Introduction to Vectors

Introduction to Vectors

• Vectors have both magnitude and direction.
• Vectors are represented by arrows, where the length indicates magnitude and direction indicates direction.

Examples of Vectors

• Displacement
• Velocity
• Acceleration
• Force

Vector Notation

• Vectors can be denoted using:
  • Boldface letters: v
  • Letters with an arrow above: $\overrightarrow{v}$
  • Component form: $\langle a, b \rangle$ or $(a, b)$ in 2D, $\langle a, b, c \rangle$ or $(a, b, c)$ in 3D
• $a$, $b$, and $c$ are the components in the $x$, $y$, and $z$ directions, respectively.

Magnitude of a Vector

• The magnitude (or length) of a vector v is denoted by $||v||$ or $|v|$.
• In 2D: $||\mathbf{v}|| = \sqrt{a^2 + b^2}$, where $\mathbf{v} = \langle a, b \rangle$
• In 3D: $||\mathbf{v}|| = \sqrt{a^2 + b^2 + c^2}$, where $\mathbf{v} = \langle a, b, c \rangle$

Direction of a Vector

• Direction is described by the angle relative to the positive $x$-axis.
• In 2D: the direction angle $\theta$ can be found using $\tan(\theta) = \frac{b}{a}$ (where $\mathbf{v} = \langle a, b \rangle$).
• Adjust the angle by quadrant.

Vector Operations

Addition

• $\mathbf{v} + \mathbf{w} = \langle a_1 + a_2, b_1 + b_2 \rangle$, where $\mathbf{v} = \langle a_1, b_1 \rangle$ and $\mathbf{w} = \langle a_2, b_2 \rangle$

Subtraction

• $\mathbf{v} - \mathbf{w} = \langle a_1 - a_2, b_1 - b_2 \rangle$, where $\mathbf{v} = \langle a_1, b_1 \rangle$ and $\mathbf{w} = \langle a_2, b_2 \rangle$

Scalar Multiplication

• $k\mathbf{v} = \langle ka, kb \rangle$, where $\mathbf{v} = \langle a, b \rangle$ and $k$ is a scalar.

Unit Vectors

• A unit vector has a magnitude of 1.
• $\hat{\mathbf{u}} = \frac{\mathbf{v}}{||\mathbf{v}||}$ finds a unit vector in the direction of vector $\mathbf{v}$.

Standard Unit Vectors

• In 2D: $\mathbf{i} = \langle 1, 0 \rangle$ and $\mathbf{j} = \langle 0, 1 \rangle$
• In 3D: $\mathbf{i} = \langle 1, 0, 0 \rangle$, $\mathbf{j} = \langle 0, 1, 0 \rangle$, and $\mathbf{k} = \langle 0, 0, 1 \rangle$

Dot Product (Scalar Product)

• $\mathbf{v} \cdot \mathbf{w} = ||\mathbf{v}|| \cdot ||\mathbf{w}|| \cdot \cos(\theta)$, where $\theta$ is the angle between $\mathbf{v}$ and $\mathbf{w}$.
• If $\mathbf{v} = \langle a_1, b_1 \rangle$ and $\mathbf{w} = \langle a_2, b_2 \rangle$: $\mathbf{v} \cdot \mathbf{w} = a_1a_2 + b_1b_2$

Properties of the Dot Product

• $\mathbf{v} \cdot \mathbf{w} = \mathbf{w} \cdot \mathbf{v}$ (Commutative)
• $\mathbf{v} \cdot (\mathbf{w} + \mathbf{u}) = \mathbf{v} \cdot \mathbf{w} + \mathbf{v} \cdot \mathbf{u}$ (Distributive)
• $k(\mathbf{v} \cdot \mathbf{w}) = (k\mathbf{v}) \cdot \mathbf{w} = \mathbf{v} \cdot (k\mathbf{w})$ (Scalar Multiplication)
• $\mathbf{v} \cdot \mathbf{v} = ||\mathbf{v}||^2$

Cross Product (Vector Product)

• $\mathbf{v} \times \mathbf{w} = \langle (b_1c_2 - c_1b_2), (c_1a_2 - a_1c_2), (a_1b_2 - b_1a_2) \rangle$, where $\mathbf{v} = \langle a_1, b_1, c_1 \rangle$ and $\mathbf{w} = \langle a_2, b_2, c_2 \rangle$.
• $||\mathbf{v} \times \mathbf{w}|| = ||\mathbf{v}|| \cdot ||\mathbf{w}|| \cdot \sin(\theta)$, where $\theta$ is the angle between $\mathbf{v}$ and $\mathbf{w}$.

