FEC Financial Engineering Club

1. FECFinancial Engineering Club

2. Welcome! • Facebook: http://www.facebook.com/UIUCFEC • LinkedIn: http://www.linkedin.com/financialengineeringclub • Email: uiuc.fec@gmail.com • Please Welcome the MSFE Director, Morton Lane! • fecuiuc.com • is up!

3. Probability & Statistics Primer

4. Discrete Random Variables • Definition: The cumulative distribution function(CDF), of a random variable X is defined by • Definition:A discrete random variable, X, has probability mass function (PMF) if and for all events we have • Definition: The expected value of a function of a discrete random variable X is given by • Definition: The variance of any random variable, X, is defined as

5. Bernoulli & Binomial RVs • Bernoulli RV: • Let X=Bernoulli(p) • Pdf: • Binomial RV: • PDF: • Models: • The probability that we achieve successes after trials, each with probability of success

6. Poisson RVs • Let • Models: • The probability that some event occurs times in a fixed time period if the event is known to occur at an average rate of times per time period, independently of the last event.

7. Geometric Distribution • Let • Models: • The probability that it takes successive independent trials to get first success with probability of success for each event

8. Continuous Random Variables • Definition:A continuous random variable, X, has probability density function (PDF) if and for all events we have • Definition: The cumulative distribution function (CDF), of a continuous random variable X is related to the PDF by: • Definition: The expected value of a function of a continuous random variable X is given by

9. Exponential • Let • PDF: • Models: • The time between events occurring independently and continuously at a constant average rate

10. Normal/Gaussian Distribution • Let • Central Limit Theorem: Let be a sequence of independent random variables with mean and variance . Then:

11. Brownian Motion

12. Brownian Motion

13. Simulating Random Variables • For continuous, use inverse CDF method: if F(x) is cdf of random variable X then to simulate X, • Generate U~Uniform(0,1) • X = • Easy example: simulate an exponential with parameter λ • CDF if x ≥  • Simulate U~Uniform(0,1), note that (1-U)~Uniform(0,1) • Set X = , X is exponential(λ)

14. Conditional Probability • Definition: The probability that X occurs given Y occurred is: • Bayes’s Theorem says that:

15. Multivariate Random Variables • We have two RVs, X and Y • Let the joint PDF of X and Y be • Definition: The joint cumulative distribution function (CDF) of satisfies • Definition: The marginal density function of is:

16. Multivariate Random Variables

17. Multivariate Random Variables

18. Independent Random Variables

19. Independent Random Variables

20. Covariance • Covariance is the measure of how much two variables change together. • Cov(X,Y)>0 if increasing X increasing Y • Cov(X,Y)<0 if increasing X decreasing Y

21. Correlation Coefficient • Definition:The correlation of two RVs, X and Y, is defined by: • If X and Y are independent, they are uncorrelated:

22. Variance and Covariance

23. Variance and Covariance

24. Linear REGRESSION • Least Squares Method: • The minimizing is: • The minimizing is:

25. Combinations of Random Variables • Examples, portfolio mean and variance: Equations (1) and (3) generalized to N variables (assets in the portfolio) with coefficients as weights: see boxed info in http://en.wikipedia.org/wiki/Modern_portfolio_theory

26. Moment Generating Functions • …..

27. Geometric Brownian Motion

28. Maximum Likelihood Estimator • Likelihood function = • Let represent all parameters to the RV • is a function of , fixed • is the maximum likelihood estimator (MLE)

29. Thank you! • Facebook: http://www.facebook.com/UIUCFEC • LinkedIn: http://www.linkedin.com/financialengineeringclub • Email: uiuc.fec@gmail.com • Next Meeting: • “Trading and Market Microstructure” • Wed. 26thFeb. 6-7pm • 165 Everitt • fecuiuc.com • is up!