Honours Finance Advanced Topics in Finance: Nonlinear Analysis

1
Honours Finance (Advanced Topics in Finance
Nonlinear Analysis)

• Lecture 3 Introduction to ODEs continued

2
Recap

• Last week got a bit hairy and apparently distant
  from economics and finance
• Lets bring it back home with an apposite
  example compound interest
• Imagine that your ancestor deposited 1 in the
  year 0 in an account which was continuously
  compounded at a rate of 2 p.a.
• How much would be in the account in the year
  2000?
• Work out the formula

Change in Asset
Time period
Rate of interest
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An Example

• Work out the solution for A

So what is the value of C? Work it out
4
An Example

• Now lets use the formula
• How much would that 1 invested at 2 p.a. be
  worth in the year 2000?
• Have a guess...
• Now work it out

5
An Example

• Get out the calculators what is this in decimal
  format?

• How much gold is that at, say, 300 an ounce?

• So how much space would that much gold occupy?
  (Gold weighs 19,300 kg per cubic metre)

6
An Example
Thats 1.15 billion cubic metresof gold

• So how large is that exactly... say, compared to
  the volume of the earth? (The earths radius is
  6370 km)

So its not that bigjust how big is it?
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An Example

• So one dollar, invested at 2 p.a., turns into a
  ball of gold 1300 metres across in 2000 years
• And I bet you thought 2 was a lousy rate of
  return!
• What do you think 4 yields?
• 250,000 balls of gold the size of the earth, or a
  sphere of gold 400,000km across!
• With the knowledge imparted by this ODE, you
  should now be sceptical about the long term
  viability of growth rates which are currently
  taken as desirable in the modern world
• 10 p.a. for China, etc.
• World history hasnt been one of continuous
  accumulation!
• Current expected yields (4-6 p.a. min.)
  unsustainable

8
Back to Solutions to ODEs

• This week well consider how to solve
• first order separable nonlinear ODEs
• second order linear ODEs
• As a prelude to
• systems of ODEs and complexity
• chaotic behaviour in deterministic systems
• (our real interest in a course on economics and
  finance)
• But before considering more solution techniques,
  a reminder
• most ODEs are insoluble impossible to find a
  closed form for y(t) from an expression for y(t)

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(In)solubility of ODEs (A)

• Last example in last weeks lecture is one of the
  many ODEs which cannot be (completely) solved
  analytically
• The vast majority of ODEs cannot even be solved
  to this level
• The general rule is that we can solve all ODEs
  which can be put into the form

• Shortly well prove that this severely limits the
  ODEs which can be solved
• Next however yet another technique for dealing
  with first order (but now nonlinear as well as
  linear) ODEs

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First order separable ODEs

• Just as we could convert

into

• We can convert

into

• These are known as separable equations
• we can solve some of these even though they are
  nonlinear in y
• We use the fractional form because it emphasises
  the chain rule aspect of this class of
  equations

So G(y) is the solutionwere after
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First order separable ODEs

• An example

Try it
Easy, huh?
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First order separable ODEs

• A more relevant example
• Exponential growth is described by the formula

Where P can be population, deposit in a bank, etc.

• But nothing grows exponentially for ever in the
  real world (though many things do exponentially
  decay)
• the number of instances of an organism tend to
  limit its numbers (overcrowding adoption of a
  new product by all consumers, etc.)
• This is captured in the logistic equation

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First order separable ODEs

• Solve this equation

So how to handle this?
Method of partial fractionsbreak difficult
inverse polynomialintegration into sum of two
inverses
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First order separable ODEs

• The technique is to guess a set of fractions of
  single factors of P that when added together
  equal this fraction of a polynomial of P
• This works because of the same trick that lets
  you add two fractions together when they have
  different denominators

• The method of partial fractions just runs this in
  reverse
• start with the composite fraction with known
  factors (7 and 11) and work out what the separate
  numerators have to be

15
The Logistic Equation

• First of all, break the denominator down into two
  factors

• Then provide guess values for the numerators
  and expand them out

• Next equate them to the original numerator

• Finally work out values of A and B that are valid
  for all P

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The Logistic Equation

• This reduces to two pretty simple equations in
  two unknowns

• Solving for these gives us

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The Logistic Equation

• Now we replace the original difficult integral
  with this pair of integrals

• The first one is easy

• The second one requires a bit more work to put it
  into the du/u form needed to extract a log

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The Logistic Equation

• Substituting back into the original equation

• Finally, substituting back in terms of P

• On it goes now we have to combine this with the
  solution for the first term, and see what we have

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The Logistic Equation

• And all of this is equal to the integral of the
  original RHS, which was simply the integral of dt

We simplify thisusing logs
Next take exponentials
20
The Logistic Equation

• Finally, a bit of fancy footwork to rearrange
  this as an expression in P

• So whats the use?
• Compare this equation as a predictor of world
  population to a straight exponential

21
The Logistic Equation

• Ecological estimates for a (births-deaths)/popula
  tion give a value of 0.029 (Braun 1993 31)
• When world population was 3.34 billion in 1965,
  it was growing at 2 p.a.
• We can put these values into the equation for P

• Now lets compare a simple Malthusian estimate of
  population of population in the year 2100 with a
  logistic estimate, given a population of 3.34
  billion in 1965

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The Logistic Equation

• Simple Malthusian growth is

• Logistic growth is

• According to Malthus, world population will be
  145 billion in 2100
• According to the logistic equation, world
  population will be 10.24 billion in 2100
• a/b ratio estimates of maximum world population
  of 10.76 billion (how does this tell us the
  maximum?)

