Algorithmic Trading: An Overview

Introduction to Algorithmic Trading

• Algorithmic trading uses computer programs to automate trading decisions based on pre-defined instructions.
• It's also known as automated trading, black-box trading, or algo-trading.

Humans vs. Algorithms Comparison

Humans

• Emotional, slow decision-makers with limited resources.
• Prone to errors but have intuitive skills.

Algorithms

• Unemotional, fast, and accurate.
• Possess unlimited resources, but lack intuition.

Algorithmic Trading Statistics

• Algorithmic trading made up 60-73% of overall U.S. equity trading volume in 2009.
• About 80% of the U.S. stock market in 2018 was attributed to algorithmic trading.
• HFT is around half of all algorithmic trading.

Types of Algorithmic Trading Strategies

• Execution algorithms reduce transaction costs.
• Market making captures the spread by quoting bid and ask prices.
• Arbitrage exploits pricing inefficiencies.
• Directional strategies profit from the direction of prices.

Execution Algorithms

Volume-Weighted Average Price (VWAP)

• VWAP breaks up a large order into smaller pieces released dynamically.
• Aims to trade close to the volume-weighted average price, calculated as: $$\textrm{VWAP} = \frac{\sum_{i=1}^{n} p_i \times q_i}{\sum_{i=1}^{n} q_i}$$.
• $p_i$ is the price of trade $i$ and $q_i$ is the quantity of trade $i$.

Time-Weighted Average Price (TWAP)

• TWAP divides a large order into smaller pieces and releases them dynamically.
• Seeks to trade near the time-weighted average price.
• It's good to use when you don't want to be in sync with the market

Implementation Shortfall

• Implementation shortfall measures the difference between the hypothetical and actual performance, considering market impact and opportunity cost.
• Calculation: $$\textrm{Implementation shortfall} = \textrm{Paper Return} - \textrm{Actual Return}$$.

Market Making

• Market making involves quoting both buy (bid) and sell (ask) prices on an order book.
• It generates revenue by collecting the bid-ask spread, which is calculated as: $$\textrm{Spread} = \textrm{Ask Price} - \textrm{Bid Price}$$.

Arbitrage

Classic Arbitrage

• Classic arbitrage exploits the same asset's price differences across different markets or forms.
• An example, one could buy a stock in New York and simultaneously sell it in London.

Statistical Arbitrage

• Statistical arbitrage uses statistical models to find mispricings.
• It's more complex and riskier than classic arbitrage, e.g., pairs trading

Directional Strategies

Trend Following

• Trend following "follows the trend" - "Buy high and sell higher!", or "Sell low and buy lower!".
• Usually requires a longer time horizon to be effective.

Mean Reversion

• Mean reversion identifies when a security has deviated from its average price.
• Bets that the price will revert to its mean, usually requiring a shorter time horizon.

Building Blocks of an Algorithmic Trading System

• Data Feed enables real-time or historical market data
• Strategy utilizes the algorithm that generates trading signals
• Risk Management sets rules for limiting losses
• Order Execution sends orders to Market and enables backtesting
• Backtesting enables testing the strategy on historical data

Algorithmic Trading: Pros and Cons

Pros

• Faster execution, reduced transaction costs, and 24/7 trading.
• Removes emotion, and allows scalability and diversification.

Cons

• Requires technical expertise, and can pose system failure.
• Over-optimization and data quality can decay the model

Cross Product

Definition

• Given two vectors, $$\overrightarrow{A} = A_x\hat{i} + A_y\hat{j} + A_z\hat{k}$$, and $$\overrightarrow{B} = B_x\hat{i} + B_y\hat{j} + B_z\hat{k}$$.
• The cross product is: $$\overrightarrow{A} \times \overrightarrow{B} = (A_yB_z - A_zB_y)\hat{i} + (A_zB_x - A_xB_z)\hat{j} + (A_xB_y - A_yB_x)\hat{k}$$.

