Algorithmic Trading Strategies

Algorithmic Trading Overview

• Algorithmic trading uses computer programs to automate trading processes.

Human Traders vs. Algorithmic Traders

• Speed: Algorithmic traders execute trades faster than human traders.
• Objectivity: Algorithmic traders are unbiased, while human traders can be biased.
• Scalability: Algorithmic trading offers high scalability, while human trading has low scalability.
• Consistency: Algorithmic trading is consistent, whereas human trading can be inconsistent.
• Cost: Algorithmic trading has lower operation costs than human trading.

Algorithmic Trading Strategies

• Execution Algorithms: Designed to execute large orders without impacting market prices.
  • VWAP (Volume Weighted Average Price) aims to execute orders at the average price weighted by volume.
  • TWAP (Time Weighted Average Price) aims to execute orders at the average price over a specified time period.
  • POV (Percentage of Volume) algorithms execute trades based on a percentage of the current market volume.
• Market Making Algorithms: Provide liquidity by placing buy and sell orders in the market.
• Statistical Arbitrage Algorithms: Identify and exploit statistical inefficiencies in the market.
• Machine Learning Algorithms: Employ machine learning techniques to find trading opportunities.
  • Supervised learning algorithms learn from labeled data to make predictions.
  • Unsupervised learning algorithms find patterns in unlabeled data.
  • Reinforcement learning algorithms learn to make decisions by maximizing a reward signal.

High Frequency Trading (HFT)

• A subset of algorithmic trading characterized by very high speed and turnover rates.
• HFT firms often use co-location to minimize latency.

Algorithmic Trading Challenges

• Overfitting: Creating a model performing well on historical data but poorly on new data.
• Latency: The time delay between order placement and execution.
• Market Impact: The influence of trading on the market price.
• Regulation: Compliance with various regulations is required.

Steps for Implementing Algorithmic Trading Strategies

• Researching and developing a trading strategy
• Backtesting the strategy on historical data
• Paper trading to test the strategy in a simulated environment
• Live trading with real capital
• Continuously monitoring the strategy's performance

Backtesting Considerations

• Data Quality: Accuracy and completeness of historical data are essential.
• Transaction Costs: Include commissions and slippage in backtesting simulations.
• Overfitting: Avoid creating overly complex models that perform well only on historical data.

Paper Trading Purpose

• Paper trading allows for testing a strategy in a simulated environment.
• It is a crucial step to test infrastructure and strategy execution before live trading.

Risk Management Elements

• Position Sizing: Determining capital allocation for each trade.
• Stop-Loss Orders: Closing positions automatically if the price moves unfavorably.
• Diversification: Spreading capital across different assets to reduce risk.

Key Textbooks

• "Advances in Financial Machine Learning" by Marcos Lopez de Prado, 2018
• "Algorithmic Trading: Winning Strategies and Their Rationale" by Ernst P. Chan, 2017

Graphical Models

• Graphical models blend probability theory and graph theory for addressing uncertainty and complexity.
• Probability theory ensures system consistency and interfaces with data.
• Graph theory offers an intuitive interface for modeling interacting variables and enables efficient inference.

Types of Graphical Models

• Bayesian Networks (Directed Graphical Models).
• Markov Random Fields (Undirected Graphical Models).

Bayesian Networks Defined

• Bayesian Network is a directed graphical model that represents probabilistic relationships among random variables.
• It consists of a directed acyclic graph (DAG)
• A set of conditional probability distributions (CPDs) quantifies the strength of dependencies between variables.

Bayesian Networks Key Concepts

• Nodes represent random variables.
• Edges indicate direct influence from variable A to variable B.
• Each node is associated with a conditional probability distribution (CPD), denoted as $P(X_i \mid Parents(X_i))$.
• The graph structure encodes conditional independence assumptions.

Bayesian Networks Joint Probability Distribution

• The joint probability distribution is factorized as the product of conditional probabilities.
• Represented as $P(X_1, X_2,..., X_n) = \prod_{i=1}^{n} P(X_i \mid Parents(X_i))$
  • $X_1, X_2,..., X_n$ are the random variables.
  • $Parents(X_i)$ are the parents of variable $X_i$.

