Algorithmic Trading Explained

• Algorithmic trading uses computer programs to automate trading decisions.
• It's also called automated trading, black-box trading, or algo-trading.

How Algorithmic Trading Works

• Define a set of rules for buying and selling.
• Translate these rules into a computer program.
• Backtest the algorithm using historical data to evaluate its performance.
• Deploy the algorithm to a live trading account.
• Monitor the algorithm to ensure it functions as expected.

Advantages of Algorithmic Trading

• Algorithms execute trades faster than humans.
• Algorithms adhere to predefined rules without emotional influence.
• Backtesting allows performance evaluation on historical data.
• Algorithms can oversee multiple markets simultaneously.

Disadvantages of Algorithmic Trading

• Algorithms can be complex and hard to understand.
• Overfitting to historical data leads to poor future performance
• Technical issues can disrupt algorithms.
• Continuous monitoring is needed to ensure algorithms work properly.

Popular Algorithmic Trading Strategies

• Trend Following: Buying during price increases, selling during price decreases.
• Mean Reversion: Buying when prices are low, selling when prices are high.
• Arbitrage: Profiting from price differences by simultaneously buying and selling assets in different markets.
• Market Making: Providing market liquidity through buy and sell orders.

Example: Simple Moving Average (SMA) Crossover

• Calculate SMA:
  • Formula: $$SMA = \frac{Sum : of : prices : over : a : period}{Number : of : periods}$$
• Generate Signals:
  • Buy Signal: When the short-term SMA crosses above the long-term SMA.
  • Sell Signal: When the short-term SMA crosses below the long-term SMA.
• Execute Trades: Automatically execute buy and sell orders per the signals.
• Monitor: Continuously monitor strategy performance and adjust parameters.

Resources for Algorithmic Trading

• Books: "Algorithmic Trading: Winning Strategies and Their Rationale" by Ernest P. Chan
• Online Courses: Platforms like Coursera, Udacity, and Quantopian.
• Software: e.g. MetaTrader 4, Python (with libraries like Pandas, NumPy, and Matplotlib)

Algorithmic Trading Conclusion

• Algorithmic trading can be a powerful tool
• It is important to understand risks and to use it responsibly

Algorithmic Complexity: Definition and Measurement

• Algorithmic complexity measures the time (time complexity) or space (space complexity) an algorithm requires as input size grows.
• It uses Big O notation, focusing on the dominant term while ignoring constants and lower-order terms.

Importance of Algorithmic Complexity

• Performance: Predicts algorithm performance with large datasets.
• Scalability: Determines if an algorithm can manage increasing data volumes.
• Optimization: Guides selection of efficient algorithms for specific tasks.

Common Algorithmic Complexities

• O(1): Constant time; Example: Accessing an array element.
• O(log n): Logarithmic time; Example: Binary search.
• O(n): Linear time; Example: Simple search.
• O(n log n): Linearithmic time; Example: Merge sort.
• O(n^2): Quadratic time; Example: Bubble sort.
• O(2^n): Exponential time; Example: Recursive Fibonacci.
• O(n!): Factorial time; Example: Permutation generation.

Visual Explanation of Algorithmic Complexities

• Compares how different algorithmic complexities will scale with the number of elements
• O(1) constant
• O(log n) logarithmic
• O(n) linear
• O(n log n) linerithmic
• O(n^2) quadratic
• O(2^n) exponential

Key Takeaways on Algorithmic Complexity

• Big O Notation describes the upper bound of an algorithm’s complexity.
• Big O usually indicates the worst-case scenario.
• Understanding complexity is very helpful in creating efficient and scalable code.

Definition of Complexity

• Algorithmic complexity measures resources (time, space, etc.) an algorithm needs to solve a problem.

Time Complexity

• Time complexity measures the time an algorithm requires to solve a problem.

Common Time Complexities

• $O(1)$ Constant: Same time regardless of input size.
• $O(log n)$ Logarithmic: Time increases logarithmically with input size.
• $O(n)$ Linear: Time increases linearly with input size.
• $O(n log n)$ Linearithmic: Time increases linearly and logarithmically with input size.

