Algorithmic Trading

Algorithmic Trading

• Involves using computer programs to automate trading strategies based on pre-determined rules.
• Can execute orders quickly
• Aims to improve efficiency
• Aims to reduce transaction costs
• Aims to enhance anonymity

Benefits of Algorithmic Trading

• Can provide speed and efficiency in trade execution.
• May reduce overall transaction costs of trades.
• May improve order execution quality.
• Can provide anonymity in the market

Challenges of Algorithmic Trading

• Involves model risk, as the algorithm relies on the accuracy of the model.
• Includes technical issues such as software and connection problems.
• Potential for over-optimization, where the algorithm performs well on historical data but poorly in live trading.
• Can have market impact, where the algorithm's trades affect the price of the asset.

Order Types

• Market Order: Executed immediately at the best available price.
• Limit Order: Executed at a specified price or better.
• Stop Order: Triggered when the price reaches a certain level.
• Stop-Limit Order: A combination of stop and limit orders.
• Iceberg Order: Large order displayed in smaller tranches.
• VWAP Order: Executes based on the Volume Weighted Average Price.
• TWAP Order: Executes based on the Time Weighted Average Price.
• Percentage of Volume (POV) Order: Participates in the market volume.

Order Execution Strategies

• Direct Market Access (DMA): Direct access to the exchange order book.
• Smart Order Routing: Automatically routes orders to the best available market.
• Dark Pools: Private exchanges for large block orders.
• Implementation Shortfall: Aims to minimize the difference between the actual execution price and the decision price.

Market Impact Definition

• Market impact is the effect of a trader's actions on the price of an asset.

Factors Influencing Market Impact

• Order Size: Impact increases with the size of the order.
• Market Liquidity: Impact is higher in less liquid markets.
• Order Urgency: Aggressive orders have a higher impact
• Trading Venue: Impacts vary depending on the trading venue.

Mitigation Strategies

• Smaller Order Sizes: Reduces the impact of each individual trade.
• Using Stealth Algorithms: Minimizes the visibility of the algorithm's trades.

Regulatory Compliance

• Key Regulations: Include MiFID II, Dodd-Frank Act, and Market Abuse Regulation (MAR).
• Compliance Requirements: Algorithmic testing and certification, order audit trail, risk management controls, surveillance, and monitoring.

Differential Equations

• Differential equations of order 1 are equations in the form of $F(x, y, y') = 0$, where 'x' is an independent variable, 'y' is a function y(x), and y' is its derivative dy/dx.

Solution

• A solution to a differential equation, $F(x, y, y') = 0$, is a function $y = \phi(x)$ reducing the equation to an identity for 'x' in an interval I.

General vs Particular solutions

• The general solution of a differential equation of order 1 is a family of solutions $y = \phi(x, C)$ dependent on an arbitrary constant $C$.
• Conversely, a particular solution is obtained by defining the value of C.

Initial Values

• An initial value problem (IVP) for a differential equation finds a solution $y = \phi(x)$ satisfying the differential equation.
• It satifies an initial condition in the form $y(x_0) = y_0$, defined by the constants $x_0$ and $y_0$.

Separable Differential equations

• A separable differential equation of order 1 is written in the form $M(x) + N(y) \frac{dy}{dx} = 0$, or $M(x) dx + N(y) dy = 0$.
• To resolve a separable differential equation, integrate the terms separately, so $\int M(x) dx + \int N(y) dy = C$, where C is a constant for the integration.

