Monte Carlo Simulation and Historical Simulation

Simulation
Problem solving technique used to approximate the probability of certain outcomes by running multiple trial runs, using random variables.

Monte Carlo simulation
Involve the use of a computer and models to find approximate solutions for complex problems.
In finance, Monte Carlo simulation involves in identifying risk factors associated with the problem and specifying probability distributions for them. Repeated random sampling from these distributions is then used to simulate the risk factors.

Limitation

• Difficult and costly to implement

Historical simulation
Involve the use of a repeated sampling from historical data series to establish the behavior of important risk factors in generating returns.

Limitation

• Only reflect the risks that exist in the historical series being used and therefore it is difficult to use it for “what-if” experiments or to investigate the impact of “rare” events.

Discretely and continuously compounded rates of return

Discretely compound rate of return

R=P1/P0-1

Where: R is so-called as holding period return.

Continuously compounded rate of return

r=ln(1+R)=ln(P1/P0),

Where: r always less or equal to holding period return (R), never be greater

Lognormal dististbution

A random variable is lognormally distributed if the natural log of the random variable follows a normal distribution. It is usually used to model asset prices.

Y=e^x

Properties of lognormal dististbution :

• Y is never be zero or negatve
• Y is not symmetrical, skewed to the right
  

Notes:

• E(Y)≠e^E(X), E(Y)>^eE(X) , Var(Y)≠e^var(X) but not clear which is greater
• Increase in E(X) leads to increase in E(Y) & var(Y)
• Increase in Var(X) leads to increase E(Y) & Var(Y)
• (Y)=e^{E(X)+Var(x)/2}, Var(Y)=e^2E(X)+Var(X) - (e^Var(X)-1)

My tips:

• No need to memorize the above ugly formula, but need to understand the indication of the direction of change.

Shortfall risk, the safety-first ratio

Shortfall risk
The risk that the value of portfolio will fall below a minimum threshold, that is:
P(Rp ◁ min. of R)
Where:◁ - smaller than

Safety-first ratio
It is used to monitor shortfall risk, aims to minimize P(E(Rp) ◁ min. of R) and maximize SFRatio=[E(Rp) – min. of R] /σ , which is just similar to sharpe={E(Rp)-Rf }/ σ ,)

Steps to choose an optimal portfolio using the safety-first criterion

1. calculate SFRatio
2. calculate P(E(Rp) smaller than min. of R)=F(-SFRatio)
3. select portfolio with largest SFRatio or lowerest F(-SFRatio)

Roy's safety-first criterion

• Choose portfolio with the highest SFR.
  
  
  

Univariate vs.multivariate distribution

Univariate distribution
A distribution of a single random variable

Multivariate distribution
2 or more random variables, with n means & n variances and n(n-1)/2 pair-wise correlations which describe how the probability behaviors of the random variables are related to one another.

Discrete vs.Continuous random variable, and their distributions

Discrete random variable
The possible value is a countable number, such as coin flip or rolling dice.

Continuous random variable
The possible value is an uncountable (infinite) number, such as asset returns or temperatures.

Discrete distribution
No. of possible outcomes are countable

• If x cannot occur, P(x) =0
• If x can occur, P(x) >0

Discrete Uniform Random Variable

• Probability of all possible outcomes are equal p(x)=1/n
  Examples: Coin flip, die

Binomial Random Variable

• A binary variable that takes on one of two values, usually 1 for success or 0 for failure. Example: coin flip

Binomial probability

• Calcualte the no. of ways to choose x “success” from n-independent trials
  p(x)=P(X=x)={n!/(n-x)!x!}p^x(1-p)^n-x
  where: p is the probability of success on any one trial

Expected value and variance of a binomial random variable

• Expected value = np for n trial, i.e. p for a single trial
• Variance = np(1-p) for n trials. I.e. p (1- p) for single trial,

Continuous distribution:

• No. of possible outcomes are uncountable.

Note:

• Even x can occur, P(x)=0. Therefore, only meaningful to consider P(x1≦X≦x2) rather than a specific value.

Continuous uniform distribution

• X is a random variable that has equal probabilities for taking on values in the interval [a,b]
• f(x) = 1/(b-a), if a≦x≦b
• f(x) = 0, if xb

Continuous Uniform Cumulative Distribution:

• F(x) = 0 if x≦a
• F(x) = (x-a)/(b-a), if a≦x≦b
• F(x) = 1 if x≧b
• For all a≦x1≦x2≦b, then P(X or X>b)=0 and P(X≧b)=1
  
  

Probability distribution and Probability Density function (pdf)

Probability distribution
Probability distribution specifies the probabilities of the possible outcomes of a random variable. There are many different classifications of probability distributions such as the chi square, and normal and binomial distributions.

Probability Density function (pdf)
Specify the probability that a continuous random variable takes on a specific value.

f(x)=P(x1≦X≦x2)=(x2-x1)/(b-a)

Notes:

• Pdf of a number is a function 0≤ p(x) <1>
• Integral over the range of the random variable equals 1,∫p(x)dx = 1

Probability function
Probability function specifies the probability that the random variable takes on a specific value: P(X = x).

Properties of probability function
· 0 ≤ p(x) ≤ 1 because a probability lies between 0 and 1.
· The sum of probabilities p(x) over all values of X equals 1.

Cumulative Distribution function
Specify the probability that the random variable X is less than or equal to a particular value x.

F(x)=P(X≦x)