analysis and linear algebra for finance part i

Analysis and Linear Algebra for
Finance: Part I
Patrick Roger

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Patrick Roger

Analysis and Linear Algebra for Finance:
Part I

Patrick ROGER
LaRGE Research Center
EM Strasbourg Business School
University of Strasbourg

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Analysis and Linear Algebra for Finance: Part I
First edition
© 2013 Patrick Roger & bookboon.com (Ventus Publishing ApS)
ISBN 978-87-403-0412-1

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3

             

           

 

Analysis
and Linear
Algebra for 
Finance:
Part I

Contents

              
              
          

Contents

Introduction

6

   

1Preliminaries
  
       

7

1.1  
Sets
and subsets
   
       
Binary relations
   1.2
 

8
17

1.3Mappings

20

1.4  
Topology
of 25
  
 
         
                   
Functions of one variable

43

  
   
  

2.1
Definitions
and
notations

2

44

2.2

Limits and continuity

50

         
2.3Differentiation

61

Logarithms
     2.4
  
 and
 exponential
  functions

   
   












2.5
Polynomial approximations and Taylor
formula

70



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Analysis and Linear Algebra for Finance: Part I

Contents

3Integrals

81

3.1

Integral of a step function

81

3.2

General case

86

3.3Computations

94

3.4

98

Improper integrals

4Matrices

101

4.1Definitions

101

4.2

Elementary algebra on matrices

104

4.3

Linear equations

115

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Analysis and Linear Algebra for Finance: Part I






             
             
          
           
          
          
            
         
          
          
            
           
       
         
          
             
         

           
            
          
         
         
             



          

  

          

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6

Introduction


Analysis and Linear Algebra for Finance: Part I

Preliminaries

 

         
          
            
        
          
           
             
          
 
        
          
            
            

          
            
             
              
            


          



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7


Analysis and Linear Algebra for Finance: Part I



  

            
          
            
        



  

            
           
            
              
          
           
       
             
          
            
           
            
    



  

             
      
                   




            
   
               
      

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8

Preliminaries


Analysis and Linear Algebra for Finance: Part I

   



              
      

 
          

               
  
                  
                
      
          
       




 

              
       


           



              
             
             
              
  
             
    
               
               
             

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9

Preliminaries


Analysis and Linear Algebra for Finance: Part I



  

Preliminaries

             
              
     
                
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10

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Analysis and Linear Algebra for Finance: Part I

   





     
  

                
                 
         

               

               
           
   
               
        
             


               
                
             
    
          
                  
   

            
           
           

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11

Preliminaries


Analysis and Linear Algebra for Finance: Part I



  

          
            
            
              

           
  
             
           
           

             
               
 
                

               
                
              
 
        
             
                   
 
          

              

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12

Preliminaries

Analysis and Linear Algebra for Finance: Part I

Preliminaries

Download free eBooks at bookboon.com

13


Analysis and Linear Algebra for Finance: Part I



  

             
              
             
       
               
          



                   
                   
                   
          
                 
                   


                  
                 
 
                  
              
            
          
    





          
            

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14

Preliminaries


Analysis and Linear Algebra for Finance: Part I

   



Preliminaries

              
                  

   
                
            
             
               
             
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15

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Analysis and Linear Algebra for Finance: Part I





  

     

          
             
             
              
             
             
           

             
     
            

     
          
                
  
            
             
              
              
               
            
             
              
                


       
              
                   
               


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16

Preliminaries


Analysis and Linear Algebra for Finance: Part I

  



      
          



 

             
                    
 
           
 
              
        
       

           

             

              

        
           
         
        





                
  

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17

Preliminaries


Analysis and Linear Algebra for Finance: Part I



  

            

              
            
            
                 
 
         
           

         
                
           
            
            
            
          
         
           
            
 
                 
                      
            
             
              
              
                 
  

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18

Preliminaries


Analysis and Linear Algebra for Finance: Part I

  



           
          
            
            
          
                     



                  
     



 

            
  
             
             
  
  

           
             
                        
                       

            
            
       


          
           
           


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19

Preliminaries


Analysis and Linear Algebra for Finance: Part I






  


   

             

              
          
             
        
               
              
            
       



                 
     
        
                 
             
                 
        
               
                
               
            

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20

Preliminaries

Analysis and Linear Algebra for Finance: Part I

 



Preliminaries

              
               
       
           


           


           

              
               
 

DTU Summer University
– for dedicated international students

Application deadlines and programmes:

Spend 3-4 weeks this summer at the highest ranked
technical university in Scandinavia.

DTU’s English-taught Summer University is for dedicated
international BSc students of engineering or related
natural science programmes.

31
15
30
3

March Arctic Technology
March & 15 April Chemical/Biochemical Engineering
April Telecommunication
June Food Entrepreneurship

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21

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Analysis and Linear Algebra for Finance: Part I



  

               
             

              
                 
         
      
             
             


              


              
       

           



   

         
  
                
  
           
             
 
        
           

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22

Preliminaries


Analysis and Linear Algebra for Finance: Part I

 



             
              
                
   
               
                 
         
               
     

     
  
                
             
               
 

1

f(x)

0.8
0.6
0.4
0.2
0
-1

-0.5

0
x

0.5

1

         

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23

Preliminaries


Analysis and Linear Algebra for Finance: Part I





  

 

                 
                 
         
         
      
              
              
         
              
       
      
              
                 
                      
       
               
                  

         
          
               

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24

Preliminaries


Analysis and Linear Algebra for Finance: Part I

   



               
      



  

           
          
                
          
           
             
             
          
          

          
          
          
           
            
          
            
           
  



    

       

          

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25

Preliminaries