Algorithmic Trading and VWAP

Algorithmic Trading

• Known as automated trading, black-box trading, or systematic trading.
• The process uses a computer program to automatically execute trades.
• Trading is based on a predefined set of instructions.

Applications of Algo Trading

• Order Execution minimizes market impact and transaction costs by splitting large orders.
• Arbitrage exploits price differences in different markets or exchanges.
• Trend Following capitalizes on identified market trends
• Market Making provides liquidity by placing buy and sell orders.

Order Execution Algorithms

Volume-Weighted Average Price (VWAP)

• VWAP executes orders to match the volume-weighted average price of a security over a specified period.
• VWAP's suitability lies in large orders and liquid securities.
• $VWAP = \frac{\sum_{i=1}^{n} P_i \times V_i}{\sum_{i=1}^{n} V_i}$
• $P_i$ = Price of the $i^{th}$ trade
• $V_i$ = Volume of the $i^{th}$ trade
• $n$ = Number of trades during the period

Time-Weighted Average Price (TWAP)

• TWAP executes orders to match the time-weighted average price of a security over a specified period.
• TWAP is similar to VWAP but assigns equal weight to each time interval.
• $TWAP = \frac{\sum_{i=1}^{n} P_i}{n}$
• $P_i$ = Price at time $i$
• $n$ = Number of time intervals

Percentage of Volume (POV)

• POV participates by trading a fixed percentage of the market volume.
• POV aims to blend in with market activity to minimize market impact

Implementation Shortfall

• Aims to minimize the difference between the actual execution price and the decision price.
• Considers both market impact and opportunity cost.
• $\text{Implementation Shortfall} = (\text{Actual Execution Price} - \text{Decision Price}) \times \text{Number of Shares}$

Advantages of Algo Trading

• Reduced transaction costs
• Improved execution speed
• Increased order accuracy
• Reduced market impact
• Ability to trade in multiple markets simultaneously
• 24/7 trading

Risks of Algo Trading

• Technical glitches
• Model risk
• Over-optimization
• Market manipulation
• Regulatory scrutiny

High-Frequency Trading (HFT)

• A subset of algorithmic trading with high speed, high turnover, and high order-to-trade ratios.
• Used for market making, arbitrage, and short-term price discrepancies.
• Relies on co-location, direct market access, and proprietary algorithms.

Concerns about HFT

• Potential for market manipulation exists
• Offers unfair advantage over other market participants
• Contribution to market volatility

Algorithmic Trading Regulations

• Regulatory bodies, such as the SEC and FINRA, have implemented rules to address the risks.
• Regulation focuses on:
  • Algorithm testing and certification
  • Orderly market making
  • Prevention of market manipulation
  • Risk management controls

Open Loop Instability Example 1

• $P(s) = \frac{1}{s-1}$
• The Closed loop T.F. is $\frac{P}{1+kP} = \frac{1}{s-1+k}$
• The system is stable for $k>1$

Open Loop Instability Example 2

• $P(s) = \frac{s+1}{(s-1)(s+6)}$
• $\frac{P}{1+kP} = \frac{s+1}{(s-1)(s+6) + k(s+1)} = \frac{s+1}{s^2 + 5s - 6 + ks + k} = \frac{s+1}{s^2 + (5+k)s + k - 6}$
• which is stable if $k > 6$ because coefficients must be positive

The Nyquist Condition Theorem

• Let $P(s) = \frac{N(s)}{D(s)}$
• The $# \text{ of roots of } D(s) \text{ in RHP} = n_p$
• The closed loop system is stable iff the Nyquist plot of $P(s)$ encircles the point $-1$ exactly $n_p$ times counter-clockwise.

