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.

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