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So the question asks: L et $S(0) = 120$ dollars, $u = 0.2$, $d = −0.1$ and $r = 0.1$. Consider a call option with strike price $X = 120$ dollars and exercise time $T = 2$. Find the option price and the replicating strategy.

So the solution is:

The option price at time $0$ is $22.92$ dollars. (Yes, I got the same answer)

In addition to this amount, the option writer should borrow $74.05$(?) dollars and buy $0.8081$ (?) of a share.

At time 1, if $S(1) = 144$, then the amount of stock held should be increased to 1 share, the purchase being financed by borrowing a further $27.64$ dollars, increasing the total amount of money owed to $109.09$ dollars. (Understood. $144*(1-0.8081) = 27.64$, $27.64+74.05+7.4=109.09$)

If, on the other hand, $S(1) = 108$ dollars at time $1$, then some stock should be sold to reduce the number of shares held to $0.2963$(why?), and $55.27$ (?) dollars should be repaid, reducing the amount owed to $26.18$ (what?) dollars. (In either case the amount owed at time 1 includes interest of 7.40 dollars on the amount borrowed at time 0.)

So basically, I really can't see why it decides to buy $0.8081$ share at time 0?

Also, where does the number $0.2963$ come from? Even though the share should be reduced to $29.63%$, how does it come out with the repaid amount, $55.27$ and the amount owned, $26.18$?

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You are at the beginning of a period and the stock price, worth $S$, can evolve in either of the 2 states: $S_u = u S$ or $S_d = d S$.

The part you don't understand is related to forming so-called replicating portfolios. More specifically, using only the stock and a (risk-less) cash account, the question is 'How can one build a portfolio allowing to perfectly replicate the option's behaviour over a given time frame' (here a period of the binomial tree).

Let $\Pi$ denote such a portfolio. Let $\Pi$ consists of $\alpha$ shares of the stock and $\beta$ in cash. Both $\alpha$ and $\beta$ can be positive or negative depending on if you own/sold shares or borrowed/lent cash.

$$\Pi = \alpha S + \beta$$

At the moment we don't know $\alpha$ nor $\beta$. But we can determine them easily. Indeed, at the end of a period, suppose that the option is worth $V_u$ in the up state and $V_d$ in the down state. Then, because we want our portfolio to be replicating, we want its value $\Pi$ to evolve to the exact same states i.e. $\Pi_u = V_u$ and $\Pi_d = V_d$. This yields two equations:

$$\Pi_u = \alpha S_u + \beta (1 + R) = V_u$$ $$\Pi_d = \alpha S_d + \beta (1 + R) = V_d$$

  • The shares' component of portfolio $\Pi$ evolved from $\alpha S$ to $\alpha S_u$ or $\alpha S_d$, because the stock price evolved.

  • The amount of cash $\beta$ borrowed/lent at the beginning of the period, has cost/earned us some interest, hence the factor $1+R$.

Solve these 2 equations for the 2 uknowns $\alpha$ and $\beta$ to end up with:

\begin{align*} \alpha &= \frac{V_u - V_d}{S_u-S_d} \\ \beta &= \frac{1}{1+R} \frac{u V_d - d V_u}{(u-d)} \end{align*}

This gives you the number of shares ($\alpha$) you need to buy/sell and cash ($\beta$) you need to borrow/lend to perfectly replicate the option. Of course because the portfolio is replicating $\Pi = V$ the option value at the beginning of the period, which is a good way to check whether you made a mistake.

If you've built your option tree in the right way (which seems to be the case since you agree on the option premium) then on the first period you can compute $\alpha$ and $\beta$ from the above equations and it should give you the numbers that puzzle you, i.e. $0.8081$ and $-74.05$ respectively. Note that if you do the computations $\Pi = 0.8081\times120 - 74.05 = 22.92 = V$, hence ok.

Now you can repeat the calculation for any future period to get the amount of shares and cash you should borrow/lend to further replicate the option.

A subtle point here: the exercise considers that you are hedging the option. In other words, you are long the option but short the replication portfolio. This is why although you'll find $\alpha = +0.8081$ (buy shares) and $\beta=-74.05$ (lend cash) over the first period, you should actually reverse that position (because you are short the replicating portfolio when hedging a long option position).

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