Research and Development

Research and Development

• R&D represents an investment towards future technology and capabilities.
• R&D is conducted to develop new products, processes, and improvements to existing products.

Research

• Research is the systematic investigation into a topic or issue.
• The purpose is to discover or revise facts, theories, and applications.

Basic Research

• The primary objective is to gain knowledge.

Applied Research

• The primary objective is solving a specific practical problem.

Development

• It is the use of research findings to plan, design, and produce new or improved products and processes.

New Product Development Process

• Consists of idea generation, feasibility study, prototype creation, pilot plant testing, and the launch of the new product.

New Product Development Considerations

• Involves the market demand, required investment and potential return.

R&D Intensity

• R&D Intensity is R&D Expenditure divided by Revenue.
• R&D intensity gauges a company's commitment to R&D.
• R&D intensity can be used for comparisons between companies in the same industry.
• Increased innovation corresponds with higher R&D intensity.

Sources of Innovation

• Include universities, government labs, competitors, customers, and suppliers.

Reglas de inferencia

• Inference rules are logical forms that allow valid conclusions to be derived from given premises.
• Inference rules are fundamental in propositional and predicate logic for constructing arguments

Modus Ponens (MP)

• If $P \rightarrow Q$ is true, and $P$ is true, then $Q$ is true

Modus Tollens (MT)

• If $P \rightarrow Q$ is true, and $Q$ is false (¬Q), then $P$ is false (¬P)

Silogismo Hipotético (SH)

• If $P \rightarrow Q$ is true, and $Q \rightarrow R$ is true, then $P \rightarrow R$ is true

Silogismo Disyuntivo (SD)

• If $P \vee Q$ is true, and $P$ is false (¬P), then $Q$ is true

Adición (Ad)

• If $P$ is true, then $P \vee Q$ is true

Simplificación (Simp)

• If $P \wedge Q$ is true, then $P$ is true

Conjunción (Conj)

• If $P$ is true, and $Q$ is true, then $P \wedge Q$ is true

Dilema Constructivo (DC)

• If $(P \rightarrow Q) \wedge (R \rightarrow S)$ is true, and $P \vee R$ is true, then $Q \vee S$ is true

Algorithmic Trading

• Algorithmic trading is also known as automated trading, black-box trading, or systematic trading.
• Algorithmic trading involves executing orders according to pre-programmed instructions based on time, price, and quantity.
• Algorithms are typically designed to exploit opportunities or to lower transaction costs.

Why Algorithmic Trading?

• For traders, algorithmic trading reduces transaction costs, improves order execution, and provides access to new markets.
• Algorithmic trading also increases execution speed, diversification and the ability to back-test strategies.
• For markets, algorithmic trading increases liquidity, reduces impact costs, and reduces operational risk.
• Markets become more efficient through algorithmic trading.

Types of Trading Strategies

Execution Algorithms

• Execution Algorithms include VWAP (Volume Weighted Average Price) and TWAP (Time Weighted Average Price).
• Other Execution Algorithms include Implementation Shortfall and POV (Percentage of Volume).
• Execution algorithms execute large orders automatically to reduce transaction costs market impact.

Direct Market Access

• Direct Market Access (DMA) provides direct access to the order book.
• DMA allows for faster execution with more control over orders.

Statistical Arbitrage

• Statistical Arbitrage involves pairs trading, index arbitrage, and triangular arbitrage.
• Statistical Arbitrage involves exploiting statistical relationships between different assets.

High-Frequency Trading

• Ultra-fast execution speeds and high turnover rates are characteristics of High-Frequency Trading (HFT)
• Complex algorithms and advanced technology is commonly used in HFT.

Market Making

• Providing liquidity through bid and offer orders result in earning the spread between these prices

Event-Driven Strategies

• Event-Driven Strategies involve trading based on specific events or news releases.
• News Analytics and Earnings Announcements are examples of Event-Driven Strategies.

Trend Following

• Trend Following includes moving averages and breakout strategies.
• Identifying and capitalizing on market trends is key

Machine Learning

• Machine Learning includes supervised learning, unsupervised learning, and reinforcement learning.
• Machine learning algorithms identify patterns and make predictions.

Common Algorithmic Trading Platforms

• Bloomberg EMSX, Fidessa, Charles River IMS, TOMS, CQG, TT and MetaTrader are common platforms.

