a2sol - ACTSC/STAT 446/846 Assignment 2 Solutions 1...

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ACTSC/STAT 446/846 – Assignment 2 – Solutions 1. Binomial Model [10pts] See the file calculA2.xls for the details of the computations. The Delta of the option is -0.417, -0.332, -0.200, -0.150, -0.108 for S 0 = 80, 90, 110, 120, 130. As the initial stock price increases, the 95-strike put option is increasingly out of the money. With everything else equal, it is more likely that the option finishes out of the money. A hedger, e.g., a market maker, must therefore sell fewer shares initially to be able to cover the obligation she will have to meet at expiration. This number of shares in the replicating portfolio is measured by delta. The initial put delta thus tends towards zero when the initial stock price increases. 2. Binomial Model [10pts] See calculA2.xls. The answers are 7.602 and 7.877 for the put (Euro. and Amer.) and 15.029 and 15.042 for the call (Euro. and Amer.). 3. Discrete-Time model I [10pts] Solution: a) We use the asset 1 and the call on asset 1 to set up two equations involving the state-price vector. Note that since the call has a price, it means its attainable and can therefore be priced using the state- price vector. Hence we must have [10 . 8 2 . 1] = ~ ψ 12 3 8 0 . So then ~ ψ = [10 . 8 2 . 1] . 0 - 3 - 8 12 . ( - 1 24 ) = [0 . 7 0 . 3]. Then we can find a so that 10 = 0 . 7 × a + 0 . 3 × 3 that is a = 13. b) Let’s find θ 1 , θ 2 so that 12 3 8 0 θ 1 θ 2 = 1 1 After solving the two equations, one finds that θ 1 = 1 / 8 and θ 2 = - 1 / 6. The value at time 0 is θ 1 × 10 . 8 - θ 2 × 2 . 1 = 1 . 35 - 0 . 35 = 1. So the interest rate must be r = 0%.

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