Call Option Value from Two Approaches¶
Suppose today the stock price is \(S\) and in one year time, the stock price could be either \(S_1\) or \(S_2\). You hold an European call option on this stock with an exercise price of \(X=S\), where \(S_1<X<S_2\) for simplicity. So you'll exercise the call when the stock price turns out to be \(S_2\) and leave it unexercised if \(S_1\).
1. Replicating Portfolio Approach¶
Case 1 | Case 2 | |
---|---|---|
Stock Price | \(S_1\) | \(S_2\) |
Option: 1 Call of cost \(c\) | ||
Exercise? | No | Yes |
Payoff (to replicate) | 0 | \(S_2-X\) |
Stock: \(\delta\) shares of cost \(\delta S\) | ||
Payoff | \(\delta S_1\) | \(\delta S_2\) |
Borrowing PV(K) | ||
Repay | K | K |
So we have:
Therefore, the call option value is given by the difference between the cost of \(\delta\) units of shares and the amount of borrowing:
When \(\delta\) is defined as \(\frac{(S_2-X)-0}{S_2-S_1}\) as in the textbook (at introductory level),
2. Risk Neutral Approach¶
Without too much trouble, we can derive the call value using risk neutral approach as
We know that
so
Therefore,
Identical Result from the Two Methods¶
It's easy to find that
Hence, the call option value from replicating portfolio is the same as from risk neutral approach.