Algorithmic Trading Strategies

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Questions and Answers

What is a 'many-one function'?

A function where multiple elements from set A map to the same element in set B. (correct)
A function where each element from set A maps to multiple elements in set B.
A function where each element in set A maps to a unique element in set B.
A function where no two elements in set A map to the same element in set B.

What is the range of an Onto function?

An empty set.
Always smaller than codomain.
Always larger than co-domain.
Equal to its co-domain. (correct)

What defines an 'into' function?

The codomain is equal to its range.
The codomain is a proper subset of its range.
The domain and the codomain are equal
The codomain is not equal to the range. (correct)

What is the graphical test to check if a function is one-one onto?

A line parallel to the x-axis cuts the graph at exactly one point. (B) Signup and view all the answers

What does $f(f(x)) = x$ imply?

$f(x) = f^{-1}(x)$ (D) Signup and view all the answers

In a many-one function, what is the relationship between elements?

More than one element from set A maps to a single element in set B. (B) Signup and view all the answers

What is the condition for a function to be considered 'onto'?

Its range is exactly equal to its co-domain. (C) Signup and view all the answers

Graphically, how can you identify an 'into' function?

A line parallel to the x-axis in the co-domain region does not cut the graph (A) Signup and view all the answers

What happens if $R = {(x,y) : x, y \text{ divisible by } 2}$?

Partitions the set of Z into three pairwise disjoints. (A) Signup and view all the answers

What is the number of remainders of $R = {(x,y) : x, y \text{ divisible by } 2}$?

3 (C) Signup and view all the answers

Flashcards

Many-one function

More than one element from set A maps to a single element in set B.

Onto function

Every element 'y' in set B has a corresponding element 'x' in set A such that y = f(x). The range equals the co-domain.

Into function

A function which is not surjective (onto). The co-domain of 'f' is not equal to the range of 'f'.

Graphical test for One-One Onto

A line parallel to the x-axis cuts the graph of y = f(x) at exactly one point within the co-domain region.

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Number of Onto functions

Number of Onto functions equals the sum from k=1 to n of (-1)^(n-k)nCkk^m. If m > n and f(n) = 0 if m < n

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Involution Property

If f(f(x)) = x, then f(x) = f⁻¹(x). Each element maps back to itself after two applications of f.

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Study Notes

Algorithmic trading uses computer programs to execute orders based on predefined rules.
It exploits computer speed and automation for efficient, effective trading.

Why Algorithmic Trading?

Computers react faster than humans.
It reduces manual errors.
Algorithms execute exactly as programmed.
Strategies can be tested on historical data.
It handles intricate strategies beyond human capability.
Executes large orders without market impact.

Types of Algorithmic Trading Strategies

Execution Algorithms

Minimizes market impact and transaction costs.
VWAP (Volume Weighted Average Price) targets the average price weighted by volume.
VWAP = (sum of (Price at transaction i * Volume at transaction i)) / (sum of Volume at transaction i).
TWAP (Time Weighted Average Price) targets the average price over a period.
TWAP = (sum of Price at time i) / (Number of prices)
Percentage of Volume (POV) participates in a fixed percentage of market volume.
Implementation Shortfall balances the cost of delay vs. the cost of immediacy.

Market Making

Profits from bid-ask spreads while providing liquidity.
Posts bid and ask orders, capturing the spread.
Faces challenges in inventory management and adverse selection.

Statistical Arbitrage

Exploits statistical mispricings.
Pairs Trading identifies correlated assets and trades on temporary divergence.
Index Arbitrage exploits price differences between an index and its constituent stocks.

Trend Following

Identifies and capitalizes on price trends.
Uses technical indicators to detect trends and place orders accordingly.
Moving Averages involve buying when the price crosses above a moving average and selling when it crosses below.

High-Frequency Trading (HFT)

Exploits very short-term market inefficiencies.
It is characterized by high speed, high turnover, and co-location.
Strategies includes; order anticipation, quote stuffing, and arbitrage.
Is subject to market manipulation and unfair advantage controversies.

Order Execution

Order Types

Market Order executes immediately at the best available price.
Limit Order executes only at a specified price or better.
Stop Order becomes a market order when the stop price is reached.
Stop-Limit Order becomes a limit order when the stop price is reached.
Hidden (Iceberg) Order displays only a portion of the order to reduce market impact.
Fill or Kill (FOK) executes the entire order immediately or cancels it.
Immediate or Cancel (IOC) executes any portion of the order immediately and cancels the rest.
All or None (AON) executes the entire order at once, if possible.

Order Placement Considerations

Higher market liquidity reduces the risk of large price movements.
Larger orders require more careful execution strategies.
Short-term vs. long-term goals influence strategy selection.
Risk tolerance affects the aggressiveness of the execution.
Volatility, news events, and economic indicators affect strategy.

Market Microstructure

Market Microstructure is the fine-grained details of how a market operates.
Key elements; Order Book, Quote Sizes, Tick Size and Market Participants.
Order Book is defined as a list of outstanding buy and sell orders.
Quote Sizes is defined as the number of shares available at each price level.
Tick Size is defined as the minimum price increment.
Market Participants are different types of traders .

Impact of Algorithmic Trading on Market Microstructure

It provides liquidity and narrow spreads which increases liquidity.
Efficient execution lowers costs, which reduces transaction costs.
HFT can exacerbate price swings, which increases volatility.
Rapid order execution can trigger sudden market drops, which leads to flash crashes.
Order flow is spread across multiple exchanges and dark pools, which leads to market fragmentation.

Dark Pools

Dark Pools are the private exchanges where orders are matched anonymously.
Dark Pools allow institutions to trade large blocks of shares without revealing their intentions.
Dark Pools reduce market impact and price discovery which is an advantage.
Dark Pools lack transparency and have potential for abuse which is a disadvantage.

Regulatory Considerations

Regulations prohibit strategies that artificially inflate or deflate prices, which is market manipulation.
Illegal to trade on non-public information, which is insider trading.
Regulators require exchanges to disseminate market data to increase order book transparency.
Brokers must seek the most favorable terms for their clients to attain best execution.
SEC Rule 611 and Market Access Rule are examples of regulatory considerations.
SEC Rule 611 (Order Protection Rule) requires brokers to route orders to the exchange with the best price.
Market Access Rule (Rule 15c3-5) requires brokers to have risk management controls in place for algorithmic trading.

Challenges and Risks

The risk that the algorithm is based on flawed assumptions is Model Risk.
Software bugs or hardware failures cause technical glitches.
Inaccurate or incomplete data can lead to poor decisions, which are data errors.
Strategies that perform well in backtesting may fail in live trading, which is Overfitting.
Strategies must be adapted to evolving market conditions because of changing market dynamics.
New rules can render strategies obsolete, which causes regulatory changes.

Best Practices

Test strategies on a variety of historical data using Robust Backtesting.
Implement controls to limit potential losses using Risk Management.
Continuously monitor performance and adjust strategies as needed using Monitoring.
Stay up-to-date with regulatory requirements to ensure compliance.
Document the logic and assumptions behind the algorithm to ensure transparency.
Have strategies reviewed by an independent party to ensure independent review.