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The Logistic Equation
The logistic curve also has a verydistinctive
shape, which is bestseen with a linear scale
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The Logistic Equation
Years from 1965 till 1990 were period of most
rapidacceleration in worldpopulation
Recent estimatesindicate taperinghas already
begun
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The Logistic Equation

• The logistic can also be written in discrete form

(dont worry, we wont try to solve it!)

• The equilibrium, as before, is a/b
• but funny things can happen on the way to
  equilibrium
• for values of a much less than 3, a smooth
  transition
• for a 3 and slightly higher, cycles between 2
  values
• for agt3.5, cycles between 4 values, then 8,
  then...
• For agt3.7, chaos...
• Over to a dynamic simulation of lemmings
  populations in lemmings.vsm and lemmings.mcd

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The Logistic Equation

• Interpretation?
• Low growth rates, population smoothly tapers to
  equilibrium
• Higher growth rates, population overshoots
• medium-high values, overshoot is cyclical
• first a 2-cycle, then a 4-cycle
• equilibrium is a 1-cycle higher order cycles
  are quite possible
• eventually an infinite-cycle for values around
  3.7, then chaos
• Aperiodic cycles always fluctuates but
  fluctuations never repeat scale or period
• apparent randomness from a deterministic process

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The Logistic Equation

• Does this have any applicability to economics and
  finance?
• Logistic curve suitable form of relationship
  which has minimum and maximum levels of
  saturation used to model
• diffusion of inventions (e.g. computer) through
  population from near zero (early adopters) to
  100 (or less)
• growth in share ownership
• relationship of asset price index to consumer
  prices...
• Apparently chaotic output of discrete logistic
  equation taken as simile for behaviour of finance
  markets
• superficially random numbers in finance stats
  could be generated by self-referential
  deterministic processes in finance markets

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Back to ODEs in general

• Clear role of differential/difference equations
  in economics and finance
• Most such models sufficiently complex that cant
  be solved at all
• But to appreciate them, need to know how to solve
  the solvable ones.
• However, we can prove that most ODEs cant be
  solved!

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Why most ODEs cant be solved

• The general technique of solving an ODE is to
  take something in the form of

• And work on it till it is in the form

• Integration of this (with respect to t) yields

• The function f is then reworked to provide an
  expression for y in terms of t.
• The question now is, how many functions of the
  form F can we rework into a function of the form
  f?
• The answer is, not many

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Why most ODEs cant be solved

• It turns out that we can only process F into this
  form if we can break F down into two parts (M and
  N) which obey the condition that the differential
  of M with respect to y is the same as the
  differential of N with respect to t
• This is, as it sounds, a highly restrictive
  condition. The next couple of slides proves this,
  but are background only.
• We start with a general ODE

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Why most ODEs cant be solved

• Can this be put into the integrable form?
• Only if

• The RHS of this can be expanded using the chain
  rule for partial differentiation

• This lets us equate M and N to the partial
  derivatives of f

• But this immediately imposes conditions on the
  forms that M and N can take

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Why most ODEs cant be solved

• In (partial) differentiation, the order of
  differentiation is irrelevant. Thus

• But the LHS of the above is the differential of M
  with respect to y, and the RHS is the
  differential of N with respect to t

• So, for a valid M and N to exist, it must be true
  that

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Why most ODEs cant be solved

• This condition will be true of the general
  relation

• Only in a very small minority of cases
• In some others (for example, the equations we
  solved using the integrating factor), initially
  unsuitable equations can be processed to be in a
  more suitable form
• But in general most ODEs cannot be solved
• and its worse for higher order ODEs

34
Second order Linear ODEs

• Functions of the second derivative of a variable
• Of the form

• Even less of these can be solved than first order
  ODEs
• One general rule used
• Differentiation is a linear operator on these
  functions
• Define

• Many equations solvable using characteristic
  equation

35
Second order Linear ODEs

• Consider equation

• Where a, b, c are constants
• Solution has to be function which, when
  differentiated, returns itself times a constant
• the exponential try ert

• Use quadratic formula

36
Second order Linear ODEs

• So general solution is

• Try it for

Roots to these are realsystem either converges
toor diverges from equilibrium
Roots to this are complexgenerates cycles
(which eitherdiverge or converge)
37
Complex Numbers!

• It seems you havent been introduced to these
  blighters yet!
• Consider the quadratic

• We know that this has solution

• This is no big deal when

(Known as the discriminant)

• But its a problem when

• Solution? Simple! Invent

38
Complex Numbers!

• Try this with that last equation

• Convert to quadratic using characteristic
  equation

• Solve quadratic

• Define

and substitute
39
Complex Numbers!

