Honours Finance (Advanced Topics in Finance: Nonlinear Analysis)

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Honours Finance (Advanced Topics in Finance
Nonlinear Analysis)

• Lecture 2 Introduction to OrdinaryDifferential
  Equations

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Why bother?

• Last week we considered Minskys Financial
  Instability Hypothesis as an expression of the
  endogenous instability explanation of
  volatility in finance (and economics)
• The FIH claims that expectations will rise during
  periods of economic stability (or stable
  profits).
• That can be expressed as
• rate of change of expectations f(rate of
  growth), or in symbols

This is an ordinary differential equation (ODE)
exploring this model mathematically (in order to
model it) thus requires knowledge of ODEs
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Why bother?

• In general, ODEs (and PDEs) are used to model
  real-life dynamic processes
• the decay of radioactive particles
• the growth of biological populations
• the spread of diseases
• the propagation of an electric signal through a
  circuit
• Equilibrium methods (simultaneous algebraic
  equations using matrices etc.) only tell us the
  resting point of a real-life process if the
  process converges to equilibrium (i.e., if the
  dynamic process is stable)
• Is the economy static?

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Economies and economic methodology

• Economy clearly dynamic, economic methodology
  primarily static.Why the difference?
• Historically the KISS principle
• If we wished to have a complete solution ... we
  should have to treat it as a problem of dynamics.
  But it would surely be absurd to attempt the more
  difficult question when the more easy one is yet
  so imperfectly within our power. (Jevons 1871
  1911 93)
• ...dynamics includes statics... But the statical
  solution is simpler... it may afford useful
  preparation and training for the more difficult
  dynamical solution and it may be the first step
  towards a provisional and partial solution in
  problems so complex that a complete dynamical
  solution is beyond our attainment. (Marshall,
  1907 in Groenewegen 1996 432)

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Economies and economic methodology

• A century on, Jevons/Marshall attitude still
  dominates most schools of economic thought, from
  textbook to journal
• Taslim Chowdhury, Macroeconomic Analysis for
  Australian Students the examination of the
  process of moving from one equilibrium to another
  is important and is known as dynamic analysis.
  Throughout this book we will assume that the
  economic system is stable and most of the
  analysis will be conducted in the comparative
  static mode. (1995 28)
• Steedman, Questions for Kaleckians The general
  point which is illustrated by the above examples
  is, of course, that our previous 'static'
  analysis does not 'ignore' time. To the contrary,
  that analysis allows enough time for changes in
  prime costs, markups, etc., to have their full
  effects. (Steedman 1992 146)

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Economies and economic methodology

• Is this valid?
• Yes, if equilibrium exists and is stable
• No, if equilibrium does not exist, is not stable,
  or is one of many...
• Economists assume the former. For example, Hicks
  on Harrod
• In a sense he welcomes the instability of his
  system, because he believes it to be an
  explanation of the tendency to fluctuation which
  exists in the real world. I think, as I shall
  proceed to show, that something of this sort may
  well have much to do with the tendency to
  fluctuation. But mathematical instability does
  not in itself elucidate fluctuation. A
  mathematically unstable system does not
  fluctuate it just breaks down. The unstable
  position is one in which it will not tend to
  remain. (Hicks 1949)

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Lorenzs Butterfly

• So, do unstable situations just break down?
• An example Lorenzs stylised model of 2D fluid
  flow under a temperature gradient
• Lorenzs model derived by 2nd order Taylor
  expansion of Navier-Stokes general equations of
  fluid flow. The result

x displacement
y displacement
temperature gradient

• Looks pretty simple, just a semi-quadratic
• First step, work out equilibrium

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Lorenzs Butterfly

• Three equilibria result (for bgt1)

• Not so simple after all! But hopefully, one is
  stable and the other two unstable
• Eigenvalue analysis gives the formal answer (sort
  of )
• But lets try a simulation first

