumstances.
In particular:  Suppose I borrow $12.50/(1+r)$ dollars and purchase $1/4$ share.  Next period, my portfolio is worth $-12.50+(1/4)S$ where $S$ is the new share price.  (Why $12.50$?  Because that's what it costs to pay back what I borrowed.)  Notice that if the share price goes up to $S=\$150$, your portfolio is worth \$25, but if it goes down to $S=\$50$, your portfolio is worth zero --- just like the call option.
So the call option price must equal the price of this portfolio.  And what is that price?  Well, 1/4 of a share costs \$25. Of this, you're paying $12.50/(1+r)$ with borrowed money.  So the cost out of your pocket is $25-(12.50/(1+r)$ --- and that's, therefore, the call option price.
How did I come up with the idea of borrowing exactly 12.50 and buying exactly 1/4 share?  Well, I assumed you borrowed $x$ and bought $y$ shares, then calculated the future value of your portfolio in terms of $x$ and $y$, and set this value to $25$ when the stock price goes up and $0$ when it goes down, in order to make the example work.  That gave me two equations in the two unknowns $x$ and $y$, which I solved.