Mathematical Preliminaries

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Mathematical Preliminaries
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Math Preliminaries

• Our first order of business is to develop
  mathematical models of asset prices and random
  factors.
• For most of this course, we will model prices as
  continuous time stochastic processes and
  stochastic differential equations.

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Math Preliminaries
Building Blocks
Stochastic Differential Equations
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Math Preliminaries
Building Blocks
Brownian Motion
Poisson Processes
Stochastic Differential Equations
Solutions to SDEs
Itos Lemma
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Gaussian (Normal) Random Variable
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Brownian Motion
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Facts about Brownian Motion

• Sample paths of Brownian motion are (can be
  chosen to be) continuous with prob. 1.
• (2) Brownian motion is nowhere differentiable.

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Simulation of Brownian Motion
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Math Preliminaries
Building Blocks
Brownian Motion
Poisson Processes
Stochastic Differential Equations
Solutions to SDEs
Itos Lemma
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Math Preliminaries
Building Blocks
Brownian Motion
Poisson Processes
Stochastic Differential Equations
Solutions to SDEs
Itos Lemma
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Poisson Random Variable
Remark X is the number of events that occur in
one time unit when the time between events is
exponentially distributed with mean 1/l.
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Poisson Process
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Simulation of a Poisson Process (l1)
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Poisson processes Jump!
Hence, they are a good building block for
modeling

• Market crashes or jumps.
• Bankruptcy.
• Other unexpected discontinuous price movements.

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Brownian Motion and the Poisson Process as limits
Chop time interval in half
Chop finer and finer.
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Brownian Motion and the Poisson Process as limits
1
1/2
1/2
-1
time 1
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Brownian Motion and the Poisson Process as limits
Chop finer and finer.
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Brownian Motion and the Poisson Process as limits
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Generalized Poisson Process (Jump Processes)
Poisson processes jump by 1. We can
generalize this and allow them to jump randomly
Let pt be a Poisson process with jump time t1,
t2, ...
Construct a new process, ptY, by assigning
jump Y1 at time t1, Y2 at time t2, ... where
Y1,Y2,... are iid random variables.
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Poisson Process
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Generalized Poisson Process
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Simulation of a Generalized Poisson Process (l1,
jumps N(0,1))
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Math Preliminaries
Building Blocks
Brownian Motion
Poisson Processes
Stochastic Differential Equations
Solutions to SDEs
Itos Lemma
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Math Preliminaries
Building Blocks
Brownian Motion
Poisson Processes
Stochastic Differential Equations
Solutions to SDEs
Itos Lemma
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(No Transcript)
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Ito Stochastic Differential Equations
where
One way to think of this is as an update formula
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Drift is determined by a(x,t).
Volatility is determined by b(x,t).
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xt
Time
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xt
Time
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Ito Stochastic Differential Equations
You should think
where
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Drift is affected by Poisson Drift.
Volatility is determined by b(x,t).
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Math Preliminaries
Building Blocks
Brownian Motion
Poisson Processes
Stochastic Differential Equations
Solutions to SDEs
Itos Lemma
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Math Preliminaries
Building Blocks
Brownian Motion
Poisson Processes
Stochastic Differential Equations
Solutions to SDEs
Itos Lemma
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Itos Lemma for Brownian Motion
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Derivation of Itos Lemma
Taylor Expand
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Example
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Itos Lemma for Poisson Processes
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Derivation of Itos lemma for Poisson Processes
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Derivation of Itos lemma for Poisson Processes
Add and subtract
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Derivation of Itos lemma for Poisson Processes
No jumps here! Use ordinary calculus.
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Derivation of Itos lemma for Poisson Processes
No jumps here! Use ordinary calculus.
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Derivation of Itos lemma for Poisson Processes
wp ldt
Jump
wp 1-ldt
No jump
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Derivation of Itos lemma for Poisson Processes
wp ldt
Jump

wp 1-ldt
No jump
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Derivation of Itos lemma for Poisson Processes
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Example
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Itos Lemma for Generalized Poisson Processes
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Derivation of Itos lemma for Generalized
Poisson Processes
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Derivation of Itos lemma for Generalized
Poisson Processes
Add and subtract
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Derivation of Itos lemma for Generalized
Poisson Processes
No jumps here! Use ordinary calculus.
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Derivation of Itos lemma for Generalized
Poisson Processes
No jumps here! Use ordinary calculus.
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Derivation of Itos lemma for Generalized
Poisson Processes
wp ldt
Jump
wp 1-ldt
No jump
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Derivation of Itos lemma for Generalized
Poisson Processes
wp ldt
Jump

wp 1-ldt
No jump
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Derivation of Itos lemma for Generalized
Poisson Processes
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Math Preliminaries
Building Blocks
Brownian Motion
Poisson Processes
Stochastic Differential Equations
Solutions to SDEs
Itos Lemma
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Math Preliminaries
Building Blocks
Brownian Motion
Poisson Processes
Stochastic Differential Equations
Solutions to SDEs
Itos Lemma
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Some basic stochastic differential equations.
The simplest Gaussian Process
a and b constants
Solution just integrate
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Some basic stochastic differential equations.
Solution
multiply by factor so left hand side is d( ).
integrate
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Some basic stochastic differential equations.
Solution
Integrate
Rearrange
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Some basic stochastic differential equations.
Poisson Stochastic Differential Equation
Y-1
0
a constant
Solution
Integrate
Rearrange
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References
Probability and Stochastic Processes
Breiman, L. Probability, SIAM, Philadelphia,
1992. Hoel, P. G., Port, S. C., and Stone, C. J.
Introduction to Stochastic Processes, Waveland
Press, Prospect Heights, Illinois,
1972. Oksendal, B. Stochastic Differential
Equations an introduction with applications,
5th ed. Springer, NY, 1998. Gillespie, D.
Markov Processes an introduction for physical
scientists, Academic Press, 1997. Vickson, R.
G. An Intuitive Outline of Stochastic
Differential Equations and Stochastic Optimal
Control, In Stochastic Models in Finance,
Edited by Ziemba and Vickson, 1975.