Algorithmic Trading Strategies

Algorithmic Trading

• Uses computer programs to execute orders based on predefined instructions.
• Referred to as automated trading, black-box trading, or algo-trading.
• Algorithms leverage speed and accuracy for high-frequency trading.
• Institutional investors utilize it to minimize market impact and execution costs.

Benefits of Algorithmic Trading

• Reduces transaction costs.
• Improves order execution speed.
• Increases trading efficiency.
• Reduces human error.
• Enhances market access.
• Increases transparency.

Drawbacks of Algorithmic Trading

• Complex to develop and maintain.
• Requires robust infrastructure.
• Carries potential for system failure.
• Risk of over-optimization exists.
• Needs continuous monitoring.
• Faces regulatory compliance requirements.

Algorithmic Trading Strategies

• Trend Following: Identifies and capitalizes on market trends using moving averages and trend lines.
• Mean Reversion: Bets on prices reverting to their mean, utilizing statistical analysis.
• Arbitrage: Exploits price differences in different markets through simultaneous buying and selling.
• Market Making: Provides liquidity by placing buy and sell orders and capturing the spread between them.
• Execution Algorithms: Optimizes order placement and timing to minimize market impact.

Order Execution

• Order execution is completing a buy or sell order in the market to achieve the best possible price.
• It involves choosing the right order type and routing strategy.

Order Types

• Market Order: Executed immediately at the best available price, guaranteeing execution but not price.
• Limit Order: Executed only at a specified price or better, guaranteeing price but not execution.
• Stop Order: Turns into a market order when a specified price is reached, used to limit losses or protect profits.
• Stop-Limit Order: Turns into a limit order when a specified price is reached, combining features of stop and limit orders.
• Iceberg Order: A large order displayed in smaller portions to reduce market impact.

Execution Venues

• Exchanges: Centralized marketplaces for trading securities providing price discovery and transparency.
• Electronic Communication Networks (ECNs): Electronic systems that match buy and sell orders offering faster execution and lower fees.
• Dark Pools: Private exchanges that do not display order information, helping institutional investors minimize market impact.
• Over-the-Counter (OTC) Markets: Decentralized markets for trading securities that offer greater flexibility but less transparency.

Order Routing Strategies

• Direct Routing: Orders are directly sent to a specific venue, offering greater control over execution.
• Smart Order Routing (SOR): Orders are automatically routed to the best available venue, aiming for the best price and execution quality.
• Broker Algorithms: Algorithms offered by brokers to optimize order execution, including volume-weighted average price (VWAP) and time-weighted average price (TWAP).

Factors Affecting Order Execution

• Market Liquidity: The ability to buy or sell securities quickly and easily; higher liquidity leads to better execution prices.
• Order Size: The number of shares or contracts traded; large orders may have a greater market impact.
• Timing: The time of day when the order is placed, which can impact execution prices.
• Volatility: The degree of price fluctuation in the market; higher volatility can lead to less predictable execution prices.

Best Execution

• The duty of brokers to execute orders at the best possible price, considering factors such as price, speed, and likelihood of execution falls under this duty.
• Regulatory agencies oversee best execution practices.

Regulations

• Market Access Rule (Rule 15c3-5): Requires brokers to have risk management controls to prevent erroneous orders and market disruptions.
• Regulation National Market System (Reg NMS): A set of rules designed to modernize and strengthen the U.S. equity markets, including provisions on order protection, access to quotations, and market data fees.
• MiFID II: A European Union regulation that aims to increase transparency and investor protection, including provisions on best execution and algorithmic trading.

Energy of a Uniform String

• Setup: Considers a string with mass density ( \mu ) and tension ( T ), with vertical displacement given by ( \eta(x, t) ).