Backtesting

• Backtesting involves evaluating the strategy on simulated historical data.
• Backtesting is important for validating a strategy, optimizing parameters, and assessing risk.
• Overfitting, data mining bias, and failure to account for transaction costs may cause potential pittfalls.

Key Performance Indicators

• Sharpes Ratio, Maximum Drawdown, Profit Factor, and Win Rate are KPI's.

Regulations

• MiFID II (Markets in Financial Instruments Directive), Regulation SHO (U.S. Securities and Exchange Commission), and Market Abuse Regulation (MAR) aim to maintain fair markets.
• The goal of these regulations is to prevent market abuse, to protect investors, while ensuring fair and transparent markets.

Risks of Algorithmic Trading

Technical Risks

• Technical Risks include software bugs, hardware failures, and connectivity issues.

Model Risks

• Model Risks include overfitting, incorrect assumptions, and data errors.

Market Risks

• Market Risks include flash crashes, unexpected events, and liquidity issues.

Regulatory Risks

• Regulatory Risks include changes in regulations, compliance issues, and legal challenges.

The Future of Algorithmic Trading

• The future involves increased use of machine learning and AI, along with a greater focus on data analytics.
• More sophisticated risk management techniques and increased regulatory scrutinity are likely to develop.
• Expansion into new markets and asset classes is expected in the future.

Summary

• Algorithmic trading involves using pre-programmed instructions to execute orders.
• It improves order execution, offers transaction costs reduction, and various strategies such as execution algorithms.
• Robust risk management is needed as it enhances market efficiency, yet, it poses risks.
• Advances in Machine Learning, data analytics, and regulatory developments are likely to shape the future.

Chemistry Careers

High School Teacher

• A bachelor's degree in chemistry and a teaching certification are required.
• A high school teacher teaches chemistry.
• The annual salary ranges between $45,000-$75,000 per year.
• The job outlook is good.

Professor

• A Ph.D. in chemistry and research experience are required.
• A professor teaches college level students and conducts chemistry research.
• The annual salary ranges between $80,000-$150,000.
• The job outlook is good.

Researcher

• A master's or Ph.D. in chemistry and research experience are required.
• Researchers conduct chemistry research in a lab environment
• The annual salary ranges between $60,000-$120,000 per year.
• The job outlook is good.

Lab Technician

• Associate's or bachelor's in chemistry and lab experience are required.
• A lab technician assists chemists in the laboratory with experiments and analysis.
• The annual salary ranges between $35,000-$60,000 per year.
• The job outlook is good.

Chemical Engineer

• A bachelor's degree in chemical engineering and an engineering license are required.
• Chemical engineers are responsible for chemical processes and equipment design.
• The annual salary ranges between $70,000-$130,000 per year.
• The job outlook is good.

Pharmacist

• A Doctorate of Pharmacy (Pharm.D.) and a pharmacy license are required.
• Pharmacists distribute medications and provide drug information to patients.
• The annual salary ranges between $90,000-$140,000 per year.
• The job outlook is good.

Algorithmic Game Theory

Game Theory Definition

• Games are strategic interactions between two or more players.
• Each player can choose from a set of possible actions.
• Players' selection of choices affects the game's outcome.
• Each player has preferences about the outcome.

Example: Prisoner's Dilemma

• If both players cooperate they each face a value of -1.
• If player 1 cooperates and player 2 defects they face values of -3 and 0, respectively.
• If player 1 defects and player 2 cooperates they face values of 0 and -3, respectively.
• If both players defect they each face a value of -2.

Algorithmic Game Theory Definition

• Traditional game theory assumes players are rational with unlimited resources.
• Algorithmic game theory considers the computational complexity of strategic interactions.
• Topics include representing games, computing solution concepts, mechanism design, and learning in games.

Representing Games

Normal-Form Games

• A set of players $N = {1, \dots, n}$
• A set of actions $A_i$ for each player $i \in N$
• A utility function $u_i: A \to \mathbb{R}$ for each player $i \in N$, where $A = \prod_{i \in N} A_i$ is the set of action profiles.

