Financial Markets

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Summery of risk neutral pricing
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Complete markets

• Can a simple European derivative security always
  be hedged?

- If the answer is Yes, such a model is said to
be complete.

• Is the binomial model complete?

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Risk neutral pricing on the binomial model
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Use Markov property to compute this!
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Markov processes

• The distribution of Xk1 conditioned on X0, X1,
  , Xk is
  
• the same as the distribution of Xk1
  conditioned on Xk

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• More generally, Conditions (a) (d) can be
  stated with the
  
• process at time k and multiple future times.

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Independence of increments

• Xk is said to have independent increments if
  X0 , X1- X0,
  
• X2 - X1, , Xk - Xk-1 are independent.

- More precisely,
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Connections to Markov processes
Proof
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In particular, we have
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Corollary
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Independence and conditional expectation
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Independence lemma
- X independent of G
- Y G-measurable
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• f(X, Y) X Y,

• f(X, Y) XY,

• If f(X, Y) h(XY),

This is the case for the binomial model with
independent coin tosses
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Binomial lattice model
Sk1 Xk1 Sk
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Binomial lattice model
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European call option pricing formula
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Summery

• Under risk neutral probability measure, the
  discounted stock
  
• price process and portfolio process are
  martingales

• The APT value process of a European derivative
  security is
  
• a martingale

• The binomial lattice model is complete, i.e., it
  is hedgeable.

• The APT value of a European derivative security
  is computed
  
• on lattice backward based on the Markov property

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- Stopping times and American options
- Properties of value processes of American
options

• The Radon-Nikodym Theorem and the state price
  density
  
• process

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American options

• European option (with an expiration N and a
  Strike price K)

You can exercise the option at the expiration only

• American option

You can exercise the option at any time until the
expiration
- You have an additional choice
When is the optimal to exercise?
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Pricing and hedging American options
- g(Sk) the payoff of an American option
- Vk the value of an American option

• We solve an backward recursion algorithm on the
  binomial
  
• lattice with the above constraint