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Simulating a dynamic system

• Many modern tools exist to simulate a dynamic
  system
• All use variants (of varying accuracy) of
  approximation methods used to find roots in
  calculus
• Most sophisticated is 5th order Runge-Kutta
  simplest Euler
• The most sophisticated packages let you see
  simulation dynamically
• Well try simulations with realistic parameter
  values, starting a small distance from each
  equilibrium

So that the equilibria are
Over to Vissim...
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Lorenzs Butterfly

• Now you know where the butterfly effect came
  from
• Aesthetic shape and, more crucially
• All 3 equilibria are unstable (shown later)
• Probability zero that a system will be in an
  equilibrium state (Calculus Lebesgue measure)
• Before analysing why, review economists
  definitions of dynamics in light of Lorenz
• Textbook the process of moving from one
  equilibrium to another. Wrong
• system starts in a non-equilibrium state, and
  moves to a non-equilibrium state
• not equilibrium dynamics but far-from equilibrium
  dynamics

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Lorenzs Butterfly

• Founding father mathematical instability does
  not in itself elucidate fluctuation. A
  mathematically unstable system does not
  fluctuate it just breaks down. Wrong
• System with unstable equilibria does not break
  down but demonstrates complex behaviour even
  with apparently simple structure
• Not breakdown but complexity
• Researcher static analysis allows enough time
  for changes in prime costs, markups, etc., to
  have their full effects. Wrong
• Complex system will remain far from equilibrium
  even if run for infinite time
• Conditions of equilibrium never relevant to
  systemic behaviour

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When economists are right

• Economist attitudes garnered from understanding
  of linear dynamic systems
• Stable linear systems do move from one
  equilibrium to another
• Unstable linear dynamic systems do break down
• Statics is the end point of dynamics in linear
  systems
• So economics correct to ignore dynamics if
  economic system is
• linear, or
• nonlinearities are minor
• one equilibrium is an attractor and
• system always within orbit of stable equilibrium
• Who are we kidding?Nonlinearity rules

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Nonlinearities in economics

• Structural
• monetary value of output the product of price and
  quantity
• both are variables and product is quasi-quadratic
• Behavioural
• Phillips curve relation
• wrongly maligned in literature
• clearly a curve, yet conventionally treated as
  linear
• Dimensions
• massively open-multidimensional, therefore
  numerous potential nonlinear interactions
• Evolution
• Clearly evolving system, therefore even more
  complex than simple nonlinear dynamics

So Economists "have to" do dynamics
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Why bother?
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Why Bother?

• Lorenzs bizarre graphs indicate
• Highly volatile nonlinear system could still be
  systemically stable
• cycles continue forever but system never exceeds
  sensible bounds
• e.g., in economics, never get negative prices
• linear models however do exceed sensible bounds
• linear cobweb model eventually generates negative
  prices
• Extremely complex patterns could be generated by
  relatively simple models
• The kiss principle again perhaps complex
  systems could be explained by relatively simple
  nonlinear interactions

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Why Bother?

• But some problems (and opportunities)
• systems extremely sensitive to initial conditions
  and parameter values
• entirely new notion of equilibrium
• Strange attractors
• system attracted to region in space, not a point
• Multiple equilibria
• two or more strange attractors generate very
  complex dynamics
• Explanation for volatility of weather
• El Nino, etc.

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Why bother?
Tiny errorin initialreadingsleads
toenormousdifferencein time pathof
system.And behindthe chaos,strangeattractors..
.
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Why bother?
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Why Bother?

• Lorenz showed that real world processes could
  have unstable equilibria but not break down in
  the long run because
• system necessarily diverges from equilibrium but
  does not continue divergence far from equilibrium
• cycles are complex but remain within realistic
  bounds because of impact of nonlinearities
• Dynamics (ODEs/PDEs) therefore valid for
  processes with endogenous factors as well as
  those subject to an external force
• electric circuit, bridge under wind and shear
  stress, population infected with a virus as
  before and also
• global weather, economics, population dynamics
  with interacting species, etc.

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Why Bother?

