Algorithmic Trading Strategies

Involves using computer programs to execute trades automatically based on pre-set rules.
Algorithms dictate when to generate, cancel, or modify orders.
It is also known as automated trading, black-box trading, or algo-trading.

Reduces transaction costs via optimal pricing.
Improves order execution speed, enabling faster response to market changes.
Can diversify trading, facilitating the implementation of numerous strategies simultaneously.
Reduces manual errors by automating the trading process.
Increases profits by identifying and exploiting profit opportunities.
Can be used to backtest to check the performance of the strategy.

Strategy involves entering a long position when the price exceeds a certain level.
Strategy exits when the price drops below another level.
Example: If the current price is above the 20-day high, then buy; if below the 20-day low, then sell.

Strategy seeks to exploit price discrepancies of the same asset across different markets.
Example: Buying in market B and selling in market A if the price in market A exceeds the price in market B plus transaction costs.

Market makers place buy and sell orders to profit from the bid-ask spread.
This involves placing buy orders at the bid price and sell orders at the ask price.
When buy orders are executed, a corresponding sell order is placed at a higher price, and vice versa.

Overfitting: Strategy performs well on historical data but poorly on new data.
Data Mining Bias: Involves identifying patterns in historical data that are not real.
Model Decay: The strategy becomes less profitable over time due to market changes.
Technical Glitches: Software bugs, hardware failures, and network outages.
Black Swan Events: Unexpected events that can cause large losses.

A procedural query language consisting of operations that take one or two relations as input, and produce a new relation as a result.

Selection: Selects a subset of tuples from a relation that satisfy a given condition denoted as $\sigma_{p}(r)$, where $\sigma$ is the selection symbol, $p$ is the predicate formula, and $r$ is the relation. Example: $\sigma_{department="Finance"}(employee)$
Projection: Selects a subset of attributes from a relation using the notation $\prod_{A_{1}, A_{2},..., A_{k}}(r)$, where $\prod$ is the projection symbol, $A_{1}, A_{2},..., A_{k}$ is the list of attributes to keep, and $r$ is the relation. Example: $\prod_{name, salary}(employee)$
Union: Combines two relations with the same schema, denoted as $r \cup s$. Relations $r$ and $s$ must have the same schema.
Set Difference: Finds tuples present in one relation but not in another, denoted as $r - s$. Both relations $r$ and $s$ must have the same schema.
Cartesian Product: Combines each tuple of one relation with each tuple of another relation, denoted as $r \times s$.

Set Intersection: Finds the tuples common to two relations, denoted as $r \cap s$, and can be expressed using set difference: $r \cap s = r - (r - s)$.
Conditional Join: Joins two relations based on a condition, denoted as $r \Join_{c} s$, and can be expressed using Cartesian product and selection: $r \Join_{c} s = \sigma_{c}(r \times s)$.
Natural Join: Joins two relations based on equality of attributes with the same name, denoted as $r \Join s$. It can be expressed using conditional join.
Division: Finds the tuples in one relation that match all tuples in another relation, denoted as $r \div s$.

Contains tables named employee (with attributes like name, employee_id, department, salary) and department (with attributes like department and location).
Example Queries:
- Find all employees in the Finance department.
- Find the names and salaries of all employees.
- Find all employees who earn more than $65,000.
- Find the department name and location of all departments.
- Find the names of all employees in the Finance department earning more than $65,000.
- Find the names of all employees and the location of their department.
- Find the names of all employees who work in a department located in New York.

A matrix A is a rectangular array of numbers with m rows and n columns, denoted as $A = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \ a_{21} & a_{22} & \cdots & a_{2n} \ \vdots & \vdots & \ddots & \vdots \ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix}$
$a_{ij}$ refers to the element in the i-th row and j-th column.
$m \times n$ is the dimension of the matrix.
Matrices are often denoted with capital letters.

Square Matrix: $m=n$
Zero Matrix: All elements are 0.
Identity Matrix: A square matrix with 1s on the main diagonal and 0s everywhere else.
Diagonal Matrix: A square matrix with all elements outside the main diagonal being 0.
Symmetric Matrix: $A = A^T$, that is, $a_{ij} = a_{ji}$.

Addition: Two matrices A and B can be added if they have the same dimension where $(A + B){ij} = a{ij} + b_{ij}$.
Scalar Multiplication: A matrix A can be multiplied by a scalar $c$ where $(cA){ij} = c \cdot a{ij}$.
Matrix Multiplication: The product of two matrices $A (m \times n)$ and $B (n \times p)$ is a matrix $C (m \times p)$, where $(AB){ij} = \sum{k=1}^{n} a_{ik}b_{kj}$.

The transpose of a matrix A, denoted as $A^T$, is obtained by swapping rows and columns, where $(A^T){ij} = a{ji}$.

The inverse of a square matrix A, denoted as $A^{-1}$, satisfies the condition $A \cdot A^{-1} = A^{-1} \cdot A = I$.
Not every matrix has an inverse; a matrix is invertible if its determinant is non-zero.

The determinant is a function that assigns a number to a square matrix.
The determinant is a matrix function.
For a $2 \times 2$ matrix: $$A = \begin{pmatrix} a & b \ c & d \end{pmatrix}$$. where $\det(A) = ad - bc$.