Quantum Mechanics Introduction

• In 1900, Planck's quantum hypothesis stated $$E = h\nu$$
• In 1905, Einstein proposed the light quantum hypothesis stating $$E = h\nu = \hbar \omega$$
• In 1913, Bohr theorized atomic spectra
• In 1923, De Broglie stated his hypothesis $$p = \hbar k$$
• In 1925, Heisenberg and in 1926, Schrödinger created matrix mechanics and wave mechanics
• In 1927, the Solvay Conference met
• Light and matter can exhibit properties of both waves and particles
• Wave Function: $$\Psi(x,t)$$
• Probability Density: $$P(x,t) = |\Psi(x,t)|^2$$
• Normalization Condition: $$\int_{-\infty}^{\infty} |\Psi(x,t)|^2 dx = 1$$

Schrodinger Equations

• The Time-dependent Schrodinger equation stated as $$i\hbar \frac{\partial}{\partial t} \Psi(x,t) = \hat{H} \Psi(x,t)$$ and $$\hat{H} = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V(x,t)$$ where $\hat{H}$ is Hamiltonian Operator
• The Time-independent Schrodinger Equation as: $$\hat{H} \psi(x) = E \psi(x)$$, where E is energy of the system
• Stationary States: $$\Psi(x,t) = \psi(x) e^{-iEt/\hbar}$$

Operators and Observables

• Operators are mathematical operators that act on wave functions to represent physical quantities.
• Eigenvalues and Eigenfunctions $$\hat{A} \psi(x) = a \psi(x)$$ a is eigenvalue when corresponding eigenfunction is $\psi(x)$
• Expectation Values: $$\langle A \rangle = \int_{-\infty}^{\infty} \Psi^*(x,t) \hat{A} \Psi(x,t) dx$$

Uncertainty Principles

• Heisenberg uncertainty principle $$\Delta x \Delta p \ge \frac{\hbar}{2}$$
• It is impossible to know both the position and momentum of a particle with perfect accuracy.

Examples of Physics

• Free Particle is a particle not subject to external forces when constant potential energy $$V(x) = 0$$.
• Particle in a Box, contained in a finite space, implies $V(x) = 0$ when $0 < x < L$ and then the potential outside the box leads to $V(x)= \infty $ otherwise
• Harmonic Oscilator when the potential energy is quadratic: $$V(x) = \frac{1}{2} m \omega^2 x^2$$

List of Physics Operators

• Position: $\hat{x} = x$
• Momentum: $\hat{p} = -i\hbar \frac{\partial}{\partial x}$
• Kinetic Energy: $\hat{T} = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2}$
• Total Energy: $\hat{H} = \hat{T} + \hat{V}$

Scope of Population Genetics

• Microevolution (population genetics): studies changes in allele frequencies within populations.
• Population: interbreeding individuals of the same species that produce fertile offspring.
• Gene pool: all alleles for all genes in a population.
• Genotype: the genetic makeup, or set of alleles, of an organism.
• Population genetics uses mathematical models to study microevolution.
• Hardy-Weinberg Theorem indicates that the frequencies of alleles and genotypes will remain constant over generations without external factors.

Conditions for Equilibrium

• The Hardy-Weinberg theorem describes a population that is not evolving.
• These conditions must be met for a population to be in Hardy-Weinberg equilibrium:
• No mutations
• Random mating
• No natural selection
• Extremely large population size
• No gene flow

Calculations for Frequency

• The allele frequency indicates Frequency of $C^R = p = \frac{970}{1000} = 0.97$ Frequency of $C^W = q = \frac{30}{1000} = 0.03$
• For two alleles, the sum of the frequencies must equal 1 (i.e., $p + q = 1$)

Hardy Weinberg Equation

• When population is in equilibrium $$p^2 + 2pq + q^2 = 1$$ where: $$p^2$$ is freq of $C^RC^R$ $$2pq$$ is freq of $C^RC^W$ $$q^2$$ is freq of $C^WC^W$

Testing for Equilibrium

• Determine population equilibrium by comparing known genotype frequencies to expected genotype frequencies
• The observed frequency of $C^RC^R$ is slightly higher than expected
• Frequencies of $C^RC^W$ and $C^WC^W$ are lower than expected

Statistical Thermodynamics

• Statistical Thermodynamics connects microscopic characteristics and energy to thermodynamic systems.

Ensembles

• An ensemble is the set of all possible systems under a given parameter.
• Different types include microcanonical (NVE constant), canonical (NVT constant), grand canonical ensemble($\mu$VT constant).

Roles of Partition Function

• Partition function (Q) connects microscopic states and macroscopic properties.
• Q equation: $$Q = \sum_i e^{-\beta E_i}$$.