Example: Prisoner's Dilemma

• $N = {1, 2}$
• $A_1 = A_2 = {\text{Cooperate, Defect}}$
• $u_1(\text{Cooperate, Cooperate}) = -1$
• $u_1(\text{Cooperate, Defect}) = -3$
• $u_1(\text{Defect, Cooperate}) = 0$
• $u_1(\text{Defect, Defect}) = -2$

Representations

• Normal-form representation is good for small games, but exponential as the number of players and actions increase.
• Graphical games represent players as nodes in a graph, where a player's utility depends only on the actions of their neighbors.
• Compact representations are useful for congestion games, voting games, and mechanism design settings.

Solution Concepts

Nash Equilibrium

• A Nash equilibrium is an action profile $a \in A$ such that for all players $i \in N$ and for all actions $a_i' \in A_i$: $u_i(a_i', a_{-i}) \leq u_i(a_i, a_{-i})$
• $a_{-i}$ is the vector of actions of all players except $i$.
• No player has an incentive to unilaterally deviate in a Nash equilibrium.

Example: Prisoner's Dilemma

• (Defect, Defect) is a Nash equilibrium.

Computing Nash Equilibria

• Two-player zero-sum games can be solved in polynomial time using linear programming.
• Two-player general-sum games are PPAD-complete.
• Multi-player games are even more complex.
• Approximate Nash equilibria are often easier to compute.

Mechanism Design

• It is designing games to achieve a desired outcome.

Example: Auctions

• The goal is to allocate an item to the player with the highest valuation.
• Challenge arises when players may lie about true valuations.
• Solution is the Vickrey auction (second-price sealed-bid auction).
• Incentive compatibility describes a mechanism where it is in each player's best interest to report their true valuation.
• Revenue maximization involves designing mechanisms to maximize the revenue of the seller.

Learning in Games

• The goal is to design strategies that converge to a Nash equilibrium.

Example: Fictitious Play

• Iterative game playing by players.
• Each player maintains a belief about the other's actions.
• Each player plays a best response to their beliefs.

Regret Minimization

• Designing strategies to minimize the difference between a player's actual payoff and the alternative best option is best described at Regret Minimization.

Summary

• Algorithmic game theory merges game theory and computer science.
• It studies the computational complexity of games and the design of algorithms to solve them.
• The field has applications in economics, computer science and political science.

Derivatives

Definition and Notation

• The derivative of a function $f(x)$ is $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$.
• Derivatives can be represented through the notations, $f'(x)$, $y'$, $\frac{dy}{dx}$, $\frac{d}{dx}f(x)$, or $D f(x)$.

Basic Rules

• The derivative of a constant is zero: $\frac{d}{dx}(c) = 0$.
• Power Rule: $\frac{d}{dx}(x^n) = nx^{n-1}$.
• Constant Multiple Rule: $\frac{d}{dx}(cf(x)) = c f'(x)$.
• Sum/Difference Rule: $\frac{d}{dx}(f(x) \pm g(x)) = f'(x) \pm g'(x)$.

Product and Quotient Rules

Product Rule

• $\frac{d}{dx}(f(x)g(x)) = f'(x)g(x) + f(x)g'(x)$

Quotient Rule

• $\frac{d}{dx}(\frac{f(x)}{g(x)}) = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}$

Chain Rule

• $\frac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x)$

Derivatives of Trigonometric Functions

• Sine: $\frac{d}{dx}(sin(x)) = cos(x)$
• Cosine: $\frac{d}{dx}(cos(x)) = -sin(x)$
• Tangent: $\frac{d}{dx}(tan(x)) = sec^2(x)$
• Cosecant: $\frac{d}{dx}(csc(x)) = -csc(x)cot(x)$
• Secant: $\frac{d}{dx}(sec(x)) = sec(x)tan(x)$
• Cotangent: $\frac{d}{dx}(cot(x)) = -csc^2(x)$

Derivatives of Exponential and Logarithmic Functions

• Exponential: $\frac{d}{dx}(e^x) = e^x$
• Logarithmic: $\frac{d}{dx}(ln(x)) = \frac{1}{x}$
• Exponential (General Base): $\frac{d}{dx}(a^x) = a^x ln(a)$
• Logarithmic (General Base): $\frac{d}{dx}(log_a(x)) = \frac{1}{x ln(a)}$

MVE487 Multivariable Analysis V2024

Lecture 1

Vector algebra

• An element $v$ of $\mathbb{R}^n$ is a vector of length $n$.
• We write it as $v = (v_1,..., v_n)$, where $v_i \in \mathbb{R}$.
• $\mathbb{R}^n$ - row vector and $\mathbb{R}^{n x 1}$ - column vector represents notations.

