sol1 - ACTSC/STAT 446/846 Midterm 1 Fall 2011 Solutions 1....

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ACTSC/STAT 446/846 – Midterm 1 – Fall 2011 – Solutions 1. [12 points] Consider an equity-linked contract as seen in class, in which an investor receives $1000 max p 1 , 1 + 0 . 6 p S ( T ) S (0) - 1 PP ( * ) at T = 2 years, where S ( t ) is the value of the index at time t . The value of the index is modeled by a binomial model with two time-steps of length h = 1 year, u = 1 . 05, d = (1 /u ) and S (0) = 20. The continuously compounded interest rate is 3% per year. (a) [6pts] Compute the value of the equity-linked contract at time 0. Solution: The possible index prices at time 2 are S (2 , 0) = 18 . 141 , S (2 , 1) = 20 and S (2 , 2) = 22 . 05. Hence the contract’s possible values at time 2 are V (2 , 0) = V (2 , 1) = 1000 and V (2 , 2) = 1061 . 5. We then compute the risk-neutral probability q = e 0 . 03 - 1 / 1 . 05 1 . 05 - 1 / 1 . 05 = 0 . 79978 . So that the value of the contract at time 0 is V (0) = e - 0 . 03 × 2 ( q 2 × 1061 . 5 + (1 - q 2 ) × 1000) = 978 . 81 . Note: another solution is to write V (0) = 1000 e - rT +30 C 0 , where C 0 is the price of a call option on the index with strike 20, using the decomposition 1000 max p 1 , 1 + 0 . 6 p S ( T ) S (0) - 1 PP = 1000 + 600 /S (0) × max(0 , S ( T ) - S (0)) of the payoF. One then needs to price the call using the binomial tree, getting C 0 = 1 . 23, and thus V (0) = 1000 e - 0 . 06 + 30 × 1 . 23 = 978 . 81. (b) [5pts] Determine the composition at time 1—in the case S (1) = uS (0), i.e., in node (1,1)—of the portfolio (trading strategy) that replicates the payoF (*). Solution: Want θ S (1 , 1) and θ B (1 , 1) such that θ S (1 , 1) × 22 . 05 + θ B (1 , 1) e 0 . 06 =1061 . 5 θ S (1 , 1) × 20 + θ B (1 , 1) e 0 . 06 =1000 therefore θ S (1 , 1) = 61 . 5 / 2 . 05 = 30 and θ B (1 , 1) = (1061
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This note was uploaded on 03/17/2012 for the course ACTSC 446 taught by Professor Adam during the Spring '09 term at Waterloo.

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sol1 - ACTSC/STAT 446/846 Midterm 1 Fall 2011 Solutions 1....

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