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Call Option Value from Two Approaches

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Suppose today the stock price is SS and in one year time, the stock price could be either S1S_1 or S2S_2. You hold an European call option on this stock with an exercise price of X=SX=S, where S1<X<S2S_1<X<S_2 for simplicity. So you'll exercise the call when the stock price turns out to be S2S_2 and leave it unexercised if S1S_1.

1. Replicating Portfolio Approach

Case 1Case 2
Stock PriceS1S_1S2S_2
Option: 1 Call of cost cc
Exercise?NoYes
Payoff (to replicate)0S2XS_2-X
Stock: δ\delta shares of cost δS\delta S
PayoffδS1\delta S_1δS2\delta S_2
Borrowing PV(K)
RepayKK

So we have:

δS1K=0\begin{equation} \delta S_1-K=0 \end{equation}
δS2K=S2X\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 δ\delta units of shares and the amount of borrowing:

cREP=δSPV(K)=δSKerf=δSδS1erf\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 δ\delta is defined as (S2X)0S2S1\frac{(S_2-X)-0}{S_2-S_1} as in the textbook (at introductory level),

cREP=S2XS2S1(SS1erf)\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

cRN=p(S2X)+(1p)×0erf=p(S2X)+0erf=p(S2X)erf\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

p×S2S+(1p)S1S=erf\begin{equation} p\times \frac{S_2}{S} + (1-p)\frac{S_1}{S} = e^{r_f} \end{equation}

so

p=erfS1SS2SS1S=SerfS1S2S1\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,

cRN=p(S2X)erf=SerfS1S2S1(S2X)erf=SS1erfS2S1(S2X)\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

cRN=cREPc_{RN} = c_{REP}

Hence, the call option value from replicating portfolio is the same as from risk neutral approach.