Simple Harmonic Motion (SHM)

14.1 Simple Harmonic Motion

• Oscillation or Vibration refers to a motion that repeats itself.
• Simple Harmonic Motion (SHM) is a specific type of periodic motion.
• For SHM, restoring force needs to be directly proportional to the displacement from the equilibrium position.
• In SHM, the restoring force has to act towards the equilibrium position.
• $F = -kx$ mathematically defines SHM, where $F$ is the restoring force, $k$ is the spring constant, and $x$ is the displacement from equilibrium.
• The minus sign in the formula for restoring force indicates it opposes the displacement.
• Amplitude (A) stands for the maximum displacement from equilibrium.
• Cycle defines one complete oscillation.
• Period (T) is the duration for one cycle, measured in seconds.
• Frequency (f) measures the number of cycles per second, in Hz.
• Frequency and period have an inverse relationship : $f = \frac{1}{T}$ and $T = \frac{1}{f}$
• SHM can be visualized as a single component of circular motion.
• Displacement at time $t$ can be defined as $x = Acos(\omega t)$, where $x$ is displacement, $A$ is amplitude, and $\omega$ is angular frequency.
• Angular frequency relates to frequency/period by $\omega = 2\pi f = \frac{2\pi}{T}$.
• Velocity, maximum speed, and acceleration in SHM are represented by the formulas $v = -A\omega sin(\omega t)$, $v_{max} = A\omega$, and $a = -A\omega^2 cos(\omega t)$ or $a = -\omega^2 x$, respectively.
• Maximum acceleration in SHM is given by $a_{max} = A\omega^2$
• Applying Newton's Second Law:
  • $F = ma$
  • $-kx = ma$
  • $a = -\frac{k}{m}x$
  • Since $a = -\omega^2 x$
  • $\omega = \sqrt{\frac{k}{m}}$
• Therefore, the period and frequency are:
  • $T = 2\pi \sqrt{\frac{m}{k}}$
  • $f = \frac{1}{2\pi} \sqrt{\frac{k}{m}}$

14.2 Applications of SHM

• When a spring hangs vertically, gravity stretches it to a new equilibrium, but oscillations about this point are still SHM.
• For small angles (less than 10 degrees), the period of a simple pendulum is $T = 2\pi \sqrt{\frac{L}{g}}$, where $L$ is the length of the pendulum, and $g$ is the acceleration due to gravity.
• The period of a pendulum is independent of the mass.
• For physical pendulums, the period is derived using the formula $T = 2\pi \sqrt{\frac{I}{mgd}}$, where I is the moment of inertia about the pivot, m is mass, g is gravity, and d is the distance from the pivot to the center of mass.

14.3 Energy in Simple Harmonic Motion

• The potential energy of a spring is $U = \frac{1}{2}kx^2$
• Potential energy in SHM varies with time described as $U = \frac{1}{2}kA^2cos^2(\omega t)$.
• Kinetic energy is $K = \frac{1}{2}mv^2$.
• Kinetic energy in SHM varies with time described as $K = \frac{1}{2}mA^2\omega^2sin^2(\omega t)$.
• Total mechanical energy $E = K + U$ is a constant: $E = \frac{1}{2}kA^2$

14.4 Damped Oscillations

• Damping involves the loss of energy in an oscillating system due to friction or resistive forces.
• Underdamping occurs when the system oscillates with a decreasing amplitude.
• Critical damping is where the system returns to equilibrium as quickly as possible without oscillating.
• Overdamping occurs when the system returns to equilibrium slowly without oscillating.

