# Math You Need to Know in Day Trading

Day trading requires a basic foundation in mathematics to make informed decisions and manage risk effectively.

This guide will cover the essential mathematical knowledge and skills needed to succeed in day trading.

## Key Takeaways – Math You Need to Know in Day Trading

- Expected value (EV) is an essential mathematical skill. A high probability doesn’t always mean positive EV – consider both the likelihood and magnitude of potential gains and losses.
- Understand risk management math, especially position sizing and drawdown calculations. This helps protect your capital and ensures you can weather volatility.
- Trading is an exercise in applied probability. Learn basic statistics and probability, including standard deviation and probability distributions.
- Understanding correlation is key to diversifying and improve risk/reward in a portfolio.

## Basic Arithmetic

### Addition and Subtraction

Knowing basic mental arithmetic is important for day traders.

You’ll constantly need to calculate:

- Profits and losses
- Price differences
- Cumulative gains or losses throughout the day

### Multiplication and Division

These operations are crucial for:

- Calculating position sizes
- Determining leverage effects
- Estimating potential profits or losses
- Risk/reward ratios
- Expected value

### Percentages

Understanding percentages is fundamental in day trading for:

- Calculating price changes
- Determining stop-loss and take-profit levels
- Assessing risk/reward ratios

### Expected Value

Expected value (EV) deserves its own section because it’s so important for traders.

EV describes the amount of value that’s expected to be derived from taking a certain action.

Too much emphasis is often placed on just doing the thing that’s most likely and not considering the payoffs and penalties, as well as other factors (what kind of loss can be afforded, scalability of taking certain actions).

#### Example #1

For example, if something has an 80% chance of being correct that seems rather attractive.

But it might not be when accounting for the returns of being right and the loss if the bet goes wrong.

Let’s say you get a $100 reward for being right in that situation. But let’s also say you have a $500 loss if it goes wrong.

Your expected value would be:

**EV** = 0.80 * $100 – 0.20 * $500 = $80 – $100 = -$20

So your expected value is actually negative.

Most of the time, you will be right and get the $100.

But do it enough times and you’re likely facing a loss that’ll get worse over time.

#### Example #2

Now let’s say you have a 10% chance of being right and a $100 reward for being right and a 90% chance of being wrong and a $8 penalty for being wrong.

**EV** = 0.10 * $100 – 0.90 * $8 = $10 – $7.20 = +$2.80

Assuming you can cover the loss (having liquidity is important in trading), this is a bet that’s worth making if it has some level of scale to it.

#### Example #3 – EV with Other Variables

Let’s say you have a 90% chance of being right with a $1,000 reward and a 10% chance of being wrong with a $1,000 penalty.

The EV here is +$800.

But let’s say this is a one-time opportunity only and learning about this opportunity took you 40 hours of time and work (effectively your EV is $20/hour).

In other words, it’s not scalable and is time-intensive.

And there’s also the factor of needing to be able to cover the $1,000 penalty in the event you’re wrong.

Would you still take this opportunity?

In this situation, despite the clearly positive EV, you might not if you’re looking for opportunities that are recurring and scalable.

## Statistics and Probability

### Mean, Median, and Mode

These measures of central tendency help traders:

- Identify average price levels
- Understand typical trading ranges
- Spot outliers and potential opportunities

### Standard Deviation

Standard deviation – and other measures of movement – is important for:

- Measuring price volatility
- Setting realistic profit targets

### Probability Distributions

Understanding probability distributions helps in:

- Understanding the potential range of outcomes
- Assessing the likelihood of price movements
- Evaluating the potential success of trading strategies
- Making more informed risk management decisions

### Correlations

Correlation studies how positions in a portfolio move and is important for diversifying and improving the risk/reward ratio of a portfolio.

Trading is ultimately a game of applied probability.

Little is known for sure.

There’s a wide range of possibilities, all with different probabilities associated with them.

Probabilistic thinking is one of the most important skills in trading.

## Risk Management Mathematics

### Position Sizing

Proper position sizing is critical for managing risk.

Traders need to calculate:

- The appropriate number of shares or contracts to trade
- The dollar amount at risk per trade
- The percentage of their account at risk

It’s also important to manage liquidity effectively and appropriate position sizing is important in this.

You can be right on a trade, but if you don’t have the liquidity cushion to manage the volatility you might get squeezed out of a trade.

### Risk-Reward Ratio

Calculating and maintaining favorable risk-reward ratios involves:

- Determining potential profit versus potential loss
- Assessing the probability of success for each trade
- Ensuring long-term profitability through proper trade selection and portfolio structuring
- Diversification is generally the best way to improve a portfolio’s risk-reward ratio

### Drawdown Calculations

Understanding drawdowns helps traders:

- Evaluate the performance of their trading strategy
- Set realistic expectations for account fluctuations
- Determine when to adjust or pause trading activities
- When to use options in a portfolio to make sure unacceptable losses don’t happen

## Technical Analysis Math

### Moving Averages

Interpreting moving averages requires:

- Understanding different types (simple, exponential, weighted)
- Determining appropriate time periods
- Recognizing crossovers and divergences

For moving averages, it’s important to at least understand how they’re calculated, what they represent, and the quality of the signal they give off.

