2017 Study Session # 3, Reading # 10
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“COMMON PROBABILITY DISTRIBUTIONS”
Discrete Continuous
Random Variable Finite (measurable) # of possible outcomes.
Infinite (immeasurable) # of possible outcomes.
Distribution
P(x) cannot be 0 if ‘x’ can occur.
We can find the probability of a
specific point in time.
P(x) can be zero even if ‘x’ can occur.
We cannot find the probability of a
specific point in time.
Probability Distribution
Describes the probabilities of all possible
outcomes for a random variable.
Sum of probabilities of all possible
outcomes is 1.
Probability Function
Probability of a random variable being equal to a specific value.
Properties:
0 ≤ p(x) ≤ 1 Σ p(x) = 1
Probability Density Function (PDF)
It is used for continuous distribution.
Denoted by f(x).
Cumulative Distribution Function (CDF)
Calculates the probability of a random
variable ‘x’ taking on the value less than or equal to a specific value of ‘x’.
F(x) = P (X ≤ x)
Discrete uniform random variable All outcomes havethe same probability.
Uniform Probability Distribution
Discrete
Has a finite number of specified outcomes.
P(x)×k. K is the probability for ‘k’ number of
possible outcomes in a range. cdf: F(xn) = n.p(x).
Continuous
Defined over a range with parameters ‘b’
(upper limit) & ‘a’ (lower limit). cdf: It is linear over the variable’s range.
Two outcomes (success & failure).
‘n’ number of independent trials.
Probability of success remains constant.
p(x) =
Binomial Tree
Shows all possible combinations of up & down
moves over a number of successive periods. Node: Each of the possible values along the
tree.
U is up-move factor.
D is down-move factor (1/U).
p is probability of up move.
2017 Study Session # 3, Reading # 10
Copyright © FinQuiz.com. All rights reserved.
Normal Distribution
Properties of Normal Distribution:
Symmetric distribution
Mean = Median = Mode
Skewness = 0
Kurtosis = 3 & Excess Kurtosis = 0
Range of possible outcomes lie between -∞ to + ∞
Asymptotic to the horizontal axis
Described by two parameters i.e. Mean and Variance or (standard deviation)
When S.D ↑ (↓), the curve flattens (steepens)
Smaller the S.D, more the observations are centered around mean.
Not appropriate to use for options.
Not appropriate to use to model asset prices.
Central Limit Theorem⇒ Sum and mean of large no. of independent
variables in approximately normally distributed.
Linear combination of two or more normal random variables is also normally
distributed.
Confidence Interval
Range of values around the expected value within which actual outcome is expected to be some specified percentage of time.
Confidence %
Applications of Normal Distribution
[) −
σ
Roy’s Safety First Criterion
Optimal portfolio minimizes the
probability that the return of the portfolio falls below some minimum acceptable level. Minimize P(RP < RL).
SFRatio =
Choose the portfolio with greatest
SFRatio.
Shortfall Risk
Risk that portfolio value will fall below some minimum level at a future date.
Safety First Rule focuses on Shortfall Risk.
Sharpe Ratio
= [E (Rp) – Rf] / σp
Portfolio with the highest Sharpe ratio minimizes the probability that its return will be less than the Rf
(assuming returns are normally distributed).
Managing Financial Risk
Value at risk (VAR) ⇒minimum value of losses (in money terms) expected over a specified time period at a specified level of probability.
Stress testing/scenario analysis ⇒use of set of techniques to estimate losses in extremely worst combinations of events or scenarios.
A random variable
whose natural log has normal distribution
cannot be negative
is completely described by mean and variance
Log Normal distribution
is more appropriate to use to model asset prices
is used in Black Scholes Merton Model
Discrete:
Daily, annually, weekly, monthly compounding
Continuous
ln(S1/S0) = ln(1+HPR)
These are additive for multiple periods.
Effective annual rate based on continuous
2017 Study Session # 3, Reading # 10
Copyright © FinQuiz.com. All rights reserved.
4. Monte Carlo Simulation
Uses
It is used to:
Plan and manage financial risk.
Value complex securities
Estimate VAR
Examine model's sensitivity to changes in
the assumptions.
Limitations
Complex procedure.
Highly dependent on assumed distributions.
Based on a statistical rather than an
analytical method.
Random Number Generator
An algorithm that generates uniformly distributed random numbers between 0 and 1.
Use of a computer to generate a large number of random samples from a probability distribution
Simulation Procedure for Stock Option Valuation
Step 1: Specify underlying variable
Step 2: Specify beginning value of underlying variable
Step 3: Specify a time period
Step 4: Specify regression model for changes in stock price
Step 5: K random variables are drawn for each risk factor using computer program/ spreadsheet
Step 6: Estimate underlying variables by substituting values of random observations in the model specified in Step 4.
Step 7: Calculate value of call option at maturity and then discount back that value at time period 0
Step 8: This process is repeated until a specified number of trials ‘I’ is completed.
Step 9: Finally, mean value and S.D. for the simulation are calculated
Historical Simulation or Back Simulation
Based on actual values & actual distribution of the factors i.e., based on historical data.
Drawbacks
Cannot be used to perform “what if’
analysis.