Big O Notation Explained

• A mathematical notation describes function behavior as the argument tends toward a value or infinity.
• In computer science, it classifies algorithms based on running time or space needs relative to input size.
• Examples:
  • $O(3n^2 + n + 5) = O(n^2)$

Space Complexity

• Space complexity measures memory space an algorithm needs to solve a problem.

Common Space Complexities

• $O(1)$ Constant: Algorithm uses the same memory regardless of input size
• $O(n)$ Linear: Algorithms memory usage increases linearly with input size

def example(n):
    for i in range(n):
        for j in range(n):
            print(i, j)

• Time Complexity: $O(n^2)$
• Space Complexity: $O(1)$

Chemical Kinetics Definition

• Chemical kinetics examine chemical reaction rates as an area of study.

Rate of Chemical Reactions

• The rate of reaction measures change in reactant or product concentration per time unit.
  • $A \rightarrow B$
  • $Rate = -\frac{\Delta[A]}{\Delta t} = \frac{\Delta[B]}{\Delta t}$

Instantaneous Rate in Reactions

• Measures the rate at a specific moment.
• $instantaneous : rate = \lim_{\Delta t \rightarrow 0} -\frac{\Delta[A]}{\Delta t} = \frac{d[A]}{dt}$

Rate Laws

• Rate law links reaction rate, rate constant, and reactant concentrations raised to powers.
  • $aA + bB \rightarrow cC + dD$
  • $$Rate = k[A]^x[B]^y$$
• $[A]$ and $[B]$ are the concentrations of reactants
• $x$ and $y$ are reaction orders
• $k$ is the rate constant

Reaction Order

• Exponent indicates the order of reaction per reactant.
• The sum of exponents gives the overall reaction order.
  • First-order reaction: Rate = $k[A]$
  • Second-order reaction: Rate = $k[A]^2$ or Rate = $k[A][B]$
  • Zero-order reaction: Rate = $k$

Determining Rate Laws

• Rate law exponents come from experiments.

First-Order Reactions: Rate and Integrated Rate Laws

• $A \rightarrow B$
• Rate Law: $$Rate = -\frac{d[A]}{dt} = k[A]$$
• Integrated Rate Law: $$\ln[A]_t - \ln[A]_0 = -kt$$
• Rearranging: $$\ln[A]_t = -kt + \ln[A]_0$$
• Plotting ln[A]t versus t yields a straight line with slope -k and intercept ln[A]0

Reaction Half-Life

• Half-life is the time needed to halve reactant concentration.
• $$t_{1/2} = \frac{ln 2}{k} = \frac{0.693}{k}$$

Second-Order Reactions Rate and Integrated Rate Laws

• Rate Law: $$\text{Rate} = -\frac{d[A]}{dt} = k[A]^2$$
• Integrated Rate Law: $$\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$$
• Plotting 1/[A]t versus t yields a line with slope k and intercept 1/[A]0.

Second-Order Reactions: Half-Life

• Second-Order Half-Life is calculated as:
  • $$t_{1/2} = \frac{1}{k[A]_0}$$

Zero-Order Reactions

$$A \rightarrow B$$

• Rate Law: $$Rate = -\frac{d[A]}{dt} = k$$
• Integrated Rate Law: $$[A]_t = -kt {+ [A]_0}$$
• Plotting [A]t versus t yields a line with slope -k and intercept [A]0.

Zero-Order Reactions: Half-Life

• Half-Life calculation: $$t_{1/2} = \frac{[A]_0}{2k}$$

Collision Theory of Chemical Kinetics

• Reactant molecules must collide to react.
• Collisions are proportional to the product of reactant concentrations.
• Molecules must collide with proper orientation and energy for bond changes.

Activation Energy

• The minimum energy for a reaction ($E_a$)
• The activated complex (transition state) is the atomic arrangement at activation energy peak.

The Arrhenius Equation

$$k = A e^{-E_a/RT}$$

• k = rate constant
• $E_a$ = activation energy
• R = gas constant (8.314 J/mol·K)
• T = absolute temperature
• A = frequency factor.