Properties of the Cross Product

• $\mathbf{v} \times \mathbf{w} = -(\mathbf{w} \times \mathbf{v})$ (Anti-commutative)
• $\mathbf{v} \times (\mathbf{w} + \mathbf{u}) = \mathbf{v} \times \mathbf{w} + \mathbf{v} \times \mathbf{u}$ (Distributive)
• $k(\mathbf{v} \times \mathbf{w}) = (k\mathbf{v}) \times \mathbf{w} = \mathbf{v} \times (k\mathbf{w})$ (Scalar Multiplication)
• $\mathbf{v} \times \mathbf{v} = \mathbf{0}$

Applications of Vectors

• Physics: Mechanics, electromagnetism
• Engineering: Structural analysis, fluid dynamics
• Computer Graphics: 3D modeling, animations
• Navigation: GPS systems, mapping

Algorithmic Trading

Definition

• "Algo Trading" uses computer programs executing instructions (algorithms) to place trades.
• It can generate profits at speeds and frequencies impossible for human traders.

How it Works

• Trader creates or utilizes an algorithm.
• Algorithm accesses market data and a trading platform.
• The algorithm monitors market data and identifies trading opportunities.
• Identified opportunities trigger automated trade placement.
• The algorithm tracks the trade and exits when conditions are met.

Algorithmic Trading Strategies

• Trend Following: Capitalize on the persistence of trends in the market.
• Mean Reversion: Exploit the tendency of prices to revert to their average value over time.
• Arbitrage: Take advantage of price differences for the same asset in different markets.
• Statistical Arbitrage: Use statistical models to identify and exploit temporary price discrepancies.
• Execution Algorithms: For efficiently executing large orders without significantly impacting market prices.
• High-Frequency Trading (HFT): Execute a large number of orders at extremely high speeds, often holding positions for only fractions of a second.
• Machine Learning: Employ machine learning algorithms to identify patterns and make predictions.
• Natural Language Processing (NLP): Utilize NLP to analyze news and sentiment for trading signals.

Advantages of Algorithmic Trading

• Speed and Efficiency: Execute trades faster and more efficiently than humans.
• Reduced Emotional Influence: Eliminates emotional biases from trading decisions.
• Backtesting Capabilities: Allow traders to backtest their strategies on historical data.
• Diversification: Enable traders to diversify their portfolios across multiple assets and markets.
• 24/7 Trading: Can operate continuously, taking advantage of opportunities around the clock.

Disadvantages of Algorithmic Trading

• Technical Expertise: Requires technical expertise in programming and data analysis.
• Development and Maintenance Costs: Developing and maintaining algorithms can be costly.
• Risk of Technical Issues: Vulnerable to technical glitches, software bugs, and connectivity problems.
• Over-Optimization: Risk of over-optimizing strategies to fit historical data, leading to poor performance in live trading.
• Regulatory Scrutiny: Subject to regulatory scrutiny and compliance requirements.

Important Libraries (Python)

• Pandas: Data manipulation and analysis
• NumPy: Scientific computing
• TA-Lib: Technical analysis
• Alphalens: Performance analysis
• Zipline: Backtesting framework
• Statsmodels: Statistical modeling
• Scikit-learn: Machine learning
• TensorFlow/Keras: Deep learning
• VaderSentiment: Sentiment analysis
• NLTK: Natural language processing

Regulations

• SEC Rule 15c3-5 (Market Access Rule): Requires brokers to have risk management controls in place for algorithmic trading.
• MiFID II (Europe): Imposes Algorithmic Trading controls, including testing, monitoring, and risk management.
• FINRA Rule 3110 (Supervision): Addresses supervision of algorithmic trading activities.

