Algorithmic Trading & Order Execution

Algorithmic Trading Motivation

• Algorithm execution leads to cost reduction through decreased market impact and leveraging price improvement.
• Algorithms' speed enables faster order execution in fast-moving markets compared to human traders.
• Automation of strategies by algorithms improves efficiency, allowing traders to focus on other tasks.
• Programs executing orders can be optimized to minimize risk and maximize profit.
• Algorithmic trading platforms provide access to markets otherwise inaccessible.
• Complex strategies implementation becomes possible through algorithmic trading.

Order Execution Motivation

• Brokers have a legal/ethical best execution obligation for clients' orders.
• Large orders can impact market prices. Aims at minimizing the effect through order execution algorithms.
• Order execution algorithms prevent information leakage, where large orders reveal trader intentions.
• Order execution algorithms reduce slippage, the difference between expected and actual trade prices.
• Increased liquidity is possible by finding hidden orders and matching buyers and sellers using algorithms.

Single-Order Execution

• Single-order execution is the simplest form of trading algorithms.
• Effectively and efficiently executes a single order without taking further positions.

Basic Execution Algorithms

Market Order

• Market orders should be executed immediately at the best available price.
• They are suitable for small orders in liquid markets with low price impact risk.
• Has a high slippage risk.

Limit Order

• Limit Orders should be executed only at a specified price or better.
• They are suitable for larger orders in less liquid markets where the trader will wait for the desired price.
• There is a risk that the order will not be filled if the price doesn't reach the specified level.

Immediate-Or-Cancel (IOC) Order

• Immediate-Or-Cancel (IOC) Orders should be executed immediately, with any unfulfilled portion cancelled.

Fill-Or-Kill (FOK) Order

• Fill-Or-Kill (FOK) Orders should be executed in its entirety immediately, or is cancelled.

Iceberging

• Iceberging is an order splitting strategy that hides the full order quantity by displaying only a small portion, appropriate for large orders with potential market impact.
• Has a risk of information leakage if others detect a large order behind the iceberg.

Volume-Weighted Average Price (VWAP)

• Volume-Weighted Average Price (VWAP) executes orders to match the historical volume-weighted average price over a specified period.
• Formula: $$ VWAP = \frac{\sum_{i=1}^{n} P_i \times V_i}{\sum_{i=1}^{n} V_i} $$ - $P_i$ is the price of the $i$-th trade - $V_i$ is the volume of the $i$-th trade - $n$ is the number of trades in the specified period
• Suitable for large orders that need to be executed over a period of time.
• There's a risk of underperforming simple strategies should the market trend in one direction.

Time-Weighted Average Price (TWAP)

• Time-Weighted Average Price (TWAP) executes orders to match the average price over a period.
• Suitable for orders requiring execution over time, trader unconcerned about matching VWAP.
• Has a risk of underperforming simple strategies should the market trend in one direction.

Percentage of Volume (POV)

• Percentage of Volume (POV) should participate in the market as a specified percentage of the total volume being traded.
• POV is suitable for minimizing market impact and maintaining anonymity.
• It has a risk of not completing the order within desired timeframe if market volume is low.

Optimal Execution

• The goal of optimal execution is to minimize the costs of executing a large order.

The Almgren-Chriss Model

• A mathematical model for optimal execution.
• It considers the trade-off between market impact and risk aversion.
• Assumptions:
  • A trader has a fixed amount of inventory to sell over a fixed time period.
  • A trader can choose how quickly to sell the inventory.
  • Selling the inventory quickly will result in a higher market impact cost.
  • Selling the inventory slowly will result in a higher risk of adverse price movements.
• Notation:
  • $X_0$: Initial inventory.
  • $T$: Time horizon.
  • $x(t)$: Inventory at time $t$.
  • $v(t)$: Trading rate at time $t$.
  • $\sigma$: Volatility of the asset price.
  • $\lambda$: Permanent market impact.
  • $\eta$: Temporary market impact.
  • $\gamma$: Risk aversion coefficient.
• Dynamics:
  • Inventory: $$ \frac{dx(t)}{dt} = -v(t), \quad x(0) = X_0, \quad x(T) = 0 $$
  • Price: $$ dS_t = \sigma dW_t - \lambda v(t) dt $$
  • $W_t$ is a standard Brownian motion (Wiener process).
• Cost Function: $$ \mathbb{E}\left[ \int_0^T v(t) (\eta v(t) + \lambda v(t)) dt - \gamma \int_0^T v(t)^2 dt \right] $$
  • Temporary market impact ($\eta v(t)$), permanent market impact ($\lambda v(t)$) and risk aversion term ($\gamma v(t)^2$) all minimize the expected cost of trading.
• Optimal Trading Rate: $$ v^*(t) = \frac{X_0 \sinh(k(T-t))}{\int_0^T \sinh(kt) dt} $$
  • $k = \sqrt{\frac{\gamma}{\eta}}$.