• To understand systems like Lorenzs, first have
  to understand the basics
• Differential equations
• Linear, first order
• Linear, second (and higher) order
• Some nonlinear first order
• Interacting systems of equations
• Initial examples non-economic (typical maths
  ones)
• Later well consider some economic/finance
  applications before building full finance model

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Maths and the real world

• Much of mathematics education makes it seem
  irrelevant to the real world
• In fact the purpose of much mathematics is to
  understand the real world at a deep level
• Prior to Poincare, mathematicians (such as
  Laplace) believed that mathematics could one day
  completely describe the universes future
• After Poincare (and Lorenz) it became apparent
  that to describe the future accurately required
  infinitely accurate knowledge of the present
• Godel had also proved that some things cannot be
  proven mathematically

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Maths and the real world

• Today mathematics is much less ambitious
• Limitations of mathematics accepted by most
  mathematicians
• Mathematical models
• seen as first pass to real world
• regarded as less general than simulation models
• but maths helps calibrate and characterise
  behaviour of such models
• ODEs and PDEs have their own limitations
• most ODEs/PDEs cannot be solved
• however techniques used for those that can are
  used to analyse behaviour of those that cannot

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Maths and the real world

• Summarising solvability of mathematical models
  (from Costanza 1993 33)

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Maths and the real world

• To model the vast majority of real world systems
  that fall into the bottom right-hand corner of
  that table, we
• numerically simulate systems of ODEs/PDEs
• develop computer simulations of the relevant
  process
• But an understanding of the basic maths of the
  solvable class of equations is still necessary to
  know whats going on in the insoluble set
• Hence, a crash course in ODEs, with some
  refreshers on elementary calculus and algebra...

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From Differentiation to Differential

• In Maths 1.3, you learnt to handle equations of
  the form

Independent variable
Dependent variable

• Where f is some function. For example

• On the other hand, differential equations are of
  the form

• The rate of change of y is a function of its
  value y both independent dependent

• So how do we handle them? Make them look like the
  stuff we know

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From Differentiation to Differential

• The simplest differential equation is

(we tend to use t to signify time, rather than
xfor displacement as in simple differentiation)

• Try solving this for yourself

Continued...
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From Differentiation to Differential
Because log of a negative number is not defined
Because an exponential is always positive

• Another approach isnt quite so formal

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From Differentiation to Differential

• Treat dt as a small quantity
• Move it around like a variable
• Integrate both sides w.r.t the relevant d(x)
  term
• dy on LHS
• dt on RHS
• Some problems with generality of this approach
  versus previous method, but OK for economists
  modelling issues

• So whats the relevance of this to economics and
  finance? How about compound interest?

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From Differential Equations to Finance

• Consider a moneylender charging interest rate i
  with outstanding loans of y.
• Who saves s of his income from borrowers
• Whose borrowers repay p of their outstanding
  principal each year
• Then the increment to bank balances each period
  dt will be dx

Divide by y Collect terms
Integrate
Take exponentials
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From Differential Equations to Finance

• Under what circumstances will our moneylenders
  assets grow?
• C equals his/her initial assets

Known as eigenvaluetells how much the
equationis stretching space

• The moneylender will accumulate if the power of
  the exponential is greater than zero

• The moneylender will blow the lot if the power of
  the exponential is less than zero

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Back to Differential Equations!

• The form of the preceding equation is the
  simplest possible how about a more general form

Same basic idea applies

• f(t) can take many forms, and all your
  integration knowledge from Maths 1.3 can be used
  A few examples

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Back to Differential Equations!

• But firstly a few words from our sponsor
• These examples are just rote exercises
• most of them dont represent any real world
  system
• However the ultimate objective is to be able to
  comprehend complex nonlinear models of finance
  that do purport to model the real world
• so put up with the rote and well get to the
  final objective eventually!

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Back to Calculus!
Try the following

• Wont pursue the last one because
• Not a course in integration
• Most differential equations analytically
  insoluble anyway
• Programs exist which can do most (but not all!)
  integrations a human can do
• But a quick reminder of what is done to solve
  such ODEs
• Also of relevance to work well do later on
  systems of ODEs

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Back to Calculus!

