Understanding Algorithmic Trading

Algorithmic Trading

• Execution of orders uses automated, pre-programmed instructions.
• Instructions define order timing, price, and quantity.

Benefits of Algorithmic Trading

• Reduced transaction costs
• Improved order execution
• Increased trading speed
• Simultaneous order execution across multiple assets/markets
• Reduced errors
• Reduced impact of human emotions

Disadvantages of Algorithmic Trading

• Requires technical skills to create and maintain
• Constant monitoring is needed
• System failure
• Model failure from unforeseen events
• Subject to gaming if poorly designed

Common Algorithmic Strategies

• Trend Following: Capitalizes on momentum of existing trends.
• Mean Reversion: Prices revert to their average value over time.
• Arbitrage: Exploits tiny price differences for the same asset on different markets.
• Index Fund Rebalancing: Trading to realign portfolio weights to match an index.
• Mathematical Model Based: Uses sophisticated mathematical models to generate trading signals.
• Execution Algorithms: Designed to execute large orders efficiently and minimize market impact.

Execution Algorithms

• VWAP: (Volume Weighted Average Price).
• Executes orders to match the volume-weighted average price over a specified period.
• Equation: $$VWAP = \frac{\sum{Price_i \times Volume_i}}{\sum Volume_i}$$
• TWAP: (Time Weighted Average Price)
• executes orders close to the time-weighted average price over a period.
• Equation: $$TWAP = \frac{\sum Price_i}{n}$$
• POV: Percentage of Volume.
• Participates in the market at a specified percentage of the total volume.

Population, Sample, Parameter, and Statistic

Definitions:

• Population: The entire group of individuals or objects of interest.
• Sample: A subset of the population selected for study.
• Parameter: A numerical value that describes a characteristic of the population.
• Statistic: A numerical value that describes a characteristic of the sample.

Example of Population, Sample, Parameter, and Statistic

• Population: All students at the university.
• Sample: A group of 100 students selected from the university.
• Parameter: The average height of all students at the university (usually unknown).
• Statistic: The average height calculated from the sample of 100 students.

Sampling Methods

• Simple Random Sample (SRS):
• Each member of the population has an equal chance of being selected.
• Each possible sample of a given size has an equal chance of being selected.
• Stratified Sample
• The population is divided into subgroups (strata) based on shared characteristics.
• A random sample is taken from each stratum.
• Useful when you want to ensure representation from different groups within the population.
• Cluster Sample:
• The population is divided into clusters (groups).
• A random sample of clusters is selected.
• All members within the selected clusters are included in the sample.
• Useful when the population is geographically dispersed.
• Systematic Sample:
• Select a starting point and then select every kth element in the population.
• Convenience Sample:
• Select individuals that are easily accessible.
• Often leads to biased results.

Sampling Bias

• Selection Bias: Occurs when the sample is not representative of the population due to the method used to select the sample.
• Non-response Bias: Occurs when individuals selected for the sample do not respond and these non-respondents differ in a relevant way from those who do respond.
• Response Bias: Occurs when respondents provide inaccurate information, either intentionally or unintentionally.

Sampling Distribution of a Sample Mean

Definitions:

• Sampling Distribution: The distribution of a statistic (e.g., sample mean) calculated from multiple samples of the same size taken from the same population.
• Central Limit Theorem (CLT): For a sufficiently large sample size (usually n ≥ 30), the sampling distribution of the sample mean will be approximately normal, regardless of the shape of the population distribution.

Properties of the Sampling Distribution of the Sample Mean:

• Mean: The mean of the sampling distribution is equal to the population mean (μ).
• Standard Deviation (Standard Error): The standard deviation of the sampling distribution is equal to the population standard deviation (σ) divided by the square root of the sample size (n): $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$, where:
  • $\sigma_{\bar{x}}$ is the standard error of the mean
  • $\sigma$ is the population standard deviation
  • $n$ is the sample size

Implications of CLT:

• Allows us to make inferences about the population mean based on the sample mean, even when the population distribution is unknown.
• The larger the sample size, the smaller the standard error, and the more precise our estimate of the population mean.

Example of using Sampling Distribution of Sample Mean

• Suppose we have a population with a mean (μ) of 50 and a standard deviation (σ) of 10.
• If we take multiple samples of size 100 from this population, the sampling distribution of the sample mean will:
  • Be approximately normal.
  • Have a mean of 50.
  • Have a standard error of $\frac{10}{\sqrt{100}} = 1$.

Formulas

Z-score for Sampling Distributions:

• Formula: $z = \frac{\bar{x} -\mu}{\frac{\sigma}{\sqrt{n}}}$
• $\bar{x}$ = sample mean
• $\mu$ = population mean
• $\sigma$ = population standard deviation
• $n$ = sample size

Statics Scalars and Vectors

Scalars

• Scalars are any positive or negative physical quantity that can be completely specified by its magnitude.
• Examples: length, area, volume, mass, density, temperature, time, energy

Vector

• Vector is a physical quantity that has both a magnitude and a direction.
• Examples: position, displacement, velocity, acceleration, force, moment
• Represented graphically by an arrow
• Length of arrow is proportional to magnitude of vector
• Angle between vector and a fixed axis defines the direction of vector
• Characterized by its magnitude and direction
• Sense of direction is indicated by arrowhead

Vector Operations

Vector Multiplication and Division by Scalar

• Product of vector and a scalar yields vector
• Magnitude is absolute value of scalar times the magnitude
• Sense is same as original vector if scalar is positive and opposite if negative