• Solution

• Complex numbers were initially invented simply to
  solve quadratics with a negative discriminant
• So how to represent this abstract idea?
• Real numbers represented by a number line

• Complex numbers represented by a 2-dimensional
  number line an Argand diagram

40
Complex Numbers!

• Real numbers on the horizontal
• Imaginary numbers on the vertical

41
Complex Numbers!

• Represent, for example, 11i 2-3i34i on an
  Argand diagram

1i
1
Ditto for cos

• It is now easy to replace these with polar
  coordinates an angle and distance from the
  origin

• For example, 11i is

42
Complex solutions to ODEs

• For a complex number to be zero, both its real
  and imaginary parts must be zero
• The real components of the real (cos(bt)) and
  imaginary (isin(bt)) parts of complex root to a
  characteristic equation are linearly independent
• These two parts alone thus provide the two
  linearly independent solutions to a second order
  ODE the imaginary i can be dropped.
• A bit more explanation

43
Second order Linear ODEs

• Complex roots come in conjugate pairs
  (aib,a-ib)
• one of the pair gives general solution
• for a solution, (aib) must give zero when put
  into equation
• for complex number to be zero, both real (a) and
  imaginary (b) component must be zero
• a and b are independent
• a and b are thus independent solutions
• complex conjugate (a-ib) just gives same
  solutions with different signs
• complex part of solution gives cycles because
  complex power equivalent to i times cos plus sin

44
Second order Linear ODEs
Since these must both be zerofor ert to be a
root, these realcomponents provide 2
linearlyindependent solutions

• General solution is

• a gives magnitude of cycles and whether they rise
  (agt0) or fall (alt0) with time
• b gives frequency of cycles small b long cycles,
  large b short cycles
• Back to solutions to second order linear ODEs

45
Second order Linear ODEs
46
Second order Linear ODEs
47
Second order Linear ODEs

• Several other classes of 2nd order ODEs, and
  different solution methods (see Braun)
• Well focus on one method
• second order ODE can be converted into pair of
  linked first order ODEs

Then
Define
48
Second order Linear ODEs

• This converts the second order ODE into a matrix
  equation

• This can be concisely expressed as a vector
  equation

In this form, similarity to first order ODEis
evident. We use similar method presume solution
of the form eltv where v a vector rather than a
scalar (as for 1st order ODEs)
49
Second order Linear ODEs

• So we presume a solution of the form x(t) eltv,
  and manipulate the equation, following the rules
  of matrix maths

• This is only true for non-zero v if

• This is the determinant of the equation, which is
  a polynomial in l. The roots of this polynomial
  will be equivalent to the rs as worked out using
  the characteristic equation approach, and tell us
  how the equation stretches space (see Braun).

Rearranging
50
Second order Linear ODEs

• Working with the general formula for a 2nd order
  ODE

51
Second order Linear ODEs

• The roots of this are the same as the solution we
  can find using the (easier) characteristic
  equation approach

• Method useful when doing third or higher order
  ODEs (there is no formula for a 6th order
  polynomial or above 3rd above formulas are a
  mess!)

52
Coupled ODEs

• Second order linear ODEs are common in economics,
  but only a curtain raiser for us to introduce the
  concept of coupled ODEs
• These model a system in which two variables
  affect each other a feedback system
• The most relevant example for us is the
  Lokta-Volterra predator-prey model

53
Predator-Prey Systems

• Fish and Sharks
• Fish eat seagrass (assumed unlimited supply)
• Sharks eat fish
• Together, a cycle
• Low numbers of fish, sharks die off
• Less sharks, more fish reproduce
• More fish available, shark numbers rise
• More sharks, fish population declines
• Low numbers of fish, sharks die off
• How to model it?
• Use F for Fish and S for Sharks

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Predator-Prey Systems

• Rate of growth of fish is
• positive function of number of fish
• negative function of the number of sharks

• Rate of growth of sharks is
• negative function of number of sharks
  (starvation)
• positive function of the number of fish

• Together, a system

Can thisbe solved?
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Predator-Prey Systems

• Well, yes but its the last one we can solve
• any system with three or more coupled ODEs is
  insoluble
• before looking at an obvious one, well play with
  this one
• first, a numerical simulation

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Predator-Prey Systems

• Can we solve it?
• How about using the separable approach?

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Predator-Prey Systems

• Notice how each variable is a function of the
  other

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Predator-Prey Systems

• What about the systems equilibrium?
• How do you define it?
• When dF/dtdS/dt0
• Is it stable or unstable?

• There are ways to work this out (pertubation
  analysis work out the dynamics of behaviour a
  short distance from equilibrium)
• It turns out that the equilibrium is neutral
• neither attracts nor repels

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Predator-Prey Systems

• Generates a stable limit cycle
• system orbits the equilibrium but never converges
  to or diverges from it.
• Such behaviour the norm in complex systems
• as well see next lecture