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:
\[ \begin{equation} \delta S_1-K=0 \end{equation} \]
\[ \begin{equation} \delta S_2 -K = S_2-X \end{equation} \]
Therefore, the call option value is given by the difference between the cost of \(\delta\) units of shares and the amount of borrowing:
\[ \begin{align} c_{REP} &= \delta S - PV(K) \newline &= \delta S - Ke^{-r_f} \newline &= \delta S - \delta S_1e^{-r_f} \end{align} \]
When \(\delta\) is defined as \(\frac{(S_2-X)-0}{S_2-S_1}\) as in the textbook (at introductory level),
\[ \begin{equation} c_{REP}= \frac{S_2-X}{S_2-S_1}(S - S_1e^{-r_f}) \end{equation} \]
2. Risk Neutral Approach
Without too much trouble, we can derive the call value using risk neutral approach as
\[ \begin{align} c_{RN} &= \frac{p(S_2-X)+(1-p)\times0}{e^{r_f}}\newline &= \frac{p(S_2-X)+0}{e^{r_f}}\newline &= p(S_2-X) e^{-r_f} \end{align} \]
We know that
\[ \begin{equation} p\times \frac{S_2}{S} + (1-p)\frac{S_1}{S} = e^{r_f} \end{equation} \]
so
\[ \begin{align} p &= \frac{e^{r_f}-\frac{S_1}{S}}{\frac{S_2}{S}-\frac{S_1}{S}}\newline &=\frac{Se^{r_f}-S_1}{S_2-S_1} \end{align} \]
Therefore,
\[ \begin{align} c_{RN} &= p(S_2-X) e^{r_f}\newline &=\frac{Se^{r_f}-S_1}{S_2-S_1}(S_2-X) e^{-r_f}\newline &=\frac{S-S_1e^{-r_f}}{S_2-S_1}(S_2-X) \end{align} \]
Identical Result from the Two Methods
It’s easy to find that
\[ c_{RN} = c_{REP} \]
Hence, the call option value from replicating portfolio is the same as from risk neutral approach.