# Call Option Value from Two ApproachesÂ¶

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 ApproachÂ¶

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 ApproachÂ¶

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 MethodsÂ¶

It's easy to find that

Hence, the call option value from replicating portfolio is the same as from risk neutral approach.