Call Option Value from Two Approaches

Author

Affiliation

Mingze Gao, PhD

Macquarie University

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:

$$\delta S_1-K=0$$

$$\delta S_2-K=S_2-X$$

Therefore, the call option value is given by the difference between the cost of $\delta$ units of shares and the amount of borrowing:

$$\begin{array}{rl}{c}_{REP}& =\delta S-PV(K)\\ & =\delta S-K{e}^{-r_f}\\ & =\delta S-\delta S_1{e}^{-r_f}\end{array}$$

When $\delta$ is defined as $\frac{(S_2-X)-0}{S_2-S_1}$ as in the textbook (at introductory level),

$$c_{REP}=\frac{S_2-X}{S_2-S_1}(S-S_1{e}^{-r_f})$$

2. Risk Neutral Approach

Without too much trouble, we can derive the call value using risk neutral approach as

$$\begin{array}{rl}{c}_{RN}& =\frac{p(S_2-X)+(1-p)\times 0}{e^{r_f}}\\ & =\frac{p(S_2-X)+0}{e^{r_f}}\\ & =p(S_2-X)e^{-r_f}\end{array}$$

We know that

$$p\times \frac{S_2}{S}+(1-p)\frac{S_1}{S}=e^{r_f}$$

so

$$\begin{array}{rl}p& =\frac{e^{r_f}-\frac{S_1}{S}}{\frac{S_2}{S}-\frac{S_1}{S}}\\ & =\frac{S{e}^{r_f}-S_1}{S_2-S_1}\end{array}$$

Therefore,

$$\begin{array}{rl}{c}_{RN}& =p(S_2-X)e^{r_f}\\ & =\frac{S{e}^{r_f}-S_1}{S_2-S_1}(S_2-X)e^{-r_f}\\ & =\frac{S-S_1{e}^{-r_f}}{S_2-S_1}(S_2-X)\end{array}$$

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.