FINANCIAL TRADING AND MARKET MICRO-STRUCTURE

1. FINANCIAL TRADING AND MARKET MICRO-STRUCTURE MGT 4850 Spring 2011 University of Lethbridge

2. Topics • The power of Numbers • Quantitative Finance • Risk and Return • Asset Pricing • Risk Management and Hedging • Volatility Models • Matrix Algebra

3. MATRIX ALGEBRA • Definition • Row vector • Column vector

4. Matrix Addition and Scalar Multiplication • Definition: Two matrices A = [aij] and B = [bij ] are said to be equal if Equality of these matrices have the same size, and for each index pair (i, j), aij = bij , Matrices that is, corresponding entries of A and B are equal.

5. Matrix Addition and Subtraction • Let A = [aij] and B = [bij] be m × n matrices. Then the sum of the matrices, denoted by A + B, is the m × n matrix defined by the formula A + B = [aij + bij ] . • The negative of the matrix A, denoted by −A, is defined by the formula −A = [−aij ] . • The difference of A and B, denoted by A−B, is defined by the formula A − B = [aij − bij ] .

6. Scalar Multiplication • Let A = [aij] be an m × n matrix and c a scalar. Then the product of the scalar c with the matrix A, denoted by cA, is defined by the formula Scalar cA = [caij ] .

7. Linear Combinations • A linear combination of the matrices A1,A2, . . . , An is an expression of the form c1A1 + c2A2 + ・ ・ ・ + cnAn

8. Laws of Arithmetic • Let A,B,C be matrices of the same size m × n, 0 the m × n zero • matrix, and c and d scalars. • (1) (Closure Law) A + B is an m × n matrix. • (2) (Associative Law) (A + B) + C = A + (B + C) • (3) (Commutative Law) A + B = B + A • (4) (Identity Law) A + 0 = A • (5) (Inverse Law) A + (−A) = 0 • (6) (Closure Law) cA is an m × n matrix.

9. Laws of Arithmetic (II) • (7) (Associative Law) c(dA) = (cd)A • (8) (Distributive Law) (c + d)A = cA + dA • (9) (Distributive Law) c(A + B) = cA + cB • (10) (Monoidal Law) 1A = A

10. Portfolio Models • Portfolio basic calculations • Two-Asset examples • Correlation and Covariance • Trend line • Portfolio Means and Variances • Matrix Notation • Efficient Portfolios

11. Review of Matrices • a matrix (plural matrices) is a rectangular table of numbers, consisting of abstract quantities that can be added and multiplied.

12. Adding and multiplying matrices • Sum • Scalar multiplication

13. Matrix multiplication • Well-defined only if the number of columns of the left matrix is the same as the number of rows of the right matrix. If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix productAB is the m-by-p matrix (m rows, p columns).

14. Matrix multiplication • Note that the number of of columns of the left matrix is the same as the number of rows of the right matrix , e. g. A*B →A(3x4) and B(4x6) then product C(3x6). • Row*Column if A(1x8); B(8*1) →scalar • Column*Row if A(6x1); B(1x5) →C(6x5)

15. Matrix multiplication properties: • (AB)C = A(BC) for all k-by-m matrices A, m-by-n matrices B and n-by-p matrices C ("associativity"). • (A + B)C = AC + BC for all m-by-n matrices A and B and n-by-k matrices C ("right distributivity"). • C(A + B) = CA + CB for all m-by-n matrices A and B and k-by-m matrices C ("left distributivity").

16. The Mathematics of Diversification • Linear combinations • Single-index model • Multi-index model • Stochastic Dominance

17. Return • The expected return of a portfolio is a weighted average of the expected returns of the components:

18. Two-Security Case • For a two-security portfolio containing Stock A and Stock B, the variance is:

19. portfolio variance • For an n-security portfolio, the portfolio variance is:

20. Minimum Variance Portfolio • The minimum variance portfolio is the particular combination of securities that will result in the least possible variance • Solving for the minimum variance portfolio requires basic calculus

21. Minimum Variance Portfolio (cont’d) • For a two-security minimum variance portfolio, the proportions invested in stocks A and B are:

22. The n-Security Case (cont’d) • A covariance matrix is a tabular presentation of the pairwise combinations of all portfolio components • The required number of covariances to compute a portfolio variance is (n2 – n)/2 • Any portfolio construction technique using the full covariance matrix is called a Markowitz model

23. Computational Advantages • The single-index model compares all securities to a single benchmark • An alternative to comparing a security to each of the others • By observing how two independent securities behave relative to a third value, we learn something about how the securities are likely to behave relative to each other

24. Multi-Index Model • A multi-index model considers independent variables other than the performance of an overall market index • Of particular interest are industry effects • Factors associated with a particular line of business • E.g., the performance of grocery stores vs. steel companies in a recession

25. Multi-Index Model (cont’d) • The general form of a multi-index model:

26. Portfolio Mean and Variance • Matrix notation; column vector Γ for the weights transpose is a row vector ΓT • Expected return on each asset as a column vector or E its transpose ET • Expected return on the portfolio is a scalar (row*column) Portfolio variance ΓTS Γ (S var/cov matrix)