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# Matrix inverses

A market has two stocks S1 and S2. Both S1 and S2 are valued at \$1 per unit today andcan be worth either \$2 or \$0.5 per unit after one year. Assume that money M valued at \$1per unit today does not change its value during that time.(a) Suppose a convertible bond B that pays either \$2 when S1 = \$2 or \$1 whenS1 = \$0.5 in a year is also traded in the market.(i) Calculate the portfolio (S1;M) that replicates B using matrix inverses.(ii) Calculate the price of convertible bond B.(b) Suppose an option O1 that pays \$1 when both S1 = S2 = \$2 (and 0 otherwise)is traded in the market and is worth \$0.125. Also suppose option O2 that pays \$1 whenS1 = \$2 but S2 = \$0.5 (and 0 otherwise) is traded in the market as well.(i) Calculate the portfolio (S1;S2;M;O1) that replicates O2 using row reduction.(ii) Calculate the price of option O2.

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