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Copying a payoff

Linear algebraHard

There are $3$ possible outcomes. Security $S_{1}$ pays $(1, 1, 1)$ , security $S_{2}$ pays $(3, 1, 0)$ and security $S_{3}$ pays $(0, 2, 4)$ , each vector listing the payoff in outcomes $1$ , $2$ and $3$ . Their prices are $0.95$ , $2$ and $1.5$ . A trader holds $a$ units of $S_{1}$ , $b$ units of $S_{2}$ and $c$ units of $S_{3}$ , chosen so that the portfolio pays exactly $(5, 7, 10)$ . What does this portfolio cost?

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Hint

Write one equation per outcome, solve for the holdings, then price them.

Solution

Matching each outcome gives $a + 3 b = 5$ , $a + b + 2 c = 7$ and $a + 4 c = 10$ .
Subtracting the first equation from the second gives $2 c - 2 b = 2$ , so $c = b + 1$ .
Subtracting the first equation from the third gives $4 c - 3 b = 5$ , so $4 b + 4 - 3 b = 5$ and $b = 1$ .
Then $c = 2$ and $a = 5 - 3 \times 1 = 2$ .
The cost is $2 \times 0.95 + 1 \times 2 + 2 \times 1.5 = 6.9$ .

Answer: $6.9$

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