Let us assume that a portfolio consisting of going long the stock and shorting the risk free bond can be made to replicate the pay out of the option that we are trying to price. If we can obtain the price of the portfolio that it should have the same price as the option since the law of one price states that two securities with the same payout must have that same price or an arbitrage condition would exist.
To show this, let us assume that we are trying to price an option that has a strike price of 50 and expiration at the end of one period. The current price of the underlying asset is 50 and we have determined at the end of the next period the stock will either rise to 60 or fall to 40. Let us further assume that the price of the risk free bond is currently B and currently has an interest rate of 3% for the period in question. The number of shares needed will be designated as H which is the delta of the option. The option's value if the price were to rise to 50 is the intrinsic value of 10. If the price after one period is 40, the option would not be exercised and therefore the value is 0.
The two conditions that need to be met are the following:
60H- Be^(.03)(1)= 10
40H- Be^(.03)(1)= 0
20H= 10, H=.5
.5*(40)- Be^(.03)(1)= 0
B= 19.41
So, the replicating portfolio is the long .5 shares and short 19.41 of risk-free bonds. The value or cost of this portfolio at the beginning of the period is:
.5(50)-19.41= $5.59. Therefore, the price of the option is $5.59
This is the detailed explanation of how to price an option.
How do we determine whether to subtract 40H from 60H, or 60H from 40H?
After getting .5 for H, how do we know whether to plug into the 40H or 60H?
Once we get B & H, how do we know whether to plug it into 50H, the current stock price, 40H, or 60H?