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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 03/17/2012 for the course ACTSC 446 taught by Professor Adam during the Spring '09 term at Waterloo.

### Page1 / 4

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online