Common Probability Distributions in Finance

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Common Probability Distributions in Finance
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Probability Distributions

• In the previous lecture, we defined a random
  variable as a variable that can take many
  uncertain values
• In order to calculate the probability that a
  random variable takes specific values, we need to
  understand its probability distribution
• A probability distribution specifies the
  probabilities of all possible values of a random
  variable
• Example Should we be interested in probability
  distributions of asset prices or changes in asset
  prices?

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Probability Distributions

• Asset prices and changes in asset prices move
  through time and, thus, their probability
  distributions tend not to be stable over time
• However, we expect that percentage changes in
  asset prices (asset returns) should be more
  stable through time even when asset prices change
• Therefore, if we can fit a probability
  distribution to historical data, we can use it as
  a basis to forecast future values of a random
  variable

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Types of Random Variables

• Two basic types of random variables
• Discrete random variables
• Continuous random variables
• Discrete random variables are those that can take
  on only a finite number of possible outcomes
• Continuous random variables are those that can
  assume one of an infinitely large number of
  values within certain limitations

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Types of Random Variables

• Examples of discrete financial random variables
• The number of IPOs in a certain year
• Stock prices quoted in decimals (ticks of 0.01)
• The number of stock price increases in a month
• The size of the board of directors of financial
  institutions
• The number of mergers and acquisitions in a
  certain year
• Examples of continuous financial random variables
• Stock returns
• Accounting variables
• Macroeconomic variables

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Discrete and Continuous Probability Distributions

• There are several discrete and continuous
  probability distributions available
• We will introduce three distributions that have
  interesting applications in asset pricing and
  asset returns
• Binomial distribution (Discrete)
• Normal distribution (Continuous)
• Lognormal distribution (Continuous)
• We will introduce a couple of additional
  distributions during the discussion of hypothesis
  testing

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The Binomial Distribution

• The binomial probability distribution is a
  discrete distribution with interesting
  applications
• Think of a trial (an event that can be repeated
  many times) that can produce only two possible
  outcomes
• The two outcomes are mutually exclusive and can
  be placed into one of two categories success or
  failure
• The probability of a success is p and that of a
  failure 1 p those probabilities stay constant
  in each trial and the trials are independent

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The Binomial Distribution

• A binomial random variable is defined as the
  number of successes in n trials
• E.g., In n trials as described above, we can have
  between 0 and n successes
• Thus, a binomial distribution can be
  characterized only by two parameters, n (number
  of trials) and p (probability of success in each
  trial)
• Example An interesting application of this
  probability distribution is in modeling stock
  price movements another one is in binomial
  option pricing

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The Binomial Distribution
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The Binomial Distribution

• If X is a binomial random variable, the
  probability that X will take the value x, P(Xx),
  meaning the probability of x successes in n
  trials, is given by
• is the number of ways of selecting x
  successes from n trials (combination formula)

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The Binomial Distribution

• Digression
• The combination formula is as follows
• The term n! is the n factorial defined as

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The Binomial Distribution

• Properties of Binomial Distribution
• The binomial distribution has mean np and
  variance np(1-p)
• The binomial distribution is symmetric when p
  0.5
• For other values of p, the binomial distribution
  exhibits some asymmetry (skewness)