14.5 Forced Oscillations and Resonance

• Forced oscillation involves an external force driving an oscillator.
• Resonance occurs when the driving frequency nears the oscillator's natural frequency, amplifying the amplitude.
• Natural frequency equation is: $f_0 = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$

Physics

• A positively charged particle ($+q$) with velocity $\overrightarrow{\mathbf{v}}$ moves perpendicularly to a uniform blue-cross magnetic field ($\overrightarrow{\mathbf{B}}$).
• Using the right-hand rule: fingers point in $\overrightarrow{\mathbf{v}}$ direction, curl to $\overrightarrow{\mathbf{B}}$, and the thumb points in the magnetic force direction ($\overrightarrow{\mathbf{F}}_B$).
• Due to magnetic force ($\overrightarrow{\mathbf{F}}_B$), charged particle moves in circle.
• $\overrightarrow{\mathbf{F}}_B$ directed toward circle's center.
• Relevant Equations:
  • Magnetic Force: $F_B = qvB$
  • Centripetal Force: $F_c = \frac{mv^2}{r}$
  • Equating: $qvB = \frac{mv^2}{r}$
  • Radius: $r = \frac{mv}{qB}$
  • Cyclotron Frequency: $f = \frac{v}{2\pi r} = \frac{qB}{2\pi m}$
  • Angular Speed: $\omega = 2\pi f = \frac{qB}{m}$
  • Where each variables stands for: charge ($q$), velocity ($v$), strength of B ($B$), mass ($m$), radius ($r$), cyclotron frequency ($f$), angular speed ($\omega$).

Algèbre Linéaire et Géométrie Vectorielle

• Linear algebra: branch of mathematics for vector spaces and linear transformations.
• Used in physics, computer science, economics, and engineering.

Vecteurs dans $\mathbb{R}^n$

• Vector in $\mathbb{R}^n$: ordered list of $n$ real numbers.
• Represented as $\mathbf{v} = \begin{pmatrix} v_1 \ v_2 \ \vdots \ v_n \end{pmatrix}$, where $v_1, v_2, \dots, v_n$ are the components.

Opérations sur les vecteurs

• Addition of vectors $\mathbf{u} + \mathbf{v} = \begin{pmatrix} u_1 + v_1 \ u_2 + v_2 \ \vdots \ u_n + v_n \end{pmatrix}$ is defined by the component-wise addition.
• Scalar multiplication $c\mathbf{v} = \begin{pmatrix} cv_1 \ cv_2 \ \vdots \ cv_n \end{pmatrix}$: multiplies each components of $\mathbf{v}$ by $c$.

Propriétés des opérations

• Commutativity: $\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$
• Associativity: $(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})$
• Neutral element for addition: $\mathbf{v} + \mathbf{0} = \mathbf{v}$
• Inverse element for addition: $\mathbf{v} + (-\mathbf{v}) = \mathbf{0}$
• Scalar distributivity: $c(\mathbf{u} + \mathbf{v}) = c\mathbf{u} + c\mathbf{v}$
• Vector distributivity: $(c + d)\mathbf{v} = c\mathbf{v} + d\mathbf{v}$
• Compatibilité : $c(d\mathbf{v}) = (cd)\mathbf{v}$
• Neutral element for scalar multiplication : $1\mathbf{v} = \mathbf{v}$

Combinaisons linéaires

• Linear combination of vectors: $\mathbf{c_1v_1 + c_2v_2 + \dots + c_kv_k}$
• This result is with scalars $c_1, c_2, \dots, c_k$.

Algorithmic Trading and Order Execution

Part 1

• Average execution price buying 400 shares with ask volume at $100.02 (100 shares) and $100.01 (300 shares) is $100.0125.
• Formula $(100 \times 100.02)+(300 \times 100.01)/400 = 100.0125$
• Buying $10,000,000 worth of stock with $1,000,000 of capital requires borrowing $9,000,000 through margin.
• Your Leverage Ratio = Total value of Assets / Capital = 10,000,000 / 1,000,000 = 10, leverage ratio is 10.
• VWAP formula:
• $$VWAP = \frac{\sum_{i=1}^{n} P_i \times V_i}{\sum_{i=1}^{n} V_i}$$
  • Where:
• $P_i$ is the price of the trade $i$,
• $V_i$ is the volume of the trade $i$,
• $n$ is the total number of trades.
• Advantages of VWAP as a benchmark:
  • Simplicity
  • Incorporates Volume
  • Widely Used
• Disadvantages of VWAP as a benchmark:
  • Susceptible to Manipulation
  • Doesn't Account for Opportunity Cost, doesn't Provide Insights into Future Price Movements, nor Reflect Market Impact