Nyquist Condition Remarks

• It is only necessary to know $P(s)$, not the whole closed loop T.F..
• The condition is necessary and sufficient.
• The number of encirclements is counted by
  • Following the Nyquist plot
  • Keeping -1 on the left implies counter clockwise
  • Keeping -1 on the right implies clockwise
• If $P(s)$ has poles on the imaginary axis, then the Nyquist plot has infinite branches.
  • One can replace $j\omega$ by $j\omega + \epsilon$ then let $\epsilon \rightarrow 0$
  • Or add a small semicircle to the Nyquist contour, so that it avoids the pole.
  • This ensures that it avoids the pole

Nyquist Condition Example 1

• $P(s) = \frac{1}{s-1}$
  • $n_p = 1$
  • $P(j\omega) = \frac{1}{j\omega - 1}$
  • $\Re{P(j\omega)} = \frac{-1}{\omega^2 + 1}$
  • $\Im{P(j\omega)} = \frac{-\omega}{\omega^2 + 1}$
• The Nyquist plot is a circle, center at -0.5, radius 0.5.
• The Nyquist plot encircles -1 once in the clockwise direction
• For closed loop stability $-k$ must be inside the circle and to the left
• $\Re{-k} < -1 \implies k > 1$

Nyquist Condition Example 2

• $P(s) = \frac{1}{s-a}, \quad a > 0$
  • $n_p = 1$
  • $P(j\omega) = \frac{1}{j\omega - a}$
  • $\Re{P(j\omega)} = \frac{-a}{\omega^2 + a^2}$
  • $\Im{P(j\omega)} = \frac{-\omega}{\omega^2 + a^2}$
• The Nyquist plot is a circle, center at $-\frac{1}{2a}$, radius $\frac{1}{2a}$.
• For closed loop stability, -k must be inside the circle and to the left
• $\Re{-k} < -\frac{1}{a} \implies k > \frac{1}{a}$

Calculus Definitions

Defintion of a Vertical Asymptote

• A line $x = a$ where $y = f(x)$ is a vertical asymptote if either condition exists

$\qquad \lim_{x \to a^-} f(x) = \pm \infty$ or $\lim_{x \to a^+} f(x) = \pm \infty$

Definition of a Horizontal Asymptote

• A line $y = b$ where $y = f(x)$ is a horizontal asymptote if either condition exists

$\qquad \lim_{x \to \infty} f(x) = b$ or $\lim_{x \to -\infty} f(x) = b$

Definition of a Slant Asymptote

• A line $y = mx + b$ is a slant asymptote of $y = f(x)$ if

$\qquad \lim_{x \to \infty} [f(x) - (mx + b)] = 0$ or $\lim_{x \to -\infty} [f(x) - (mx + b)] = 0$

Derivative Definition

• The derivative of $f(x)$ is defined as

$\qquad f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

Alternate Form of Derivative

• An alternative definition of the derivative is

$\qquad f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$

Precise Limit Definition

• $\lim_{x \to a} f(x) = L$ if for every $\epsilon > 0$ there is a $\delta > 0$ such that whenever $0 < |x - a| < \delta$ then $|f(x) - L| < \epsilon$

Infinite Limit Definition

• $\lim_{x \to a} f(x) = \infty$ if for every $M > 0$ there is a $\delta > 0$ such that whenever $0 < |x - a| < \delta$ then $f(x) > M$

Limit at Infinity Definition

• $\lim_{x \to \infty} f(x) = L$ if for every $\epsilon > 0$ there is an $M > 0$ such that whenever $x > M$ then $|f(x) - L| < \epsilon$

Definite Integral Definition

• For $f(x)$ on $[a, b]$ the interval is divided into $n$ subintervals of width $\Delta x$ select the point, $x_i^*$, from each interval. Then the definite integral of $f(x)$ from $a$ to $b$ is

$\qquad \int_a^b f(x) , dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x$

Algorithmic Game Theory Definition

• The study of interactions among self-interested agents.
• Consists of Game Theory and Algorithm Design coming together.
  • Game Theory - the study of mathematical models of strategic interactions among rational agents
  • Algorithm Design - the design and analysis of algorithms and data structures.

Selfish Routing Example

The Model

• A network of consisting of $n$ nodes and $m$ edges.
• Every edge $e$ has a cost function $c_e(x)$ representing the cost per unit of traffic on edge $e$ when carrying $x$ traffic units.
• A set of $k$ traffic demands exists:
• Traffic demand $i$ has a source $s_i$, a destination $t_i$,
• Traffic is routed from $s_i$ to $t_i$.

Selfish Routing Defined

• Each user aims to minimize their cost.
• Users route their traffic through the cheapest route.

Questions for Selfish Routing

• What is the cost of a Nash equilibrium?
• How does the cost of a Nash equilibrium compare to the cost of a centrally planned optimum?
• How can we design networks that are less sensitive to selfish routing?

