Mathematicf for Finance

Imperial College Business School Mathematics for Finance Tutorial 2

Objective: This tutorial aims to help you practice and gain confidence with topics in linear algebra and its application in hedging

1.

Given

Linear Algebra - Vectors.      2 −3 2 a =  6 b =  4 c =  7  9 2 2 (a) (b) (c) find a + 2b find a∗ b find c∗ c 

Please note that a∗ denotes the transpose of a

2.

Given

Linear Algebra - Matrices. a= (a) (b) (c) find Ab find Ax find AB −3 4 x= 1 2 A= 1 2 3 6 B= 10 4 −5 −2

3.

Inverse Matrix Find inverse of matrix A 3 −4 (a) A= 2 −5   1 3 1.5 (b) A =  1 2 0.5  1 1 0   1 0 0 −1  3 1 2 2   (c) A=  1 0 −2 1  2 0 0 1 Note: You can consult the references at the end of Chapter 1 for the detailed exposition of Gaussian elimination or a YouTube link: http://www.youtube.com/watch?v=P2abN3P32cY. 1 Please also note that the general formula for an inverse matrix is A−1 = det(A) adj(A). For 2 x a b , the formula can be written as A−1 = c d

1 ad−bc

2 matrix A =

d −b −c a

4.

Answer questions 1.1-1.10 from Chapter 1 in the main textbook (page 22-23)

5.

Redundant securities In this question an m x n matrix A represents the pay-off of n securities in m states. In each of the markets below divide securities into linearly independent and redundant:   2 1 1 (a) A= 1 1 0  0 1 −1   2 1 0 3 1 (b) A= 1 1 1 2 1  0 1 2 1 0

6.

Hedging in complete market There is a security to be hedged with payoffs [1 2 3] in three states of the world. Considering there are also three securities available for hedging with payoffs [1 2 1], [2 5 0], [3 3 8] which are traded with prices 1.3, 1.7 and 6 respectively. Find the state prices of this economy, hedging portfolio and fair price of the payoffs [1 2 3].

7. Hedging in incomplete market. There is a security to be hedged with payoffs [0 0 0 0 50 100 200 300] in eight states of the world with following state probabilities [0.05 0.1 0.15 0.2 0.2 0.15 0.1 0.05]. Considering there are two...