• Some useful rules from differentiation and
  integration
• Product rule

• Simple to derive from first principles consider
  a function which is the product of two other
  functions

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Back to Calculus!

• These rules then reworked to give us integration
  by parts for complex integrals

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Back to Calculus!
Treat integration as a multiplication operator

• Convert difficult integration into an easier one
  by either
• reducing u component to zero by repeated
  differentiation
• repeating u and solving algebraically

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Back to Calculus!

• Practically
• choose for u something which either
• gets simpler when integrated or
• cycles back to itself when integrated more than
  once
• For our example

• Try sin
• cycles back
• formulas exist for expansion

These dont get any simpler, but do cycle
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Back to Calculus!
Reproduces this
Next differentiation of this
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Back to Calculus!

• Stage Two

• Finally, Stage Three we were trying to solve the
  ODE

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Back to Differential Equations!

• We got to the point where the equation was in
  soluble form

• Then we solved the integral

• Now we solve the LHS and take exponentials

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Back to Differential Equations!

• So far, we can solve (some) ordinary differential
  equations of the form

• These are known as
• First order
• because only a first differential is involved
• Linear
• Because there are no functions of y such as
  sin(y)
• Homogeneous
• Because the RHS of the equation is zero

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Back to Differential Equations!

• Next stage is to consider non-homogeneous
  equations

• g(t) can be thought of as a force acting on a
  system
• We can no longer divide through by y as before,
  since this yields

• which still has y on both sides of the equals
  sign, and if anything looks harder than the
  initial equation
• So we apply the three fundamental rules of
  mathematics

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The three fundamental rules of mathematics J

• (1) What have you got that you dont want?
• Get rid of it
• (2) What havent you got that you do want?
• Put it in
• (3) Keep things balanced
• Take a look at the equation again

What does this look almost like?
The product rule
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Non-Homogeneous First Order Linear ODEs

• The LHS of the expression

is almost in product rule form

• Can we do anything to put it exactly in that
  form?
• Multiply both sides by an expression m(t) so that

• Now we have to find a m(t) such that

• This is only possible if

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The Integrating Factor Approach

• This is a first order linear homogeneous ODE,
  which we already know how to solve (the only
  thing that makes it apparently messy is the
  explicit statement of a dependence on t in m(t),
  which we can drop for a while)

This is known as the integrating factor
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The Integrating Factor Approach

• So if we multiply

by
we get

• Anybody dizzy yet?
• Its complicated, but there is a light at the end
  of the tunnel
• Next, we solve the equation by taking integrals
  of both sides

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The Integrating Factor Approach

• And finally the solution is

• This is a bit like line dancing it looks worse
  than it really is.
• Lets try a couple of examples firstly, try

• (Actually, line dancing probably is as bad as it
  looks, and so is this)...

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The Integrating Factor Approach
becomes

• The first one

using the integrating factor

• Now we need a m such that

• Which is only possible if

• This is a first order homogeneous DE piece of
  cake!

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The Integrating Factor Approach

• Thus we multiply

to yield
by

• Then we integrate

Back to basics 2the Chain Rule inreverse
Next problem how to integrate this?
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The Chain Rule

• This expression

• Looks like

• Or in differential form

• That integral is elementary

• Now substituting for u and taking account of the
  constant

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The Integrating Factor Approach

• Finally, we return to

• Putting it all together

is the solution to

• Before we try another example, the general
  principle behind the technique above is the chain
  rule in reverse

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The Chain Rule
Rate of change of composite function is rate of
change of one times the other
Slope of one
slope of other

• In reverse, the substitution method of
  integration

slope of composite
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Back to Differential Equations!

• Try the technique with

• Stage One Finding m

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Linear First Order Non-Homogeneous

• Stage Two apply m

• Stage Three integrating RHS
• there is no known integral! (common situation in
  ODEs)
• Completing the maths as best we can

This can only be estimated numerically