Another way to remember

• $$\overrightarrow{A} \times \overrightarrow{B} = \begin{vmatrix}\hat{i} & \hat{j} & \hat{k} \A_x & A_y & A_z \B_x & B_y & B_z\end{vmatrix} = (A_yB_z - A_zB_y)\hat{i} + (A_zB_x - A_xB_z)\hat{j} + (A_xB_y - A_yB_x)\hat{k}$$.

Magnitude

• $$|\overrightarrow{A} \times \overrightarrow{B}| = |\overrightarrow{A}||\overrightarrow{B}|sin\theta$$, where $\theta$ is the angle between $\overrightarrow{A}$ and $\overrightarrow{B}$.

Properties

• Anticommutative: $$\overrightarrow{A} \times \overrightarrow{B} = - \overrightarrow{B} \times \overrightarrow{A}$$.
• Not associative: $$\overrightarrow{A} \times (\overrightarrow{B} \times \overrightarrow{C}) \neq (\overrightarrow{A} \times \overrightarrow{B}) \times \overrightarrow{C}$$.
• Distributive: $$\overrightarrow{A} \times (\overrightarrow{B} + \overrightarrow{C}) = \overrightarrow{A} \times \overrightarrow{B} + \overrightarrow{A} \times \overrightarrow{C}$$.
• $$\overrightarrow{A} \times \overrightarrow{A} = \overrightarrow{0}$$.
• $$c(\overrightarrow{A} \times \overrightarrow{B}) = (c\overrightarrow{A}) \times \overrightarrow{B} = \overrightarrow{A} \times (c\overrightarrow{B})$$.
• $$(\overrightarrow{A} \times \overrightarrow{B}) \cdot \overrightarrow{A} = (\overrightarrow{A} \times \overrightarrow{B}) \cdot \overrightarrow{B} = 0$$.

Examenul de bacalaureat naţional 2024

Informatică

• Limbajul C/C++
• Model

SUBIECTUL I (20 de puncte)

C/C++ expresion

• 123/10*10/10

Pseudocode Algorithm

• Reads a natural number $x$.
• Has operations:
  • a%b restul împărțirii numărului natural a la numărul natural b
  • [a] partea întreagă a numărului real a

What is Quantum Mechanics

Quantum Mechanics Definition

• Quantum mechanics is a fundamental theory in physics describing physical properties of nature at the atomic and subatomic levels.
• Includes:
  • Wave-particle duality
  • Quantization of physical quantities
  • Probabilistic nature of physical phenomena.

History of Quantum Mechanics

• 1900: Max Planck introduces energy quantization to explain blackbody radiation.
• 1905: Albert Einstein explains the photoelectric effect with photons.
• 1913: Niels Bohr develops atomic model with quantized energy levels.
• 1924: Louis de Broglie proposes matter has wave-like properties.
• 1925: Heisenberg, Born, and Jordan develop matrix mechanics.
• 1926: Erwin Schrödinger develops wave mechanics.
• 1927: Werner Heisenberg formulates the uncertainty principle.
• 1927: Max Born proposes the probabilistic interpretation of the wave function.
• 1928: Paul Dirac develops the Dirac equation, combining quantum mechanics with special relativity.

Core Concepts in Quantum Mechanics

• Quantization of energy, angular momentum, and electric charge.
• Wave-Particle Duality describes how particles like electrons and photons exhibit both wave-like and particle-like properties.
• Superposition is when a quantum system can exist in multiple states simultaneously.
• Uncertainty Principle defines a fundamental limit to the precision with which pairs of physical quantities can be known simultaneously.
• Entanglement occurs when two quantum systems become correlated in a way that cannot be explained by classical physics.

