Industrial revolution

Algorithmic Trading

• Method to execute orders using automated, pre-programmed trading instructions.
• Considers price, timing, and volume variables.

Algorithmic Trading Users

• Retail Traders: Individual investors using platforms to automate strategies.
• Hedge Funds: Employ complex algorithms and HFT for short-term profit.
• Institutional Traders: Financial institutions utilizing algorithms for large order execution.

Algorithmic Trading Pros

• Faster and more efficient than humans for market opportunities.
• Minimizes emotional bias, promoting rational decisions.
• Strategies can be backtested on historical data.
• Operates 24/7 for opportunities in global markets.

Algorithmic Trading Cons

• Requires programming skills and financial market knowledge.
• Vulnerable to technical issues and data errors.
• Over-optimization can lead to poor live trading performance.
• Subject to regulatory scrutiny, especially market manipulation and HFT.

Algorithmic Trading Future

• Increased AI and machine learning for adaptive strategies.
• Scalability and cost-effectiveness with cloud computing.
• Analysis of vast data for improved trading performance.
• Application to decentralized exchanges and DeFi platforms.

Vector Functions

• Function domain = subset of real numbers; range = set of vectors.
• Assigns a vector to each real number in its domain.

Vector Representation

• Parametric form: Vector components are functions of a parameter $t$; example $\vec{r}(t) = (x(t), y(t))$.
• Component form: Emphasizes each component as a separate function; example $\vec{r}(t) = x(t)\hat{\imath} + y(t)\hat{\jmath} + z(t)\hat{k}$.

Vector Function Calculus

• Performed component by component.

Limit of Vectors

• The limit of $\vec{r}(t)$ as $t$ approaches $a$ is a vector of the limits of its components, if those limits exist.
• Formula: $\lim_{t \to a} \vec{r}(t) = \left( \lim_{t \to a} x(t), \lim_{t \to a} y(t), \lim_{t \to a} z(t) \right)$

Derivative of Vectors

• The derivative of $\vec{r}(t)$ is found by deriving each component with respect to $t$.
• Formula: $\frac{d\vec{r}}{dt} = \left( \frac{dx}{dt}, \frac{dy}{dt}, \frac{dz}{dt} \right)$
• $\frac{d\vec{r}}{dt}$ represents a vector tangent to the curve described by $\vec{r}(t)$.

Integral of Vectors

• The integral of $\vec{r}(t)$ is calculating by integrating each component with respect to $t$.
• Formula: $\int \vec{r}(t) dt = \left( \int x(t) dt, \int y(t) dt, \int z(t) dt \right)$

Vector Function Applications

• Used for many things including Physics and Engineering
• Cinemetics: Describes object movement in space.
• Position, velocity, and acceleration as vector functions of time.
• Curves and surfaces: Parameterizes curves and surfaces, facilitates calculations of geometric properties.
• Vector fields: Represents vector fields, such as fluid velocity or electromagnetic fields.

Vector Example

• Given $\vec{r}(t) = (t^2, \sin(t), e^t)$, the derivative is $\frac{d\vec{r}}{dt} = (2t, \cos(t), e^t)$.
• This derivative shows the tangent vector to the curve described by $\vec{r}(t)$.

Bernoulli's Principle

• States that fluid speed increase occurs with pressure or potential energy decrease.

How Wings Generate Lift in Airplanes

• Wings are shaped to move air faster over the top of the wing.
• Faster air exerts less pressure.
• Top-of-wing pressure < bottom-of-wing pressure.
• Pressure difference creates lifting force.

Chemical Bonds

• An attraction between atoms.
• Enables formation of chemical substances with 2+ atoms.
• Different types exist with all bonds arising from the electromagnetic force between atomic nuclei and electrons.

Covalent Bond

• Involves electron pairs shared between atoms.
• Shared pairs (bonding pairs).
• Forms through stable balance of attractive and repulsive forces when atoms share electrons.

Ionic Bond

• Involves electrostatic attraction between oppositely charged ions.
• Ions are atoms that gained/lost valence electrons to fill valence shell.

Metallic Bond

• Electrostatic attraction between conduction electrons and positively charged metal ions.
• Sea of electrons shared between lattice of cations.
• Occurs only with metals.

Bond Length

• Average distance between nuclei of two bonded atoms
• Inversely proportional to bond order
• Inversely related to bond strength and dissociation energy.
• Other things being equal, a stronger bond will be shorter.

Bond Angles

• Determined by the repulsion between electron pairs
• Angle formed between three atoms across at least two bonds.
• Torsion angle: For a chain of four bonded atoms, it's the angle between the plane formed by the first 3 atoms and the plane formed by the last 3 atoms.

Point Estimation

• Statistic from a single value that estimates an unknown population parameter.

Estimator Bias

• $W$ point estimator of parameter $\theta$ is unbiased if $E(W) = \theta$.
• Bias for non-unbiased $W$ is $Bias(W) = E(W) - \theta$.

Estimator Efficiency

• For unbiased estimators $W_1$ and $W_2$ of parameter $\theta$:
• $W_1$ is more efficient than $W_2$ if $V(W_1) < V(W_2)$.

Theorem 9.1

• Given random sample $X_1, X_2,..., X_n$ from population:
  • Sample mean, $\bar{X}$ is unbiased estimator of $\mu$.
  • Statistic, $S^2 = \frac{1}{n-1}\sum_{i=1}^{n}(X_i - \bar{X})^2$ is unbiased estimator of $\sigma^2$.

Mean Squared Error

• $MSE(W)$ of estimator $W$ of parameter $\theta$ is:
  • $MSE(W) = E[(W - \theta)^2]$.

Theorem 9.2

• For any estimator $W$ of a parameter $\theta$:
  • $MSE(W) = V(W) + [Bias(W)]^2$

Consistent Estimator

• If $W_n$ estimates parameter $\theta$ based on sample size $n$:
  • $W_n$ is consistent if $W_n$ converges in probability to $\theta$ as $n \rightarrow \infty$.
  • Denoted as $W_n \xrightarrow{P} \theta$.