Arrhenius Equation and Graphs

• $$\ln k = -\frac{E_a}{RT} + \ln A$$
  • Plot $lnk$ versus $\frac{1}{T}$ for slope $-\frac{E_a}{R}$ and intercept $lnA$

The Equation of the Rate Constant

• Given rate constants at 2 temps:
  • $$\ln \frac{k_1}{k_2} = \frac{E_a}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right)$$

Reaction Mechanisms

• Step series showing how chemical reaction occurs.
• Elementary Step: Each is a single step in the reaction mechanism.
• Molecularity of reaction is molecule count reacting in an elementary step.
  • Unimolecular: one molecule,
  • Bimolecular: two molecules
  • Termolecular: three molecules.
• The rate-determining step is the slowest.
• An intermediate forms in the mechanism but not in balanced equation.

Catalysis

• Catalysts speed reactions without being consumed.
• They provide another reaction pathway with lower activation energy.
  • Homogeneous catalyst: in the same phase as reactants.
  • Heterogeneous catalyst: is in a different phase.

Congestion Games: Definition

• A "tuple" is (N, R, (Si)i ∈ N, (cr)r ∈ R)
  • n = {1,..., n} = Player set
  • R = {1,..., m} = Resource set
  • Si 2R = Strategy set of player i
  • cr : ℕ → ℝ = Cost function of resource r
    • cr(k) = cost for each player when k players utilize resource r

Congestion Games: Definitions 2-5

• Strategy profile: Vector s = (s1,..., sn) where si Si is strategy of player i.
• For profile s, nr(s) = |{i ∈ N | r ∈ si}| = # of players using resource r.
• Player i's cost in profile s = Ci(s) = ∑r ∈ si cr(nr(s)).
• Strategy profile s social cost = C(s) = ∑i ∈ N Ci(s) = ∑r ∈ R nr(s) · cr(nr(s)).

Congestion Game Example

• $N = {1, 2}$, Two Players
• $R = {a, b, c, d}$, Four Resources
• Strategy: $S_1 = S_2 = {{a, b}, {a, c}, {d}}$
• Cost function:
  • Resource Cost Function
  • a ca(k) = k
  • b cb(k) = 2k
  • c cc(k) = 3
  • d = cd(k) = 1

Congestion Game Profile Example

• Profile s = ({a, b}, {a, c}): na = 2, nb = 1, nc = 1, nd = 0
• C1(s) = ca(na(s)) + cb(nb(s)) = 2 + 2 = 4
• C2(s) = ca(na(s)) + cc(nc(s)) = 2 + 3 = 5
• C(s) = C1(s) + C2(s) = 4 + 5 = 9
• Alternatively, C(s) = na · ca(na) + nb · cb(nb) + nc · cc(nc) + nd · cd(nd) = 2 · 2 + 1 · 2 + 1 · 3 + 0 · 1 = 9.

Definition of Nash Equilibria

• A strategy profile s is a NE if no player can lessen its cost by one deviation.
  • Ci(si, s−i) ≤ Ci(si', s−i) to another strategy, i.e., for all i ∈ N and for all si' ∈ Si
    • si and s-i notations
• Theorem: Every congestion game admits a pure Nash equilibrium

Understanding Rosenthal's Potential Function

• Φ(s) = ∑r ∈ R∑k=1 cr(k)
• Player i’s unilateral deviation from si to si’:
• Φ(si', s−i) − Φ(si, s−i) = ∑r ∈ si' ∑si cr(nr(s) + 1) − ∑r ∈ si ∑si' cr(nr(s))
• Ci(si', s−i) − Ci(si, s−i) = ∑r ∈ si' ∑si cr(nr(s) + 1) − ∑r ∈ si ∑si' cr(nr(s))
• Concluding Statements
  • Hence, Φ(si', s−i) − Φ(si, s−i) = Ci(si', s−i) − Ci(si, s−i)
  • Player improves deviation from si to si' i.e C-functions
• Since Φ takes few values, a pure Nash equilibrium is reached on finitely many improving steps.