Direct Market Access (DMA)

• Direct Market Access (DMA) allows traders to directly access order books of exchanges and other trading venues.
• Benefits:
  • Faster execution speeds.
  • Greater control over order routing.
  • Access to deeper liquidity.
  • Lower transaction costs.
• Risks:
  • DMA requires a high level of technical expertise.
  • DMA setup and maintenance is expensive.
  • Increased risk of errors and omissions.

Smart Order Routing (SOR)

• Smart Order Routing (SOR) systems automatically route orders to the best available trading venue based on factors like price and size.
• Benefits:
  • Improved execution quality.
  • Reduced transaction costs.
  • Increased efficiency.
• Risks:
  • SOR setup and maintenance can be complex.
  • May not always find the best available trading venue.
  • Relies on the accuracy of market data.

High-Frequency Trading (HFT)

• High-Frequency Trading (HFT) uses high speed, high turnover, and high order-to-trade ratios.
• Characteristics:
  • Sophisticated algorithms and hardware are used.
  • Focus on short-term trading opportunities.
  • Often involves market making and arbitrage.
• Controversies:
  • Can exacerbate market volatility.
  • May create an uneven playing field for other market participants.
  • There are concerns about fairness and transparency.

Conclusion

• Algorithmic trading and order execution are complex and evolving fields that are becoming more important as markets become increasing electronic.
• It offers potential to improve execution quality, reduce transaction costs, and increase efficiency.
• They also pose risks that need to be carefully managed.

Rules of Potentiation

Multiplication of Powers with the Same Base

• The product of two or more powers with the same base is equal to the base raised to the sum of the exponents.
• $a^m \cdot a^n = a^{m+n}$

Examples:

• $2^3 \cdot 2^2 = 2^{3+2} = 2^5 = 32$
• $x^2 \cdot x^5 \cdot x = x^{2+5+1} = x^8$

Division of Powers with the Same Base

• The quotient of two powers with the same base is equal to the base raised to the difference of the exponents.
• $\frac{a^m}{a^n} = a^{m-n}$

Examples:

• $\frac{3^5}{3^3} = 3^{5-3} = 3^2 = 9$
• $\frac{y^7}{y^2} = y^{7-2} = y^5$

Power of a Power

• The power of a power is equal to the base raised to the product of the exponents.
• $(a^m)^n = a^{m \cdot n}$

Examples:

• $(4^2)^3 = 4^{2 \cdot 3} = 4^6 = 4096$
• $(z^4)^5 = z^{4 \cdot 5} = z^{20}$

Power of a Product

• The power of a product is equal to the product of each of the factors raised to the exponent.
• $(a \cdot b)^n = a^n \cdot b^n$

Examples:

• $(2 \cdot 3)^2 = 2^2 \cdot 3^2 = 4 \cdot 9 = 36$
• $(x \cdot y)^3 = x^3 \cdot y^3$

Power of a Quotient

• The power of a quotient is equal to the quotient of each of the terms raised to the exponent.
• $(\frac{a}{b})^n = \frac{a^n}{b^n}$

Examples:

• $(\frac{4}{2})^3 = \frac{4^3}{2^3} = \frac{64}{8} = 8$
• $(\frac{m}{n})^4 = \frac{m^4}{n^4}$

