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I have a question I am not sure how to approach: Suppose interest rates is 50%, a stock worth \$1 today can be worth \$2, \$1, \$0.5 next year.

If the option that pays \$1 only when S = \$2 is traded in the market and is worth \$0.125, calculate the price and replicating portfolio of the option that pays \$0.5 when S = \$1.

It has something to do with pay-off matrix but I don't know how to apply it?

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up vote 3 down vote accepted

If I understand well, you have a market with 3 states: up, flat or down.

You have 3 instruments:

  1. The stock
  2. The risk-free rate (50%)
  3. The option

If you can create a portfolio today with these 3 instruments that can replicate de payoff of the option you have to price, then the law of one price tells you that the price of the option should be the price of this portfolio.

So, you have to solve a system of 3 equations with 3 unknown:

Assume you will hold $a$ stock, $b$ risk-free bond, $c$ option, and using fraction notation for 0.5, 1.5, the system is:

Up State: $a \cdot 2 + b \cdot \frac{3}{2} + c \cdot 1=0$

Flat State: $a \cdot 1 + b \cdot \frac{3}{2} + c \cdot 0=\frac{1}{2}$

Down State: $a \cdot \frac{1}{2} + b \cdot \frac{3}{2} + c \cdot 0=0$

From the third equation, you get $a=-3b$.

Substituting into the second, you get $-3b+\frac{3}{2}b = \frac{1}{2}$ which yield $b=-\frac{1}{3}$ and hence $a=-3 \cdot \left( - \frac{1}{3} \right)=1$.

Finally, substituting in the first equation, you get $1 \cdot 2 + \left( - \frac{1}{3} \right) \cdot \frac{3}{2} + 1 \cdot c =0$.

This yields $c= - \frac{3}{2}$.

So the portfolio holding 1 stock, selling $\frac{1}{3}$ risk-free bond and selling $\frac{3}{2}$ options replicates perfectly the payoff of the option you have to price. The price of this option is hence $1 \cdot 1\$ - \frac{1}{3} \cdot 1\$ - \frac{3}{2} \cdot 0.125\$=0.48\$$

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In the final sentence, you mean selling $\frac{3}{2}$ options and not $12$ right? –  Mat.S Feb 15 at 20:15
    
@Mat.S oh yes of course! –  SRKX Feb 16 at 13:38
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