Discrete Continuous Random Variable Finite measurable # of possible outcomes.. Distribution specific point in time.. Probability Distribution outcomes for a random variable.. Probability
Trang 1Discrete Continuous Random Variable Finite (measurable) # of possible
outcomes
Infinite (immeasurable) # of possible outcomes
Distribution
specific point in time
specific point in time
Probability Distribution
outcomes for a random variable
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)
Cumulative Distribution Function (CDF)
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
possible outcomes in a range
cdf: F(xn) = n.p(x)
Continuous
(upper limit) & ‘a’ (lower limit)
cdf: It is linear over the variable’s range
Properties:
P ( x ≤ a) = 0 & P (x ≥ b) = 1
P( a < x < b) = ି
௫ మି௫భ
!
− ! !
௫
Binomial Distribution Properties:
Two outcomes (success & failure)
p(x) =
Binomial Tree
moves over a number of successive periods
Node: Each of the possible values along the tree
Trang 2Normal Distribution
Properties of Normal Distribution:
Kurtosis = 3 & Excess Kurtosis = 0
Central Limit Theorem⇒ Sum and mean of large no of independent
variables in approximately normally distributed
distributed
Confidence Interval
Range of values around the expected value within which actual outcome is expected to be some specified percentage of time
Confidence % Interval
x ± 1s 68.%
x ± 1.96s 95%
x ± 2s 95.45%
x ± 3s 99.73%
Applications of Normal Distribution
[) −
σ
Roy’s Safety First Criterion
probability that the return of the portfolio falls below some minimum acceptable level
Minimize P(RP < RL)
SFRatio =
SFRatio
Shortfall Risk
Risk that portfolio value will fall below some minimum level at a future date
Safety First Rulefocuses onShortfall 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
Log Normal distribution
Discrete:
Daily, annually, weekly, monthly compounding
Continuous
ln(S1/S0) = ln(1+HPR)
compounding is given as:
EAR = e Rcc-1
Lognormal Distribution Properties
Compounds Rate
of Return
Trang 34 Monte Carlo Simulation
Uses
It is used to:
the assumptions
Limitations
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
analysis