Call Option Value from Two Approaches

Teaching Notes

Mingze Gao, PhD

Macquarie University


May 25, 2020

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

\[ \begin{equation} p\times \frac{S_2}{S} + (1-p)\frac{S_1}{S} = e^{r_f} \end{equation} \]


\[ \begin{align} p &= \frac{e^{r_f}-\frac{S_1}{S}}{\frac{S_2}{S}-\frac{S_1}{S}}\newline &=\frac{Se^{r_f}-S_1}{S_2-S_1} \end{align} \]


\[ \begin{align} c_{RN} &= p(S_2-X) e^{r_f}\newline &=\frac{Se^{r_f}-S_1}{S_2-S_1}(S_2-X) e^{-r_f}\newline &=\frac{S-S_1e^{-r_f}}{S_2-S_1}(S_2-X) \end{align} \]

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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