Cautions

• Slippage: Difference between expected trade price and actual execution price.
• Latency: Order placement and execution time delay.
• Market Impact: Effects of trades on market prices.
• Data Mining Bias: Finding spurious patterns by chance.
• Backtesting Bias: Over-optimizing strategies to fit past data.
• Transaction Costs: Brokerage fees and commissions.
• Black Swan Events: Unexpected events significantly affect market prices.

Chapter 14: The Laplace Transform

Definition 14.1.1

• The Laplace Transform of $f(t)$, denoted by $F(s)$ or $\mathcal{L}{f(t)}$, is defined as $\mathcal{L}{f(t)} = F(s) = \int_{0}^{\infty} e^{-st}f(t) dt$, provided the integral converges and $t \geq 0$.

Theorem 14.1.1

• If $f$ is defined for $t \geq 0$ and satisfies:
  • $f'(t)$ is piecewise continuous on $[0, \infty)$.
  • $|f(t)| \leq Ke^{at}$ for constants $K, a > 0$ and all $t \geq 0$.
• Then $\mathcal{L}{f(t)}$ exists for $s > a$.

Example 14.1.1

• $f(t) = 1$, $\mathcal{L}{1} = \frac{1}{s}, \quad s>0$

Example 14.1.2

• $f(t) = e^{at}$, $\mathcal{L}{e^{at}} = \frac{1}{s-a}, \quad s>a$

Example 14.1.3

• $f(t) = t$, $\mathcal{L}{t} = \frac{1}{s^{2}}, \quad s>0$

UNIDAD 4: Integrales Impropias

4.1. Integrales impropias de primera especie

• Sea $f(x)$ una función continua en $[a, +\infty[$. Se define $$\int_{a}^{+\infty} f(x) d x = \lim_{R \to +\infty} \int_{a}^{R} f(x) d x$$
• Si el límite existe, la integral se considera convergente; otherwise, it's divergente.
• Análogamente, si $f(x)$ es una función continua en $]-\infty, a]$, se define $$\int_{-\infty}^{a} f(x) d x = \lim_{R \to -\infty} \int_{R}^{a} f(x) d x$$
• Si $f(x)$ es una función continua en $]-\infty, +\infty[$, se define $$\int_{-\infty}^{+\infty} f(x) d x = \int_{-\infty}^{a} f(x) d x + \int_{a}^{+\infty} f(x) d x$$
• La integral impropia $\int_{-\infty}^{+\infty} f(x) d x$ converge si y sólo si convergen ambas integrales del segundo miembro, y en tal caso, $$\int_{-\infty}^{+\infty} f(x) d x = \int_{-\infty}^{a} f(x) d x + \int_{a}^{+\infty} f(x) d x$$
• El valor de la integral no depende del valor de $a$.

4.2. Integrales impropias de segunda especie

• Sea $f(x)$ una función continua en $[a, b[$ y discontinua en $b$. Se define $$\int_{a}^{b} f(x) d x = \lim_{R \to b^{-}} \int_{a}^{R} f(x) d x$$
• Si el límite existe, la integral se considera convergente; otherwise, it's divergente.
• Análogamente, si $f(x)$ es una función continua en $]a, b]$ y discontinua en $a$. Se define $$\int_{a}^{b} f(x) d x = \lim_{R \to a^{+}} \int_{R}^{b} f(x) d x$$
• Si $f(x)$ es una función discontinua en $c \in ]a, b[$, se define $$\int_{a}^{b} f(x) d x = \int_{a}^{c} f(x) d x + \int_{c}^{b} f(x) d x$$
• La integral impropia $\int_{a}^{b} f(x) d x$ converge si y sólo si convergen ambas integrales del segundo miembro, y en tal caso, $$\int_{a}^{b} f(x) d x = \int_{a}^{c} f(x) d x + \int_{c}^{b} f(x) d x$$