Kinetic Energy

• Equation: ( KE = \int \frac{1}{2} \mu \left( \frac{\partial \eta}{\partial t} \right)^2 dx )

Potential Energy

• Derived from the work done to stretch the string, and is given by ( PE = \int \frac{1}{2} T \left(\frac{\partial \eta}{\partial x}\right)^2 dx )

Total Energy

• The sum of kinetic and potential energies: ( E = \int \frac{1}{2} \mu \left(\frac{\partial \eta}{\partial t}\right)^2 + \frac{1}{2} T \left(\frac{\partial \eta}{\partial x}\right)^2 dx )

Normal Modes

• General Solution: Displacement is ( \eta(x, t) = \sum_{n=1}^{\infty} A_n \sin(k_n x) \cos(\omega_n t - \phi_n) ), where ( k_n = \frac{n \pi}{L} ) and ( \omega_n = v k_n = \frac{n \pi v}{L} ), with ( v = \sqrt{T/\mu} )
• Energy in Each Normal Mode: ( E_n = \frac{L}{4} \mu \omega_n^2 A_n^2 = \frac{\pi^2 T}{4L} n^2 A_n^2 )
• Total Energy as Sum of Normal Mode Energies: Can be expressed as ( E = \sum_{n=1}^{\infty} E_n )

Initial Value Problem

• Initial Conditions: Given ( \eta(x, 0) = f(x) ) and ( \frac{\partial \eta}{\partial t}(x, 0) = g(x) )
• Solving for Coefficients: Coefficients ( A_n ) and ( \phi_n ) are found using initial conditions and Fourier analysis.
  • ( A_n \cos(\phi_n) = \frac{2}{L} \int_0^L f(x) \sin\left(\frac{n\pi x}{L}\right) dx )
  • ( A_n \sin(\phi_n) = \frac{2}{n \pi v} \int_0^L g(x) \sin\left(\frac{n\pi x}{L}\right) dx )

Quantum Mechanics - Postulates

• The state is defined by a wavefunction Ψ(r, t), where |Ψ(r, t)|^2 dτ is the probability of finding the particle.
• The wavefunction must be single-valued, continuous, finite, and normalized.

Observables and Operators

• Every classical observable corresponds to a linear hermitian operator in quantum mechanics.
• Hamiltonian operator (Ĥ) corresponds to the total energy: Ĥ = -ħ²/2m ∇² + V(r)

Eigenvalues and Eigenfunctions

• The only values observed when measuring an observable are the eigenvalues (ai): ÂΨi = aiΨi
• Average Value: The average value of an observable corresponding to  is <A> = ∫ Ψ* Â Ψ dτ

Time Evolution

• The Schrödinger equation: Ĥ Ψ(r, t) = i ħ ∂Ψ(r, t) / ∂t

The Hydrogen Atom

Schrödinger Equation

• Fundamental Equation: Ĥ ψ(r, θ, φ) = E ψ(r, θ, φ)
• Hamiltonian Operator (Ĥ): Ĥ = -ħ²/2μ ∇² - e²/4πε₀r

Solving the Equation

• Wave Function: ψ(r, θ, φ) = R(r)Y(θ, φ)

Quantum Numbers

• Principal Quantum Number (n): Determines energy level.
• Angular Momentum Quantum Number (l): Determines orbital shape.
• Magnetic Quantum Number (m): Determines orbital orientation.
• Spin Quantum Number (s): Determines electron spin.

Partial Differential Equations (PDEs)

Definition of a PDE

• An equation involving derivatives of an unknown function with more than one variable.
• PDEs are used in models for fluid mechanics, heat transfer, electromagnetism, acoustics, elasticity, etc.
• Heat Equation Example: ∂u/∂t = k ∂²u/∂x² (u is temperature, x is spatial coordinate, t is time, k is thermal diffusivity).
• Wave Equation Example: ∂²u/∂t² = c² ∂²u/∂x² (u is displacement, x is spatial coordinate, t is time, c is wave speed).
• Laplace's Equation Example: ∂²u/∂x² + ∂²u/∂y² = 0 (u is potential, x and y are spatial coordinates).