Zero Exponent

• Any power with an exponent of zero is equal to one.
• $a^0 = 1$

Examples:

• $5^0 = 1$
• $x^0 = 1$

Negative Exponent

• A power with a negative exponent is equal to the inverse of the base raised to the positive exponent.
• $a^{-n} = \frac{1}{a^n}$

Examples:

• $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$
• $y^{-2} = \frac{1}{y^2}$

Fractional Exponent

• A power with a fractional exponent is equal to the nth root of the base raised to the mth power.
• $a^{\frac{m}{n}} = \sqrt[n]{a^m}$

Examples:

• $4^{\frac{1}{2}} = \sqrt{4^1} = \sqrt{4} = 2$
• $z^{\frac{2}{3}} = \sqrt[3]{z^2}$

Linear Algebra and Analytical Geometry I

Chapter 1: Vectors in $\mathbb{R}^n$

1.1 The vector space $\mathbb{R}^n$

• $\mathbb{R}^n = {(x_1, x_2,..., x_n) \mid x_i \in \mathbb{R}, i = 1,..., n}$

Operations on $\mathbb{R}^n$:

• Addition: $(x_1,..., x_n) + (y_1,..., y_n) = (x_1 + y_1,..., x_n + y_n)$
• Scalar multiplication: $\lambda(x_1,..., x_n) = (\lambda x_1,..., \lambda x_n)$

Properties of operations: $\forall u, v, w \in \mathbb{R}^n$ and $\lambda, \mu \in \mathbb{R}$:

• $u + v = v + u$
• $(u + v) + w = u + (v + w)$
• $\exists 0 \in \mathbb{R}^n$ such that $u + 0 = u$
• $\exists -u \in \mathbb{R}^n$ such that $u + (-u) = 0$
• $\lambda(u + v) = \lambda u + \lambda v$
• $(\lambda + \mu)u = \lambda u + \mu u$
• $(\lambda \mu)u = \lambda(\mu u)$
• $1u = u$

1.2 Scalar product, norm and distance in $\mathbb{R}^n$

• Scalar product: $ = \sum_{i=1}^{n} u_i v_i$ where $u = (u_1,..., u_n)$ and $v = (v_1,..., v_n)$

Properties of the scalar product: $\forall u, v, w \in \mathbb{R}^n$ and $\lambda \in \mathbb{R}$:

• $ = $
• $ = + $
• $ = \lambda $
• $ \geq 0$ and $ = 0 \iff u = 0$

Norm: $||u|| = \sqrt{} = \sqrt{\sum_{i=1}^{n} u_i^2}$

Properties of the norm: $\forall u, v \in \mathbb{R}^n$ and $\lambda \in \mathbb{R}$:

• $||\lambda u|| = |\lambda| \cdot ||u||$
• $||u|| \geq 0$ and $||u|| = 0 \iff u = 0$

Cauchy-Schwarz inequality: $|| \leq ||u|| \cdot ||v||$

Triangle inequality: $||u + v|| \leq ||u|| + ||v||$

Distance: $d(u, v) = ||u - v|| = \sqrt{\sum_{i=1}^{n} (u_i - v_i)^2}$

1.3 Straight lines and planes in $\mathbb{R}^n$

• Straight line passing through $P$ of direction $v$: ${P + tv \mid t \in \mathbb{R}}$
• Plane passing through $P$ of directions $v, w$: ${P + tv + sw \mid t, s \in \mathbb{R}}$ where vectors $v$ and $w$ are non collinear.

1. Einleitung

1.1 Motivation

• Data-Mining-Verfahren aim at generalizing patterns found in the data.
• There is a trade-off between accuracy and generalization.
• High accuracy on training data but low generalization on new data results in overfitting.

1.2 Zielsetzung

• An overview of overfitting and regularization will be given. Various regularization techniques will be introduced and the advantages, as well as disadvantages of different techniques will be discussed.

1.3 Gliederung

• Section 2 defines Overfitting and its causes.
• Section 3 gives an overview of regularization techniques.
• Section 4 introduces evaluation criteria.
• Section 5 is a summary and outlook.