Connection to Microscopic Properties

• Partition Function (Q) connects microscopic states and macroscopic properties.
• Equations associated with the function include Internal Energy: $$U = -\frac{\partial}{\partial \beta} \ln Q$$ Entropy: $$S = k(\ln Q + \frac{U}{kT})$$ Helmholtz Free Energy: $$A = -kT \ln Q$$ Pressure: $$P = kT \frac{\partial}{\partial V} \ln Q$$

Molecular Partition Function

• Equations include : For independent and distinguishable particles: $$Q = q^N$$ For indistinguishable particles: $$Q = \frac{q^N}{N!}$$.
• Molecular partition function has translational, rotational, vibrational and electronic freedom given by $$q = q_{trans} \cdot q_{rot} \cdot q_{vib} \cdot q_{elec}$$.

Molecular Motion Equations

• Translational Partition Function: $$q_{trans} = \left( \frac{2\pi m}{h^2 \beta} \right)^{3/2} V$$
• Rotational Partition Function: For linear molecules $$q_{rot} = \frac{1}{\sigma \beta h c B}$$
• For non-linear molecules $$q_{rot} = \frac{\sqrt{\pi}}{\sigma} \left( \frac{1}{\beta h c A} \frac{1}{\beta h c B} \frac{1}{\beta h c C} \right)^{1/2}$$
• Vibrational Partition Function: $$q_{vib} = \frac{1}{1 - e^{-\beta h \nu}}$$
• Electronic Partition Function: $$q_{elec} = \sum_i g_i e^{-\beta E_i}$$

Transition State Theory

• Transition State Theory (TST): rate of a reaction is determined by the rate at which reactants reach a transition state $$$A + B \rightleftharpoons [AB]^{\ddagger} \rightarrow Products$$

Key TST Equation

• rate constant $k_{TST}$ is given by:$$k_{TST} = \kappa \frac{k_B T}{h} \frac{q^{\ddagger}}{q_A q_B} e^{-\beta \Delta E_0^{\ddagger}}$$

Kinetic Isotope Effect

• Change in reaction rate when one of the atoms in the reactants is replaced by an isotope, given by equation$$KIE = \frac{k_H}{k_D}$$

Arrhenius and Marcus Theory

• Arrhenius equation: $$k = A e^{-E_a/RT}$$
• Marcus Theory: $$k_{ET} = \frac{2\pi}{\hbar} |H_{AB}|^2 \frac{1}{\sqrt{4\pi\lambda k_B T}} e^{-\frac{(\Delta G^0 + \lambda)^2}{4\lambda k_B T}}$$

Reaction Rate Basics

• A Reaction rate measures change in concentration
• Laws express this rate to concentration of reactants: $$rate = k[A]^m[B]^n$$

Types of Reaction Orders

• The order of a reaction is the sum of the exponents in the rate law.
• Overall order measures m + n

Calculations for Reaction

• Zero Order: $$[A]_t = -kt + [A]_0$$
• Half Life $$t_{1/2} = \frac{[A]_0}{2k}$$
• First Order: $$ln[A]_t = -kt + ln[A]_0$$
• Half life $$t_{1/2} = \frac{0.693}{k}$$
• Second Order: $$\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$$.
• Half life $$t_{1/2} = \frac{1}{k[A]_0}$$

Reaction Mechanisms

• A is a step-by-step sequence of elementary reactions by which overall chemical change occurs
• Elementary Step occurs in a single step. Rate-Determining Step: the slowest step in a mechanism

Reactions and Temperatures

• Activation energy (E_a): minimum energy required for a reaction to occur.
• Activation energy increases with temperature.
• Arrhenius equation $$k = Ae^{-E_a/RT}$$. k relates to activation energy with temperature

Catalyst Roles

• A catalyst increases the rate of a reaction without being consumed in the overall reaction.
• Types of catalysts:
  • Homogeneous: catalyst and reactants in the same phase.
  • Heterogeneous: catalyst and reactants in different phases.