Fin500J Mathematical Foundations in Finance

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• Fin500J Mathematical Foundations in Finance
• Topic 7 Numerical Methods for Solving Ordinary
  Differential EquationsPhilip H. Dybvig
• Reference Numerical Methods for Engineers,
  Chapra and Canale, chapter 25, 2006
• Slides designed by Yajun Wang

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Numerical Solutions

• Numerical method are used to obtain a graph or a
  table of the unknown function
• Most of the Numerical methods used to solve ODE
  are based directly (or indirectly) on truncated
  Taylor series expansion
• Taylor Series methods
• Runge-Kutta methods

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Taylor Series Method

• The problem to be solved is a first order ODE

Estimates of the solution at different base
points are computed using truncated Taylor
series expansions
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Taylor Series Expansion
nth order Taylor series method uses nth
order Truncated Taylor series expansion
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First Order Taylor Series Method(Euler Method)
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Euler Method
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Interpretation of Euler Method
y2
y1
y0
x0 x1 x2
x
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Interpretation of Euler Method
Slopef(x0,y0)
y1
y1y0hf(x0,y0)
hf(x0,y0)
y0
x0 x1 x2
x
h
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Interpretation of Euler Method
y2y1hf(x1,y1)
y2
Slopef(x1,y1)
hf(x1,y1)
Slopef(x0,y0)
y1y0hf(x0,y0)
y1
hf(x0,y0)
y0
x0 x1 x2
x
h
h
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High Order Taylor Series methods
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Runge-Kutta Methods (Motivation)

• We seek accurate methods to solve ODE that does
  not require calculating high order derivatives.
• The approach is to a formula involving unknown
  coefficients then determine these coefficients to
  match as many terms of the Taylor series
  expansion

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Runge-Kutta Method
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Taylor Series in One Variable
Approximation
Error
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Taylor Series in One Variableanother look
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Definitions
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Taylor Series Expansion
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Taylor Series in Two Variables
yk
y
x
xh
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Runge-Kutta Method
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Runge-Kutta Method
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Runge-Kutta Method
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Runge-Kutta Method
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Runge-Kutta MethodAlternative Formulas
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Runge-Kutta Methods

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Runge-Kutta Methods

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Runge-Kutta Methods
Higher order Runge-Kutta methods are
available Higher order methods are more
accurate but require more calculations. Fourth
order is a good choice. It offers good accuracy
with reasonable calculation effort
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Example 1Second Order Runge-Kutta Method

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Example 1Second Order Runge-Kutta Method

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Example 1Summary of the solution

Summary of the solution
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Solution after 100 steps
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Numerically Solving ODE in Matlab

• Matlab has a few different ODE solvers, Matlab
  recommends ode45 is used as a first solver for a
  problem
• ode45 uses simultaneously fourth and fifth order
  Runge-Kutta formula (DormandPrince)

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Numerically Solving ODE in Matlab (Example 1)

• Step 1 Create a M-file for dy/dx as firstode.m
• function yprimefirstode(x,y)
• yprime1y2x3
• Step 2 At a Matlab command window
• gtgtx,yode45(_at_firstode,1,2,-4)
• gtgt x,y
• Matlab returns two column vectors, the first with
  values of x and the second with value of y.

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Numerically Solving ODE in Matlab (Example 1)
gtgt plot(x,y,'')
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Solving a system of first order ODEs

• Methods discussed earlier such as Euler,
  Runge-Kutta,are used to solve first order
  ordinary differential equations
• The same formulas will be used to solve a system
  of first order ODEs. In this case, the
  differential equation is a vector equation and
  the dependent variable is a vector variable.

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Euler method for solving a system of first order
ODEs

• Recall Euler method for solving first order ODE.

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Solving a system of n first order ODEs using
Euler method

• Exactly the same formula is used but the scalar
  variables and functions are replaced by vector
  variables and vector values functions.
• Y is a vector of length n
• F(Y,x) is vector valued function

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Example Euler method for solving a system of
first order ODEs

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Example RK2 method for solving a system of
first order ODEs

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Example RK2 method for solving a system of
first order ODEs

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The general approach to solve high order ODE
convert
solve
high order ODE
System of first order ODE
convert
solve
Second order ODE
Two first order ODEs
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Conversion Procedure
convert
solve
high order ODE
System of first order ODE

• Select of dependent variables
• One way is to take the original dependent
  variable and its derivatives up to one degree
  less than the highest order derivative.
• Write the Differential Equations in terms of the
  new variables. The equations comes from the way
  the new variables are defined or from the
  original equation.
• Express the equations in matrix form

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Example of converting High order ODE to first
order ODEs
One degree less than the highest order derivative
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Example of converting High order ODE to first
order ODEs
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Numerically Solving high order ODE in Matlab

• Step 1 First convert the second order equation
  to an equivalent system of first order ODEs, let
  z1y, z2y

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Numerically Solving high order ODE in Matlab

• Step 2 Create the following M-file and save it
  as F.m
• function zprimeF(x,z)
• zprimezeros(2,1) since output must be a
  column vector
• zprime(1)z(2)
• zprime(2)-xz(1)exp(x)z(2)3sin(2x)
• Step 3 At Matlab prompt
• gtgt x,zode45(_at_F,0,1,2,8)
• Since z1(x)y, to print out the solution y
• gtgt x,z(,1)

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Numerically Solving ODE in Matlab (Example 1)
To plot y against x gtgt plot(x, z(,1))
Because the vector z has first component
z1y