Financial Markets

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Summery

• Stopping time t is a time when you can decide
  to stop (or
• exercise an American derivative security)

• The value process of an American derivative
  security is a
• super-matingale

• It is optimal to exercise an American derivative
  security
• first time when the value and the payoff are
  the same

• The initial value of an American derivative
  security can be
• obtained on the binomial lattice by a backward
  recursion

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- The Radon-Nikodym theorem
- The state price density process

• General random variables

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Radon-Nikodym theorem
There exists a nonnegative random variable Z s.t.
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Intuitive meaning

• Discrete case

• Continuous case

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Equivalent probability measure

• Binomial stock process on

- Real probability measure P
(corresponding to p and q)
(corresponding to )
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Connection between two measures

• Radon-Nikodym theorem

In particular,
the discounted stock price process is a
martingale
there exists a stochastic process s.t.

is a martingale
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Radon-Nikodym martingales
state price density process
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Proof
(Taking in what is known)
(Martingale property)
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• The value process of a self-financing portfolio

satisfies
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Binomial lattice stock process
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- Law of a random variable
- Density of a random variable

• Marginal density, conditional density

• Normal distribution

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Law of a random variable
W sample space,
containing all possible outcomes
s-algebra of subsets of W.
P a probability measure on (W, ), a set
function s.t.

• Random variable

a -measurable function s.t.
where
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• Induced measure

X induces a measure on
s.t.
Induced measure,
or the Law of X

• Lebesgue measure m0 a measure on

providing
the length of an interval (or more generally, a
Lebesgue set)
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Density a random variable

• Two measures

m0 and mX on
If m0 gtgt mX,
there exists a random variable fX s.t.
We write

• X has a density iff m0 gtgt mX,

i.e.,