# 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.