Operations:

• $v + w = (v_1 + w_1,..., v_n + w_n)$
• $\lambda v = (\lambda v_1,..., \lambda v_n)$
• Where $v, w \in \mathbb{R}^n, \lambda \in \mathbb{R}$

Scalar product (dot product, inner product)

• $v \cdot w = \sum_{i=1}^{n} v_i w_i = v_1 w_1 +... + v_n w_n$

Properties:

• $v \cdot w = w \cdot v$
• $v \cdot (w + u) = v \cdot w + v \cdot u$
• $(\lambda v) \cdot w = \lambda (v \cdot w) = v \cdot (\lambda w)$
• $v \cdot v = \sum_{i=1}^{n} v_i^2 \geq 0$ and $v \cdot v = 0$ only if $v = 0$

Euclidean norm (length)

• $|v| = \sqrt{v \cdot v} = \sqrt{\sum_{i=1}^{n} v_i^2}$

Properties:

• $|v| \geq 0$ and $|v| = 0$ only if $v = 0$
• $|\lambda v| = |\lambda| |v|$
• $|v \cdot w| \leq |v| |w|$ (Cauchy-Schwarz inequality)
• $|v + w| \leq |v| + |w|$ (Triangle inequality)

Angle between two vectors:

• $v \cdot w = |v||w|cos(\theta)$, $v, w \neq 0$
• $cos(\theta) = \frac{v \cdot w}{|v||w|}$
• $\theta = arccos(\frac{v \cdot w}{|v||w|})$
• Two vectors $v, w$ are orthogonal if $v \cdot w = 0$ or one of them is zero.

Vector product (cross product)

• $v x w = (v_2 w_3 - v_3 w_2, v_3 w_1 - v_1 w_3, v_1 w_2 - v_2 w_1)$
• $v, w \in \mathbb{R}^3$

Properties:

• $v x w = - w x v$
• $v x (w + u) = v x w + v x u$
• $(\lambda v) x w = \lambda (v x w) = v x (\lambda w)$
• $(v x w) \cdot v = (v x w) \cdot w = 0$
• $|v x w| = |v||w|sin(\theta)$, where $\theta$ is the angle between $v$ and $w$
• $|v x w|$ is the area of the parallelogram spanned by $v$ and $w$

Mixed product (scalar triple product)

• $(u x v) \cdot w = det( \begin{bmatrix} u_1 & v_1 & w_1 \ u_2 & v_2 & w_2 \ u_3 & v_3 & w_3 \end{bmatrix} )$
• $u, v, w \in \mathbb{R}^3$

Properties:

• $(u x v) \cdot w = (v x w) \cdot u = (w x u) \cdot v$
• $|(u x v) \cdot w|$ is the volume of the parallelepiped spanned by $u, v, w$

Lines and planes

Lines:

• A line in $\mathbb{R}^n$ is given by $r(t) = a + t v$
• $a \in \mathbb{R}^n$ is a point on the line and $v \in \mathbb{R}^n$ is a direction vector, where $t \in \mathbb{R}$.

Planes:

• A plane in $\mathbb{R}^3$ is given by $n \cdot (r - a) = 0$
• Where $a \in \mathbb{R}^3$ is a point on the plane, $n \in \mathbb{R}^3$ is a normal vector and $r = (x, y, z) \in \mathbb{R}^3$.

More about functions

Mappings:

• $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ $x \mapsto f(x)$, where $x \in \mathbb{R}^n$ and $f(x) \in \mathbb{R}^m$
• Another name for mapping is transformation
• If $m = 1$, $f$ is called a real-valued function
• If $m > 1$, $f$ is called a vector-valued function
• If $n > 1$, $f$ is called a function of several variables

Examples:

• $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, $(x, y) \mapsto x^2 + y^2$
• $f: \mathbb{R}^3 \rightarrow \mathbb{R}^3$, $(x, y, z) \mapsto (x + y, y + z, x + z)$
• $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is equivalent to $m$ real-valued functions of $n$ variables: $f(x) = (f_1(x),..., f_m(x))$, where $f_i: \mathbb{R}^n \rightarrow \mathbb{R}$ for $i = 1,..., m$.