General Form of a Linear Second-Order PDE:

$$ A \frac{\partial^2 u}{\partial x^2} + B \frac{\partial^2 u}{\partial x \partial y} + C \frac{\partial^2 u}{\partial y^2} + D \frac{\partial u}{\partial x} + E \frac{\partial u}{\partial y} + F u = G $$

Classifications Based on Discriminant (Δ = B² - 4AC):

• Hyperbolic if Δ > 0 (e.g., wave equation)
• Parabolic if Δ = 0 (e.g., heat equation)
• Elliptic if Δ < 0 (e.g., Laplace's equation)

Boundary Conditions include:

Dirichlet :

• Solution value specified on boundary u(x, y) = f(x, y)

Neumann

• Normal derivative of solution specified on boundary ∂u/∂n(x, y) = g(x, y)

Robin Boundary Condition

• Linear combination of solution and normal derivative specified

Common Solution Techniques:

• Separation of Variables: Assumes solution is a product of functions each depending on only one variable.
• Fourier Series: Represents solution as a sum of sine and cosine functions.
• Laplace Transforms: Applies Laplace transform to convert PDE to ODE.
• Numerical Methods: Includes Finite Difference Method, Finite Element Method, and Finite Volume Method.

Example:

Solving Heat Equation using Separation of Variables:

Method consists:

• Separation of Variables
• Solving ODE's
• Apply boundary conditions
• Apply initial conditions to find the value ( u(x,t) )

Markdown Basics

Definition of Markdown

• It is a lightweight markup language with a plain text formatting syntax.
• Used in blogs, messaging, forums, documentation, and readme files.
• Markdown files have the extension .md.
• Cleaner method for adding format.

Headings:

Heading 1, ## Heading 2, ### Heading 3, #### Heading 4, ##### Heading 5, ###### Heading 6

Emphasis:

• Italics*, Bold, Bold and Italics

Lists:

• Unordered

1. Ordered

Links:

Link

Images:

Code:

inline code ​javascript function fancyAlert(arg) { if (arg) { $.facebox({ div : '#foo' }) } } ​

Tables:

Quote:

This is a quote

Divider:

Matrices

Dimensions:

• A matrix is an array of numbers with dimensions ( m \times n ), where ( m ) is the number of rows and ( n ) is the number of columns.

Matrix Addition:

• If $A$ and $B$ are both $m \times n$ matrices, then $C = A + B$ is also an $m \times n$ matrix, where $c_{ij} = a_{ij} + b_{ij}$.

Matrix Multiplication:

• If $A$ is an $m \times n$ matrix and $B$ is an $n \times p$ matrix, then $C = AB$ is an $m \times p$ matrix, where ( c_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj} )

MIKROE

• Pressure 2 Click is used to measure barometric pressure.
• It is a compact solution featuring the LPS22HBTR sensor.
• Ultra-compact piezoresistive absolute pressure sensor functions as a digital output barometer.
• Runs on either 3.3V or 5V power supply.
• It communicates with the microcontroller over the I2C or SPI interface.

Electrical Characteristics

• Operating Voltage: 3.3V or 5V
• Supply Current: 2mA
• Pressure Range: 260 to 1260 hPa
• Pressure Accuracy: $\pm$0.5 hPa
• Temperature Accuracy: $\pm$1.0 °C

Fourier Transform Properties

Linearity:

F{af(t) + bg(t)} = aF(f(t)) + bF(g(t))

Time Scaling:

F{f(at)} = 1/|a| F(𝜔/a)

Time Shifting:

F{f(t - t0)} = e^(-j𝜔t0)F(𝜔)

Frequency Shifting:

F{e^(j𝜔0t)f(t)} = F(𝜔 - 𝜔0)

Conjugation:

F{f*(t)} = F*(-𝜔)

Time Differentiation:

F{d/dt f(t)} = (j𝜔)F(𝜔)

Frequency Differentiation:

F{tf(t)} = j d/d𝜔 F(𝜔)

Integration:

F{∫ f(τ) dτ} = 1/(j𝜔) F(𝜔) + πF(0)δ(𝜔)

Convolution:

Properties

• Linearity
• Time Scaling
• Time Shifting
• Frequency Shifting
• Conjugation
• Time Differentiation
• Frequency Differentiation
• Integration
• Convolution
• Multiplication
• Parseval's Theorem