Part 2

• Problems with a trading strategy based on 5-day and 20-day moving averages: Transaction costs, Whipsaws, and Market impact
• Market making is when you provide market liquidity by quoting bid and ask prices.
• Profit comes from the spread between the bid and ask prices
• Main risks faced by market makers include:
  • Adverse Selection, Inventory Risk, and Volatility Risk
• Advantages of using dark pools:
  • Reduced Market Impact, Price Improvement, and Anonymity.
• Disadvantages of using dark pools:
  • Lack of Transparency, Order Exposure, and Potential for Manipulation

Física

• The average acceleration is the change rate of vector-valued velocity.
• You can determine this with the formula: $\overline{a} = \frac{\Delta v}{\Delta t} = \frac{v_f - v_i}{t_f - t_i}$
• Acceleration is measured in m/s².
• The instantaneous acceleration is the limit when the time interval tends to zero: $a = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t}$
• The slope of the velocity-time graph represents acceleration at that time.
• In Uniformly Varied Motion (MUV), acceleration is nonzero and constant ($a = cte \neq 0$).

MUV Equations:

• $v = v_0 + a \cdot t$
• $s = s_0 + v_0 \cdot t + \frac{1}{2} a t^2$
• $v^2 = v_0^2 + 2 \cdot a \cdot \Delta s$
• Acceleration-time graph for MUV: horizontal line
• Velocity-time graph for MUV: inclined line
• Position-time graph for MUV: parabola
• A car starts from rest with a constant acceleration of 2 m/s². After 5 seconds, the car speed is $v = v_0 + a \cdot t = 0 + 2 \cdot 5 = 10 m/s$

Lecture 26: The Simplex Method

• Linear Programming (LP) technique optimizes linear objective function.
  • LP is subject to linear equality and inequality constraints.
  • These constraints make a feasible region, a convex polyhedron.
  • Objective function: linear function of decision variables.
  • Goal is to find best point in the feasible region.
• Standard Form:

$$ \begin{array}{ll} \text{maximize} & c^T x \ \text{subject to} & Ax = b \ & x \geq 0 \end{array} $$

• Basic Feasible Solutions (BFS)
  • $n-m$ variables are set to zero.
  • Remaining $m$ variables are solved from the constraints.
  • $m$= basic variables, and $n-m$ non-basic variables.

The Simplex Method

• Iterative algorithm for solving linear programs.
• Steps:
  • LP to Standard Form
  • Find BFS
  • Iterate
  • Optimal?

Iterate:

• Choose non-basic variable by pricing strategy
• Choose basic via ratio test
• Update BFS by pivoting

Choose:

• Variable chosen based on reduced cost, with most positive to enter

Pivoting:

• Solve for variables
• Substituting

Algorithmic Trading

• Algorithmic trading (AT)is a method of executing orders through instructions, that use the following:
  • Price
  • Timing
  • Volume
• Reasons to use Algorithmic Trading (AT):
  • Reduce transaction costs
  • Improved order exectution
  • Access otherwise inaccessible markets

Types of AT Strategies

• Trend Following Strategies:
• Mean Reversion Strategies:
• Arbitrage:
• Market Making:
• Statistical Arbitrage:
• Time Weighted Average Price (TWAP):
• Volume Weighted Average Price (VWAP):

What to Consider

• Backtesting
• Risk Managment
• Execution Platform

How to Build a Algorithmic Trading System

1. Define Strategy
2. Data Collection
3. Backtesting
4. Paper Trading
5. Automation
6. Monitoring
7. Optimisation

Common Mistakes to avoid

• Overcomplicating Strategies
• Ignoring Transaction Costs
• Insufficient Backtesting
• Lack of Risk Management
• Neglecting Market Dynamics