Vector Addition

• Vector quantities obey the parallelogram law of addition
• To add vectors, form a parallelogram by drawing lines from the tip of each vector parallel to the other
• Resultant extends from the tail of the original vectors to the intersection of the lines
• Alternatively, can be found by a triangle construction
• Vector addition is commutative
  • A + B = B + A
• Vector addition is associative
  • A + (B + C) = (A + B) + C

Vector Subtraction

• Special case of vector addition
  • A - B = A + (-B)
• Negative vector has the same magnitude as original, but acts in the opposite direction

Vector Addition of Forces

Finding Resultant Force

• Use spring scales to measure force magnitude and direction
• Determine resultant force using parallelogram law
• Compare resultant to the force required to pull with a single force

Procedure for Analysis

• Parallelogram Law
• Determine angles between forces and diagonals
• Defines the geometry
• Sketch the parallelogram
• Trigonometry
• Redraw half of the parallelogram
• Use head-to-tail method
• Use the laws of cosines and sines
• Cosines: $C = \sqrt{A^2 + B^2 - 2AB\cos{c}}$
• Sines: $\frac{A}{\sin{a}} = \frac{B}{\sin{b}} = \frac{C}{\sin{c}}$

Partial Differential Equations

Introduction:

• A partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives.

Examples:

• Heat Equation:
• $\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$
• Wave Equation:
• $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$
• Laplace Equation:
• $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$

General Form:

• $F(x, y, u, \frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}, \frac{\partial^2 u}{\partial x^2}, \frac{\partial^2 u}{\partial x \partial y}, \frac{\partial^2 u}{\partial y^2},...) = 0$
• $x, y$ are independent variables
• $u$ is the dependent variable (unknown function)
• $F$ is a function that defines the relationship between the variables and the partial derivatives

Classifications:

• Order: The order of a PDE is the highest order of derivative that appears in the equation.
• $\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$ (2nd order)
• Linearity: A PDE is linear if the dependent variable and its derivatives appear linearly.
• Linear Example:
  • $\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$
• Non-Linear Example:
  • $\frac{\partial u}{\partial t} = u \frac{\partial u}{\partial x}$
• Homogeneity: A PDE is homogeneous if all terms contain the dependent variable or its derivatives.
• Homogeneous Example:
  • $\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$
• Non-Homogeneous Example:
  • $\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2} + f(x, t)$

Types of PDEs:

• Elliptic:
  • $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$
  • Typically describe steady-state phenomena.
• Parabolic:
• $\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$
• Typically describe diffusion processes.
• Hyperbolic:
• $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$
• Typically describe wave phenomena.

Solving PDEs:

• Analytical Methods:
• Separation of variables
• Method of characteristics
• Integral transforms (e.g., Fourier, Laplace)
• Numerical Methods:
• Finite Difference Method (FDM)
• Finite Element Method (FEM)
• Finite Volume Method (FVM)

Applications:

• Physics: Heat transfer, fluid dynamics, electromagnetism
• Engineering: Structural analysis, signal processing, control systems
• Finance: Option pricing, risk management
• Biology: Population dynamics, epidemiology

Bernoulli's Principle

• Increase in fluid speed occurs simultaneously with a decrease in pressure or potential energy.

The Bernoulli Equation

• For steady, incompressible flow:
• $P + \frac{1}{2} \rho v^2 + \rho g h = constant$
• $P$ is static pressure
• $\rho$ is density
• $v$ is speed
• $g$ is gravity
• $h$ height
• The sum is constant along a streamline

Applications

• Aerodynamics
• Fluid mechanics
• Meteorology
• Sports

Limitations

• Only applicable to inviscid flows
• Assumes steady and incompressible flow

Example of Bernoulli's Principle

• Example for solving given vars using Bernoulli's Principle

Partial Differential Equations - Introduction and Types

Definition of Heat Equation

• $\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$
• It describes how temperature u i a given region changes with time t.
  • $u$ temperature
  • $t$ time
  • $x$ position
  • $\alpha$ thermal diffusivity

Definition of Wave Equation

• $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$
• It describes the propagation of a wave.
  • $u$ displacement of the wave
  • $t$ time
  • $x$ position
  • $c$ wave speed

Laplace Equation

• $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$
• It describes the steady-state distribution of temperature or electric potential.
• $u$ temperature or electric potential
• $x, y$ spatial coordinates

Types

• Linear unknown function and derivatives appear linearly.
• Nonlinear equations are those which are not linear.
• Homogeneous: if u is a solution, then cu is also a solution for any constant c.
• Non-homogeneous eqaution is not homogeneous.

Boundary Conditions

• Dirichlet
• The value of the solution is specified on the boundary.
• $u(x,t) = f(x)$ on the boundary
• Neumann
• The normal derivative of the solution is specified on the boundary.
• $\frac{\partial u}{\partial n} (x,t) = g(x)$ on the boundary
• Robin
• A linear combination of the solution and its normal derivative is specified on the boundary.
• $\alpha u(x,t) + \beta \frac{\partial u}{\partial n}(x,t) = h(x)$ on the boundary

Methods to Solve PDEs

• Separation of Variables
  • Reduces a PDE to a set of ordinary differential equations (ODEs).
• Fourier Transform
  • Transforms the PDE into an algebraic equation.
• Laplace Transform
  • It is similar to Fourier Transform, but for time-dependent problems. Numerical Methods
  • Uses numerical approximations to solve the PDE.
  • Finite Difference Method
  • Finite Element Method