Statistiques Théoriques

Estimation Ponctuelle

Définition

• $X_1,..., X_n$ represents an i.i.d. sample of law $P_{\theta}$, where $\theta \in \Theta \subset \mathbb{R}^d$ is an unknown parameter.
• An estimator of $\theta$ is a statistic $T(X_1,..., X_n)$ that takes its values in $\Theta$.
• The quantity $\hat{\theta} = T(X_1,..., X_n)$ is defined

Biais

• The bias of an estimator $\hat{\theta}$ is defined by $biais(\hat{\theta}) = \mathbb{E}[\hat{\theta}] - \theta$
• $\hat{\theta}$ is unbiased if $biais(\hat{\theta}) = 0$, i.e., $\mathbb{E}[\hat{\theta}] = \theta$.

Risque Quadratique

• The quadratic risk of an estimator $\hat{\theta}$ is defined by: $R(\hat{\theta}) = \mathbb{E}[(\hat{\theta} - \theta)^2]$
• The following decomposition occurs: $R(\hat{\theta}) = Var(\hat{\theta}) + biais(\hat{\theta})^2$

Convergence

• $\hat{\theta}_n \xrightarrow{P} \theta$ if for all $\epsilon > 0$, $\mathbb{P}(|\hat{\theta}_n - \theta| > \epsilon) \xrightarrow{n \to \infty} 0$.
• $\hat{\theta}_n \xrightarrow{L^2} \theta$ if $\mathbb{E}[(\hat{\theta}_n - \theta)^2] \xrightarrow{n \to \infty} 0$.
• $\hat{\theta}n \xrightarrow{presque\ sûrement} \theta$ if $\mathbb{P}(\lim{n \to \infty} \hat{\theta}_n = \theta) = 1$.
• $\hat{\theta}_n \xrightarrow{D} \theta$ if $\mathbb{P}(\hat{\theta}_n \le x) \xrightarrow{n \to \infty} \mathbb{P}(\theta \le x)$ in any point $x$ where $\mathbb{P}(\theta \le x)$ is continuous.

Estimateur Consistent

• $\hat{\theta}$ is a consistent estimator of $\theta$ if $\hat{\theta}_n \xrightarrow{P} \theta$.

Estimateur Efficace

• Let $\hat{\theta}$ be an unbiased estimator of $\theta$. $\hat{\theta}$ is considered efficient if $Var(\hat{\theta})$ reaches the Cramér-Rao bound.

Information de Fisher

• Fisher information measures the amount of information that a random variable $X$ contains about the unknown parameter $\theta$ on which its distribution depends.
• Defined by: $I(\theta) = \mathbb{E}[(\frac{\partial}{\partial \theta} log f(X|\theta))^2] = - \mathbb{E}[\frac{\partial^2}{\partial \theta^2} log f(X|\theta)]$
• Where $f(X|\theta)$ is the probability density function of $X$ given $\theta$.

Borne de Cramér-Rao

• The Cramér-Rao bound provides a lower bound on the variance of any unbiased estimator of a parameter.
• It states that the variance of any unbiased estimator $\hat{\theta}$ of $\theta$ is at least as large as the inverse of the Fisher information: $Var(\hat{\theta}) \ge \frac{1}{I(\theta)}$

Estimateur du Maximum de Vraisemblance (EMV)

• The Maximum Likelihood Estimator (MLE) of $\theta$ is the value of $\theta$ that maximizes the likelihood function:
• In other words, it's the value of $\theta$ that makes the observed data most probable.
• The likelihood function is defined as the product of the probability density functions of the observations: $L(\theta) = \prod_{i=1}^n f(X_i|\theta)$
• The MLE of $\theta$ is then given by: $\hat{\theta}{EMV} = arg \max{\theta \in \Theta} L(\theta)$
• In practice, it is often easier to maximize the log of the likelihood function: $\hat{\theta}{EMV} = arg \max{\theta \in \Theta} log L(\theta) = arg \max_{\theta \in \Theta} \sum_{i=1}^n log f(X_i|\theta)$