Risk Neutral Pricing and Financial Mathematics: A Primer

Appendix B

Exercise Solutions

Chapter 1

1.1. The sum is as follows:

$\begin{array}{ccccc}\left[\begin{array}{ccc}2& 4& 9\\ 6& 4& 25\\ 0& 2& 11\end{array}\right]& +& \left[\begin{array}{ccc}3& 0& 6\\ 2& 1& 3\\ 7& 0& 4\end{array}\right]& =& \left[\begin{array}{ccc}5& 4& 15\\ 8& 5& 28\\ 7& 2& 15\end{array}\right]\\ \mathbf{A}& \mathbf{B}& \mathbf{C}\end{array}$

1.3. 

a. $\begin{array}{cc}\left[\begin{array}{ccc}1& 8& 9\\ 6& 4& 25\\ 3& 2& 35\end{array}\right]& \left[\begin{array}{ccc}1& 6& 3\\ 8& 4& 2\\ 9& 25& 35\end{array}\right]\\ \mathbf{A}& {\mathbf{A}}^{\mathrm{T}}\end{array}$

b. The transpose of a column vector is a row vector:

$\begin{array}{cc}\left[\begin{array}{c}9\\ 6\\ 3\\ 7\end{array}\right]& [\begin{array}{cccc}9& 6& 3& 7\end{array}]\\ \mathbf{y}& {\mathbf{y}}^{\mathrm{T}}\end{array}$

Similarly, the transpose of a row vector is a column vector.

c. Note that the transpose V^T of a symmetric matrix V is V:

$\mathbf{V}=\left[\begin{array}{ccc}0.09& 0.01& 0.04\\ 0.01& 0.16& 0.10\\ 0.04& 0.10& 0.64\end{array}\right]{\mathbf{V}}^{\mathrm{T}}=\left[\begin{array}{ccc}0.09& 0.01& 0.04\\ 0.01& 0.16& 0.10\\ 0.04& 0.10& 0.64\end{array}\right]=\mathbf{V}$

1.5. 

a. $2\mathbf{A}=[\begin{array}{cc}2(-2)& 2(0)\\ 2(3)\end{array}$