Key Equations in Quantum Mechanics

• Schrödinger Equation: $$i\hbar\frac{\partial}{\partial t}\Psi(r, t) = \hat{H}\Psi(r, t)$$.
  • $i$ is the imaginary unit,
  • $\hbar$ is the reduced Planck constant,
  • $\Psi(r, t)$ is the wave function,
  • $\hat{H}$ is the Hamiltonian operator.
• Heisenberg Uncertainty Principle: $$\Delta x \Delta p \geq \frac{\hbar}{2}$$.
  • $\Delta x$ is the uncertainty in position,
  • $\Delta p$ is the uncertainty in momentum.

Applications of Quantum Mechanics

• Electronics
• Lasers
• Medical Imaging
• Materials Science
• Quantum Computing.

Branches of Quantum Mechanics

• Deals with systems where particles move at speeds much lower than the speed of light.
• Relativistic Quantum Mechanics incorporates special relativity for systems with high-speed particles.
• Quantum Field Theory combines quantum mechanics with special relativity to describe fields and particles
• Quantum Information Theory focuses on quantum mechanical aspects of information processing, computation, and communication.

Measurement in Quantum Mechanics

• The act of measurement causes the wave function to collapse into a definite state.
• Measurement in quantum mechanics differs from classical mechanics.
• The observer may play a role in determining the outcome of a quantum experimen

Algorithmic Game Theory

Definition of Algorithmic Game Theory

• It's concerned with the design and analysis of algorithms in strategic environments.
• It applies game-theoretic solution concepts to algorithm design.
• It brings insights from computer science to game theory.
• It studies the performance of systems from a computational perspective and quantifies the inefficiency of selfish behavior.

Selfish Routing

Model

• A network of $$n$$ users, each controlling one unit of traffic.
• User $$i$$'s strategy is a path $$P_i$$ from $$s_i$$ to $$t_i$$.
• A strategy profile is a vector $$P = (P_1, \dots, P_n)$$.
• Let $$f_e$$ be the total flow on edge $$e$$: $$f_e = #{i : e \in P_i}$$.
• Each edge $$e$$ has a cost function $$\ell_e(x)$$, which is assumed to be non-negative and non-decreasing.
• The cost to user $$i$$ is the sum of the latencies on her path: $$c_i(P) = \sum_{e \in P_i} \ell_e(f_e)$$.

Wardrop Equilibrium

• A Wardrop equilibrium is when no user can unilaterally decrease their cost by switching strategies.
• For every user $$i$$ and path $$P'i$$ from $$s_i$$ to $$t_i$$: $$\sum{e \in P_i} \ell_e(f_e) \le \sum_{e \in P'_i} \ell_e(f_e + \mathbb{I}{e \in P'_i \setminus P_i})$$.
• In Wardrop equilibrium, all flow travels along the shortest paths.

Social Cost

• The social cost is the sum of all users' costs: $$SC(P) = \sum_i c_i(P) = \sum_e f_e \cdot \ell_e(f_e)$$.
• The social optimum $$P^$$ is what minimizes social cost: $$P^ = \arg\min_P SC(P)$$.

Price of Anarchy

• The price of anarchy (PoA) measures the worst-case loss of social welfare due to selfish behavior.
• The ratio is: $$PoA = \max_P \frac{SC(P)}{SC(P^*)}$$, where $$P$$ is a Wardrop equilibrium.

Braess's Paradox

Description

• Adding a resource can worsen network performance.
• An additional road creates a new, more costly equilibrium.

Cost Function

• Key element in both the Price of Anarchy and Coordination Ratio.
• Coordination Ratio calculation: $$\frac{\text{System Cost at Nash Equilibrium}}{\text{Optimal System Cost}}$$.

Key Concepts of Algorithmic Game Theory

• Game Theory: Studying strategic interactions among rational agents.
• Algorithm Design: Designing efficient algorithms.
• Selfish Behavior: Agents acting self-interestedly, even if it harms the overall system.
• Price of Anarchy: Efficiency loss from selfish behavior.

Applications

• Network Design where networks should be robust to selfish behavior.
• Mechanism Design where mechanisms should incentivize agents to act benefits the overall system
• E-commerce where online marketplaces must be fair and efficient