Financial Mathematics

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  FM 5012: Mathematical Background for Finance 
   Spring 2008

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• Textbook:  Numerical Methods in Finance and Economics, P. Brandimarte.
• Reading material:
• The Mathematics of Financial Derivatives: A Student Introduction,  P. Willmot, S. Howison and J. Dewynne.
• Numerical Analysis: An Introduction,  W. Gautschi.
• Difference Methods for Initial-Value Problems,  R.D. Richtmyer and K.W. Morton.
• Syllabus:
• Basic concepts in numerical analysis
• Consistency, stability and converge
  
• Real numbers, machine numbers and rounding
  
• The effect of rounding errors
  
• Solving linear systems of  equations
• Vector and matrix norms
• Condition number of a matrix
• Direct methods
  
• Classical iterative methods
• The method of steepest descent
• The method of conjugate gradients
  
• The method of generalized residuals
• Finite difference methods for European Options
  
• The terminal, boundary-value problem
  
• The relation with the heat and transport equations
  
• An explicit method for the heat equation
• An implicity methods for the hear equation
• The Crank-Nicolson method for the heat equation
• The upwinding method for the transport equation
• Finite difference methods for American options
  
• The terminal, boundary-value problem
  
• The obstacle problem
  
• A finite difference method for the obstacle problem
• The projected SOR method
  
• A finite difference method for American options 
  

• Some lecture notes:
  
• Newton's method                                                  ----> click here
• Classic iterative methods                                       ----> click here
• Minimization Algorithms                                      ----> click here
• GMRES methods                                                  ----> click here
• Finite difference methods for European options    ----> click here
• Finite difference methods for American options    ----> click here

• Grade:   Average of the N-1 best grades of the N homeworks
  
• Homeworks:
• Homework #1  (Newton's method)                                          ----> click here (With a solution of the first four questions.)
  
• Homework #2  (Iterative methods for Ax=b)                           ----> click here (With a solution.)
• Homework #3  (Finite differences, heat equation)                    ----> click here
• Homework #4  (Finite differences, European call options)       ----> click here    
• Homework #5  (Finite differences, the obstacle problem)         ----> click here 
  
• Homework #6  (Finite differences, American options)             ----> click here