Braess’s Paradox

• Using directed graph with 4 nodes and 5 edges
  • A -> B = Cost 1
  • A -> C = Cost x
  • B -> D = Cost Value x
  • C -> D = Cost 1
  • B -> C = Cost value of 0

Studying Equilibrium and Social Cost

• Cost for each player is (1+k)/n + 1 where k = edge A->B users
• In NE, equal distribution: half using paths. Cost ~ 3/2 + 1/n
• Social Cost is n(3/2 + 1/n) approximately equal to 3/2n + 1
• Equation*
• Social Cost is n(2 + 2/n) approximately equal to 2n + 2
• Equation
• Social Cost ratio approaches 4/3 for many n's

Price of Anarchy (PoA)

• Anarchy measures ratio between worst NE social cost & optimal outcome
• PoA equation
• max(s ∈ NE)C(s) / min(s ∈ ⊆S) C(s)
• Stability, the ratio between the best NE social cost & optimal outcome.
• Equation for Stability
• max(s ∈ NE)C(s) / min(s ∈ ⊆S) C(s)
• Theorem: Congestion game with affine costs (cr(x) = ar · x + br for ar, br ≥ 0), PoA ≤ 5/2.

Fonction Exponentielle: Definition

• Exp derivability on ℝ with: $\exp' = \exp$ and $exp(0) = 1$
• For any real x, exp(x) is ex by notation.

More on the Exponential Function

• Relations
  • $𝑒^{𝑥+𝑦} = e^x * e^y$
  • $𝑒^{𝑥−𝑦} = e^x / e^y$
  • 𝑒^−𝑥 = 1/𝑒^𝑥
  • (ex)y = exy
• Asymptotes
  • $lim(𝑥→∞)e^x=+∞$"
  • $lim(𝑥→−∞)e^x=0$
• Always above 0. $ex>0$
• Strict Increase on real numbers

Properties of the Function

• Derivative relation for u(x) = ef(x) is u'(x) = f'(x)ef(x) and example.
• Limits
  • As $𝑥→+∞ = 𝑒^𝑥 / 𝑥 =+∞$
  • As $x→−∞, 𝑥 ^ex = 0.$"
  • If $x->0, $ 𝑒𝑥−1/𝑥 = 1
• Compared Crescence
• Limit $L when x -> inf = ex/ x^n = inf"
• Limit $L when x -> -inf = XpowerN ex

Equations and inequations

• When a, b are real:
  • $ex power a = ex power b <=> a =b"
  • $ ex power a < ex power b

Graph of $e^x$ and $e^{-x}$

• Properties of $e^x$
  • the exp graph remains above the x at all y
  • Properties of symmetry around y
    • if f(x) = ex then its Y - symetrical line = y = e^-x

Function Lab

• Introduction: Explain the purpose of lab as measuring motion study pendulum relation to length
• Theory: The formula given relates the period to its Length under the gravitational constant
  • T = 2𝜋Root of L/G "
• Stte objective to quantify T as relation within length, also determine value of G

Materials and Procedures in the Lab

• Discuss required materials in experiment
• Set up and measure process for getting length and period value
• Calculate total time divided by oscillation rate for oscillation

Data Analysis in the Lab

• Set up table relating length vs oscillation T and T2 calculation
• Draw relation for L to T2 for slope calculation
• Use data and value from chart calculation: g = 4𝜋 ^ slope
• Value compare with accepted value

Questions on the Experiment

• Relate the variation in term and the other elements like angle
• Discuss possible variation and how improve accuracy of the process

Lab Report Structure

• Write report which has these:
  • All from above + compare, state compare result then what all variation occured in data. and what can do so to better estimate it.

Algorithmic Trading: Definition and Parameters

• Computers then investment decisions based on instructions
• Good to fill at the best price as the removes all possible element if Emotions

Algo trade vs high

Algo Trade Strategies

• Trend: the value will contune high or low as perform in past
  • Upper resitance when to higher
• Exploits the little difference of asset then markets
  • cheap in one then in other higher

Most used Algo Strategies

• Assumes the price will back return and follow that it
  • when cheap is fall lower cost then average get good one.
• Btreaks up a large and give all order over sepied time.
  • price get closer get average
• a more get volume trade on dif times day and get
  • when volume then low high the algo give that
• To get pattern and learn from the past price moments
  • learn from other time and get adaptive moments

Algo platforms

• BArubs : cloud then support
• Meta Traer : stocks and good
• Trading Net: social
• Interactive broerser wide with api

This is no long list the depend what new for erson