Financial Markets

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Yuji Yamada
Control and Dynamical Systems California Institut
e of Technology
First Term Fall 2001
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This course
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What are we going to learn?

• Basics of stochastic calculus and its
  application to finance

• Underlying theory and computational tools for
  stochastic
  
• processes and simulations in derivative
  pricing/hedging
  
• problems

• Topics basic probability theory (measure
  theory), martingale
  
• theory, Markov processes, and stochastic control

Lecture notes Steven Shreve, Stochastic
Calculus and Finance, downloadable at
http//www.cs.cmu.edu/chal/shreve.html
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Before starting

• From the preface of Thomas Mikosh, Elementary
• Stochastic Calculus with Finance in View, 1998

Ten years ago I would not have dared to write a
book like this a non-rigorous treatment of a
mathematical theory. I admit that I would have
been ashamed, and I am afraid that most of my
colleagues in mathematics still think like this.
However, my experience with students and
practitioners convinced me that there is a strong
demand for popular mathematics.
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• From the preface of Bernt Oksendal,
• Stochastic Differential Equations, 1985

There are several reasons why one should learn
more about stochastic differential equations Th
ey have a wide range of applications out side ma
thematics . Unfortunately most of the literatu
re about stochastic differential equations seems
to place so much emphasis on rigor and
completeness that it scares many nonexperts away.
These notes are an attempt to approach the
subject from the nonexpert point of view.
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• From the preface of David Williams,
• Probability with Martingales, 1990

Preface please read! . You cannot avoid mea
sure theory an event in probability is a
measurable set, a random variable is a measurable
function on the sample space, the expectation of
a random variable is its integral with respect to
the probability measure.
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Course material
Steven Shreve, Stochastic Calculus and Finance

• T. Mikosch, Elementary Stochastic Calculus with
  Finance in
  
• View, World Scientific, 1998
• B. Oksendal, Stochastic Differenctial Equations
  An Introduction
  
• with Applications, 5th ed., Springer Berlin
  Heidelberg, 1998
  
• D. Williams, Probability with Martingales,
  Cambridge Univ.
  
• Press, 1991

• D. Duffie, Dynamic Asset Pricing Theory, 2nd
  ed., Princeton
  
• Univ. Press, 1996
• D.T. Gllespie, Markov Process An Intoroduction
  for Physical
  
• Scientists, Academic Press, 1992
• J. Hull, Options, Futures, and Other Derivative
  Securities, 4th
  
• ed., Prentice-Hall, 1999

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Course outline
1. Basics of arbitrage pricing and probability
theory
2. The Markov property and American options
3. Properties of continuous models
4. Numerical techniques
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Course outline
Conditional Expectation Martingale theory Markov
processes
Ito formula
Pricing and Hedging on - European option - A
merican option
- Exotic option
Recommend BEM 105 Options
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• Using a discrete random walk model, we are going
  to
  
• learn the basic theory of pricing and hedging
  for
  
• derivative securities

Derivative an instrument whose price depends
on, or is derived from, the
price of another asset Hull
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European call option
the right to buy an asset at a strike price K

under specified terms T
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Arbitrage pricing theory
Arbitrage a strategy that is guaranteed to make
money with no initial cost, or
no future payment.
Arbitrage pricing theory the theory of asset
pricing which
permits no arbitrage opportunity
Comparison principle if you know for sure that
two securities
will have the same price, then the initial
prices have
to be the same, too.
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Replicating portfolio
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Perfect replication

• Is perfect replication XT CT possible?

Yes, if the market is complete.
Which market (or model) allows us perfect
replication? What kind of hedging strategy do
we need?
Binomial lattice model an example of complete
market
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Single period binomial model
X1(uS)DuSq(1r)W
X1(dS)DdSq(1r)W
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Single period binomial model
Solve these equations for D and q
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Multi-period binomial lattice model
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Apply one step pricing formula at each step, and
solve
backward until initial price is obtained.
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Multi-period binomial lattice model

• Perfect replication is possible

• Real probability is irrelevant

• Risk neutral probability dominates the pricing
  formula

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• Using the binomial lattice model as a guide, we
  are
  
• going to introduce Probability spaces

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Finite probability spaces

• Random experiment of 3 coin tosses

After 3 tosses, the set W of all possible
outcomes are given as
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stock price at time k
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Filtration
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Probability measure
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Probability space and filtered space
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• Random experiment of 3 coin tosses

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Adapted processes