160a09_hw8sol

# 160a09_hw8sol - 100 120 70 100(20(0-10(30-30(0 10 20 Using...

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Winter 2009 PSTAT 160a HW #8 solutions Problem 1 1) @ @ K Think (or protect against) the price will go down 2) ± ± K Think the price will not go down 3) @ @ K Think the price will not go up 4) ± ± ± @ @ @ K Think the price will go either up or down a lot (high volatility) 5) ± ± K 2 K 1 6) @ @ K 2 K 1 7) @ @ K 1 ± ± K 2 ² ² K 3 Problem 2 Here S 1 i is the price of asset i , where i = 1 is for gas, i = 2 electricity, i = 3 call option, i = 4 spread option. Recall that for options, S 1 is the payoﬀ at time 1, S 0 is the price we are looking for. In this problem, the tree takes the form (the number in parenthesis are the return of the stocks) ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ * H H H H H H H H H H j - S 1 1 (gas) S 1 2 (electricity) S 1 3 (call option) S 1 4 (spread option)
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Unformatted text preview: 100 120 100 90 130 70 100 (20) (0) (-10) (30) (-30) (0) 10 20 Using the arbitrage theorem for stock 1 and 2 we get: 20 p 1-10(1-p 1-p 2 ) = 0 30 p 1-30 p 2 = 0 ⇒ 30 p 1 = 30 p 2 40 p 1 = 10 ⇒ p * = (1 / 4 , 1 / 4 , 1 / 2) Since we were able to ﬁnd p * , there can’t be any arbitrage oportunity. Now using p * and the arbitrage theorem to ﬁnd the price of the options, we have S 3 = E p * ( S 1 3 ) = 10 × 1 4 = \$2 . 5 , and for the spread option S 4 = E p * ( S 1 4 ) = 20 × 1 4 = \$5...
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