Example Network

• A simple network with 2 nodes connected by 2 parallel edges
  • Edge 1 has cost function $c_1(x) = 1$
  • Edge 2 has cost Function $c_2(x) = x$
  • The traffic demand is 1 unit from $s$ to $t$

Centrally planned optimum

• The goal is to minimize the total cost:
  • route $1/2$ unit of traffic on each edge
  • Total cost = $1/2 \cdot 1 + 1/2 \cdot 1/2 = 3/4$.

Nash equilibrium

• All traffic on edge 2.
• The total cost 1 * 1 = 1

Price of Anarchy

• The ratio between the cost of a Nash equilibrium and the cost of a centrally planned optimun
• In the previous example Price of anarchy = $1 / (3/4) = 4/3$.

Braess's Paradox

• adding an edge to a network can increase the cost of a Nash equilibrium.
  • Example
    • A network with nodes s,a,b and t
    • Edge (s,a) has cost function = x
    • Edge (s,b) has cost function = 10 + x
    • Edge (a,t) has cost function = 10 + x
    • Edge (b,t) has cost function = x
    • Traffic demand is 1 unit fron s to t

Nash Equilibrium

• Route all traffic on path $𝑠 \rightarrow 𝑎 \rightarrow 𝑡$. Cost is $10 + 1 + 10 = 21$.
• Route all traffic on path $𝑠 \rightarrow 𝑏 \rightarrow 𝑡$. Cost is $10 + 1 + 10 = 21$.
• Total cost is 21.

Adding an Edge

• Add an edge $(a, b)$ with cost function $c(x) = 0$.

New Nash Equilibrium

• Route all traffic on path $s \rightarrow a \rightarrow b \rightarrow t$.
• Cost is $1 + 0 + 1 = 22$.
• Total cost is 22.

Price of Anarchy

• The price of anarchy is $22/21$.

Mechanism Design

• How can we design a system so that the selfish behavior of the users leads to a desirable outcome?
  • Example
    • Auctions
    • Voting
    • Matching

Algorithmic Game Theory

• Growing field applicable to computer science, economics and other fields
• It is a challenging and rewarding area of research.

Poisson Process

• $N(t)$ represents the number of events up to time t.
• ${N(t), t ≥ 0}$ is a poisson process if:
  • N(0) = 0 ; there are no initial events recorded when time has not started
  • Increments are independent : occurrences from disjoint time intervals do not influence each other
  • Number of Events follows a Poisson Distribution P(N(t+s) - N(s) = n) = $e^{-\lambda t} \frac{(\lambda t)^n}{n!}$

Poisson Distribution function is

• the probability mass function : $P(X = k) = e^{-\mu} \frac{\mu^k}{k!}; k = 0, 1, 2,...$ -The mean and variance of a poisson random variable = µ

Interarrival Times

Ti Description

• T1 is the time of the first event;
• Ti the tome between the (I-1)th and ith event
• the sequence ${Ti, i=1,2,..}$ are : the interarrival times

Interarrival Time Description

• follows an independent exponential distribution -$f_{T_i}(t) = \lambda e^{-\lambda t}, \quad t \geq 0, \quad i = 1, 2, \dots$
• cumulative distribution
• $F_{T_i}(t) = P(T_i \leq t) = 1 - e^{-\lambda t}, \quad t \geq 0, \quad i = 1, 2, \dots$

Exponential Distribution

• is memoryless:
  • $P(T > t + s \mid T > s) = P(T > t)$

Superposition and Splitting for Multiple Poisson Processes

• Superposition for independent poisson processes N1(t) and N2(t) with respective rate λ1 and λ2 , generates a new process N(t) = N(t)+N2(t); whose generated rate is 𝜆=𝜆1 + 𝜆2
• Splitting for a Poisson process N(t) with rate 𝜆, each event gets divided into type 1 with probability p and type 2 with 1-p events, independently ; where N1(t) consists of type 1 and N2(t) : Type 2 -N1 and N2 are independent poisson processes with respective rates 𝜆p and 𝜆(1-p)

Explore algorithmic trading, also known as automated or systematic trading, which uses computer programs to execute trades based on predefined rules. Learn about applications like order execution and arbitrage. Understand VWAP (Volume-Weighted Average Price) and its suitability for large orders.