Bayesian Networks Example

• Structure with A -> B and B -> C
• Joint probability distribution is $P(A, B, C) = P(A) \cdot P(B \mid A) \cdot P(C \mid B)$

Markov Random Fields Defined

• Markov Random Field (MRF) is an undirected graphical model, representing probabilistic relationships via an undirected graph.
• Nodes represent variables, and edges represent dependencies.

Markov Random Fields Key Concepts

• Each graph node represents a random variable.
• Edges denote direct dependency between linked variables.
• The graph structure encodes conditional independence assumptions.
• MRFs lack inherent directionality but utilize potential functions for defining variable relationships.

Markov Random Fields Joint Probability Distribution

• Joint distribution is factorized via potential functions over graph cliques

$$ P(X_1, X_2,..., X_n) = \frac{1}{Z} \prod_{c \in C} \phi_c(X_c) $$

• $X_1, X_2,..., X_n$ are random variables in the network.
• $C$ is the set of maximal cliques in the graph.
• $\phi_c(X_c)$ is the potential function.
• $Z$ is the normalization constant, known as the partition function.

Markov Random Fields Example

• A simple graph with four fully connected variables: A, B, C, and D.
• Joint probability distribution: $$ P(A, B, C, D) = \frac{1}{Z} \cdot \phi_{AB}(A, B) \cdot \phi_{BC}(B, C) \cdot \phi_{CD}(C, D) \cdot \phi_{DA}(D, A) $$

Graphical Models Applications

• Machine Learning (classification, regression, clustering)
• Computer Vision (image segmentation, object recognition)
• Natural Language Processing (text classification, machine translation)
• Bioinformatics (gene networks, protein interactions, disease modeling)
• Robotics (state estimation, path planning, decision-making)

Advantages of Graphical Models

• Interpretability: Clear representation of variable relationships.
• Modularity: Complex systems modeled by simpler combined models.
• Scalability: Efficient algorithms for large-scale models.
• Flexibility: Adaptability to data types and problem settings.

Disadvantages of Graphical Models

• Complexity: Model construction and learning can be expensive.
• Data Requirements: Large amounts of data needed for accurate learning.
• Model Selection: Choosing the right graph structure is challenging.
• Inference: Exact inference can be intractable, needing approximations.

Heat Equation Definition

$$ \begin{gathered} \frac{\partial u}{\partial t} - \Delta u = 0 \quad \text{in } \mathbb{R}^n \times (0, \infty) \ u(x, 0) = g(x) \quad \text{on } \mathbb{R}^n \times {t = 0} \end{gathered} $$

• Where $g: \mathbb{R}^n \rightarrow \mathbb{R}$ is bounded and continuous.

Heat Equation Solution

$$ u(x, t) = \int_{\mathbb{R}^n} \Phi(x - y, t) g(y) dy $$

$$ \Phi(x, t) = \frac{1}{(4\pi t)^{n/2}} e^{-\frac{|x|^2}{4t}} $$

• Is the heat kernel, also known as the fundamental solution of the heat equation.

Heat Kernel Properties

• $\Phi(x, t) > 0$ for all $x \in \mathbb{R}^n$, $t > 0$
• $\int_{\mathbb{R}^n} \Phi(x, t) dx = 1$ for all $t > 0$
• $\Phi(x, t) \rightarrow 0$ uniformly for $x \neq 0$ as $t \rightarrow 0$
• $\int_{\mathbb{R}^n} \Phi(x - y, t) g(y) dy \rightarrow g(x)$ as $t \rightarrow 0$

Proof Of Heat Kernal Property 2

$$ \begin{aligned} \int_{\mathbb{R}^n} \Phi(x, t) dx &= \int_{\mathbb{R}^n} \frac{1}{(4\pi t)^{n/2}} e^{-\frac{|x|^2}{4t}} dx \ &= \frac{1}{(4\pi t)^{n/2}} \int_{\mathbb{R}^n} e^{-\frac{|x|^2}{4t}} dx \ &= \frac{1}{(4\pi t)^{n/2}} \prod_{i=1}^{n} \int_{-\infty}^{\infty} e^{-\frac{x_i^2}{4t}} dx_i \ &= \frac{1}{(4\pi t)^{n/2}} \prod_{i=1}^{n} \sqrt{4\pi t} \ &= \frac{1}{(4\pi t)^{n/2}} (4\pi t)^{n/2} = 1 \end{aligned} $$

$$ \int_{-\infty}^{\infty} e^{-ax^2} dx = \sqrt{\frac{\pi}{a}} $$