Lecture 3 Types of Probability

DistributionsDr Peter Wheale

Discrete and Continuous Probability Distributions

• A probability distribution gives the probabilities of all possible outcomes for a random variable

• A discrete distribution has a finite number of possible outcomes

• A continuous distribution has an infinite number of possible outcomes

• The number of days next week on which it will rain is a discrete random variable that can take on the values {0,1,2,3,4,5,6,7}

• The amount of rain that will fall next week is a continuous random variable

Probability Functions• A probability function, p(x), gives the

probability that a discrete random variable will take on the value xe.g. p(x) = x/15 for X = {1,2,3,4,5}→ p(3) = 20%

A probability density function (pdf), f(x) can be used to evaluate the probability that a continuous random variable with take on a value within a range

A cumulative distribution function (cdf), F(x), gives the probability that a random variable will be less than or equal to a given value

A Probability Function• Denoted p(x), specifies the probability that a random

variable is equal to a specific value. It is the probability that random variable X takes on the value x, or p(x) = P(X=x).

• The two key properties of a probability function are:• 0 ≤ p(x) ≤ 1 • Σp(x) = 1

• The sum of all probabilities for all possible outcomes, x, for a random variable, X, equals 1

Probability Density Function• A pdf is a function, denoted f(x), that can be used to

generate the probability that outcomes of a continuous distribution lie within a particular range of outcomes.

• A pdf is used to calculate the probability of an outcome between two values, i.e. the probability of the outcome falling within a specified range. Calculus is used to take the integral of the function to make this calculation.

Cumulative Distribution Function

• A cdf defines the probability that a random variable, X, takes on a value equal to or less than a specific value, x. It is the sum of the probabilities for the outcomes up to and including a specified outcome.

• The cdf for random variable, X, may be expressed as F(x) = P(X ≤ x).

Discrete Uniform Random Variable

• A discrete uniform distribution has a finite number of possible outcomes, all of which are all equal.

e.g. p(x) = 0.2 for X = {1,2,3,4,5}The probability of each outcome, p(1), p(2) etc,

is equal to 0.2The cdf for the nth outcome, F(xn) = np(x), and

the probability for a range of outcomes is p(x)k, where k is the number of possible outcomes in the range.

Binomial Random Variable• The probability of exactly x successes in n

trials, given just two possible outcomes (gien either success or failure)

• Probability of success on each trial (p) is constant, and trials are independent

Binomial Random VariableExample: What is the probability of drawing exactly 2 white marbles from a bowl of white and black marbles in six tries if the probability of selecting white is 0.4 each time? (x = 2, p = 0.4, n = 6)

Binomial TreeTwo possible outcomes each period, up or

down

Prob (up move) + Prob (down move) = 1Up factor (U) > 1 Down factor (D) = 1/U

Example:Beginning stock price (S0) = $20Prob up = 60% Prob down = 40%Up factor = 1.12 Down factor 1/1.12

A Binomial Tree for Stock Price

Continuous Uniform Distribution

Defined over a range that spans between some lower limit, a, and some upper limit, b, which serve as the parameters of the distribution.Probability distributed evenly over an intervale.g. continuous uniform over the interval 2 to 10P(X < 2) = 0 P(X > 10) = 0 probability of X outside the boundaries is zeroP(3 ≤ X ≤ 5) = (5 - 3)/(10 - 2) = 2/8 = 25% defines the probability between x1 and x2

Lognormal Distribution• The lognormal distribution is generated by the function

ex, where x is normally distributed. Since the natural logorithm, ln, of ex is x, the logorithms of lognormally distributed random variables are normally distributed

• The lognormal distribution is skewed to the right, and is bounded from below by zero so that it is useful for modelling asset prices which never take negative values.

Monte Carlo Simulation• A technique based on the repeated generation of one or

more risk factors that affect security values, in order to generate a distribution of security values

• The analyst must specify the parameters of the probability distribution that the risk factor is assumed to follow

• A computer programme then is then used to generate random values for each risk factor based on its assumed probability distributions

Monte Carlo Simulation(Cont.)Simulation can be used to estimate a distribution of derivatives prices or of NPVs

1) Specify distributions of random variables such as interest rates, underlying stock prices

2) Use computer random generation of variables3) Value the derivative using those values4) Repeat steps 2 and 3 1,000s of times5) Calculate mean/variance of all values

Uses of Monte Carlo Simulation

• Value portfolios of assets that have non-normal return-distributions

• Value complex securities• Simulate the profits/losses from a trading strategy• Calculate estimates of value at risk (VAR) to determine

the riskiness of a portfolio of assets and liabilities• Simulate pension fund assets and liabilities over time to

examine the variability of the difference between the two