### Relative Strength Index (RSI)

The RSI involves calculations to:

- Measure the speed and change of price movements
- Identify overbought and oversold conditions
- Spot potential trend reversals

### Fibonacci Retracements

Using Fibonacci ratios in trading requires:

- Calculating key retracement levels (23.6%, 38.2%, 61.8%)
- Identifying potential support and resistance areas
- Combining Fibonacci levels with other technical indicators

## Advanced Mathematical Concepts

### Calculus

While not used directly in day-to-day trading, understanding basic calculus concepts can help in:

- Analyzing rate of change in price movements
- Developing and understanding more complex trading algorithms and quantitative strategies
- How to model price movements in the context of nondeterministic environments like markets (e.g., stochastic calculus)
- Interpreting certain economic models
- Summing the area under a curve when looking at probability distributions

### Linear Algebra

Basic linear algebra concepts are useful for:

- Understanding correlation between different assets
- Developing multi-factor trading models
- Analyzing portfolio performance
- Portfolio optimization uses a lot of linear algebra concepts

### Game Theory

Applying game theory principles can assist in:

- Understanding market dynamics and participant behavior
- In each market, who are the buyers and sellers, how big are they, what are they motivated to do?
- Developing strategies to counter other traders’ actions
- Making decisions in competitive trading environments
- Thinking through the first-, second-, third-, and nth-order consequences of trading decisions

## Algorithmic Trading Mathematics

### Time Series Analysis

For traders developing algorithmic strategies, time series analysis is important for:

- Identifying trends and patterns in historical data
- The process of forecasting future price movements (probability distributions, not deterministic lines)
- Developing mean-reversion or momentum-based strategies

### Stochastic Processes

Understanding stochastic processes helps in:

- Modeling random price movements
- Developing Monte Carlo simulations for risk assessment
- Creating more trading algorithms
- Understanding signal vs. noise

### Machine Learning Algorithms

While not strictly mathematical, understanding the basics of machine learning algorithms involves:

- Linear regression for trend analysis
- Decision trees and random forests for pattern recognition
- Neural networks for complex pattern identification and prediction

## Financial Mathematics

### Options Pricing

For traders dealing with options, understanding the mathematics behind options pricing:

- Black-Scholes model basics
- Factors affecting option prices (underlying price, strike price, time to expiration, volatility, interest rates)
- Greeks (delta, gamma, theta, vega) for measuring option sensitivities

### Yield Curve Analysis

For traders dealing with fixed income securities or interest rate-sensitive assets:

- Calculating and interpreting yield curves
- Understanding the relationship between short-term and long-term interest rates
- Predicting potential economic shifts based on yield curve shapes

### Value at Risk (VaR)

VaR calculations help traders:

- Estimate the potential loss in value of an asset or portfolio
- Set appropriate risk limits

Many brokers provide VaR calculations for traders directly within their trading terminal.

## Psychological Aspects of Trading Mathematics

### Discipline

- Develop resilience during losing streaks
- Set realistic performance expectations
- Ensure capital preservation for future opportunities
- Minimize risk on individual trades
- Avoid emotional decision-making from overexposure

### Expectancy

Calculating and understanding expectancy helps traders:

- Evaluate the long-term profitability of a trading strategy
- Make decisions based on probabilistic outcomes rather than individual trades
- Maintain discipline in following a proven system

### Risk of Ruin

Understanding the mathematics behind the risk of ruin helps traders:

- Avoid overleveraging their accounts
- Use OTM options to hedge and/or set appropriate stop-loss levels
- Develop a sustainable long-term trading approach
- Diversify and not make things dependent on one outcome or a handful of outcomes
- Increase return relative to risk while keeping the portfolio within acceptable risk parameters

### Kelly Criterion

While controversial in its application to trading, understanding the Kelly Criterion can help in:

- Learning different approaches to optimizing position sizes for long-term growth
- Balancing risk and reward in betting scenarios
- Understanding the mathematical limits of aggressive trading strategies

## Practical Application of Mathematics in Day Trading

### Spreadsheet Skills

Proficiency in spreadsheet software is used for:

- Tracking and analyzing trading performance
- Organizing data (e.g., measuring progress, backtesting)
- Creating custom indicators and algorithms
- Doing various forms of quick math
- Discounted cash flow and other forms of fundamental analysis

### Programming Skills

Basic programming knowledge (Python, R, or similar languages) is beneficial for:

- Automating data analysis and strategy testing
- Implementing custom indicators and alerts
- Developing and testing trading algorithms

### Data Visualization

The ability to create and interpret charts and graphs is essential for:

- Identifying patterns and trends in price data
- Communicating analysis results effectively
- Making quick, informed decisions based on visual information

## Conclusion

Mastering these fundamental concepts will provide a solid foundation for success.

Note that mathematical knowledge alone doesn’t guarantee profitable trading.

Emotional control, discipline, passion for trading and markets, and continuous learning are also important factors in becoming a successful trader.

Markets evolve over time and new trading technologies emerge, so the mathematical skills required will likely continue to expand.

Traders should stay informed about new developments and be prepared to adapt their skills accordingly.