Probability & Statistics: The Backbone of Quantitative Trading#

The Role of Uncertainty#

Uncertainty in financial markets primarily arises from incomplete information and the random nature of price movements. Traders use probability to:

• Estimate the likelihood of achieving certain returns
• Model price movements to identify profitable opportunities
• Manage risk by understanding the probabilities of large drawdowns

Key Probability Concepts#

Key ideas such as random variables, probability distributions, expected values, and variance allow us to quantify uncertainty. These abstractions form the foundation of strategies, risk models, and performance evaluations.

Random Variables#

A random variable is a numerical outcome of a random process. In finance, a random variable might be the daily return of a stock, the duration until a market crash, or the difference in prices between two instruments.

• Discrete Random Variable: Takes on countable outcomes (e.g., daily number of price up-moves).
• Continuous Random Variable: Takes on uncountably many outcomes within a range (e.g., the exact daily return of a stock).

Probability Distributions#

A probability distribution lists the possible values of a random variable and the probabilities that these values occur. Probability distributions can be discrete (like the binomial distribution) or continuous (like the normal distribution).

Expected Value and Variance#

• Expected Value (Mean): A measure of the central value of a distribution.
• Variance: A measure of how spread out the distribution is.
• Standard Deviation: The square root of the variance; used extensively in risk calculations.

For a discrete random variable X, the expected value is:

E(X) = [x?* P(X = x?]

Variance is:

Var(X) = E(X) - [E(X)]

Uniform Distribution#

The simplest distribution: all outcomes in an interval have the same probability. While not frequently useful for sophisticated financial modeling, it can be a starting point for random number generation.

Binomial Distribution#

The binomial distribution applies to cases where we have repeated Bernoulli trials (i.e., each trial has only two outcomes, success or failure). This can model:

• The number of successful trades out of N attempts
• Win/loss streaks in trading simulations

Example probability mass function (PMF) for binomial with parameters n (trials) and p (probability of success):

P(X = k) = (n choose k) p^k (1 - p)^(n - k)

Poisson Distribution#

Useful for modeling the number of occurrences within a fixed interval. In finance, it can model:

• The arrival of market orders
• The number of price jumps in a given time

Normal Distribution#

The normal (Gaussian) distribution is ubiquitous in finance:

• Often used to model returns (though real returns can exhibit heavier tails).
• Central Limit Theorem ensures many processes tend toward normality when aggregated.

PDF of a normal distribution with mean and standard deviation :

f(x) = (1 / (?2))) * e^(-(x - ) / (2))

Lognormal Distribution#

For asset prices, the lognormal distribution is often more appropriate than the normal distribution, because prices cannot be negative. If X is lognormally distributed, log(X) is normally distributed.

Descriptive Statistics#

Descriptive statistics summarize data:

• Mean
• Median
• Mode
• Variance and Standard Deviation
• Skewness and Kurtosis

By computing descriptive statistics on returns, drawdowns, or trading performance metrics, traders identify general trends and potential anomalies.

Inferential Statistics#

Inferential statistics use sample data to make inferences about a larger population of interest. In trading, we might have limited data for a particular asset or strategy, so we rely on statistical inference to:

• Estimate true means and volatility
• Make predictions about future performance
• Test the validity of trading hypotheses

Sampling and the Central Limit Theorem#

The Central Limit Theorem (CLT) states that, given a sufficiently large sample size, the sample mean of a random variable will be approximately normally distributed, regardless of the variables underlying distribution. This supports why many risk models assume normality for aggregated data.

Correlation and Covariance#

Correlation and covariance measure how two variables (e.g., asset returns) move together.

• Covariance: E[(X - ?(Y - ?]
• Correlation: (X, Y) = Cov(X, Y)/()

When building multi-asset portfolios, covariance and correlation matter for diversification decisions.

Simple Linear Regression#

In trading, simple linear regression is used to analyze relationships between two variables (e.g., one assets return as a function of anothers):

Y = + X +

Where:

• Y is the dependent variable (e.g., return on asset A)
• X is the independent variable (e.g., return on asset B)
• is the slope (how much asset A moves relative to asset B)
• is the intercept
• is the error term

Multiple Regression#

Multiple regression extends simple linear regression to include multiple independent variables. This is crucial when analyzing factors (e.g., market risk, size, value, momentum) that drive returns.

Null and Alternative Hypotheses#

• Null Hypothesis (H): Usually posits no effect or no difference. Example: Trading strategy has zero alpha.?
• Alternative Hypothesis (H?: Indicates the presence of an effect. Example: Trading strategy has non-zero alpha.?

Type I and Type II Errors#

• Type I Error (False Positive): Rejecting H when it is true.
• Type II Error (False Negative): Failing to reject H when it is false.

T-tests, Z-tests, and Chi-Square Tests#

• T-test: Used when the sample size is small or the population variance is unknown.
• Z-test: Used when sample size is large and population variance is known.
• Chi-Square Test: Often used for categorical data or testing variances.

Risk Management#

A single misstep in managing risk can wipe out years of profits. Probabilistic and statistical tools help traders:

• Estimate Value at Risk (VaR)
• Calculate drawdowns
• Understand tail-risk events

Portfolio Optimization#

By modeling each assets expected return, volatility, and correlation with other assets, traders can use techniques like Modern Portfolio Theory (MPT) to optimize the portfolios risk-return tradeoff.

Time Series Analysis#

Most financial data is inherently sequential. Time-series models and statistical processes such as ARIMA and GARCH help traders:

• Forecast future prices
• Model volatility clustering
• Identify seasonal or cyclical trends

Statistical Arbitrage#

Statistical arbitrage relies on identifying pricing anomalies and temporary mispricings. Probability and statistics help measure the strength and reliability of these anomalies, guiding trade entries and exits.

ARIMA, GARCH, and Other Time Series Models#

• ARIMA (Autoregressive Integrated Moving Average) models capture autocorrelations in data.
• GARCH (Generalized Autoregressive Conditional Heteroskedasticity) models volatility clustering.

They are widely employed in forecasting returns and volatility for risk management and strategy development.

Machine Learning Approaches#

Machine learning techniques like random forests or neural networks often incorporate or extend statistical assumptions:

• Feature Engineering: Statistical transformations of data (e.g., rolling means, volatility calculations).
• Model Evaluation: Metrics like RMSE, MAE, precision, recall, and backtesting results.

Factor Models#

Factor models break down returns into components attributed to various risk factors such as market, size, value, momentum, or sector exposure. Traders gauge the factor sensitivities (betas) to manage portfolio risk and identify alpha sources.

Simulating a Random Walk#

A random walk can be used for basic modeling of asset price movements.

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import numpy as np
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import matplotlib.pyplot as plt
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np.random.seed(42)  # For reproducibility
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# Parameters
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T = 1000  # number of steps
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mu = 0.0005  # drift
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sigma = 0.01  # volatility
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# Simulate returns
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returns = np.random.normal(mu, sigma, T)
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# Price starts at 100
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price = 100
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price_path = [price]
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for r in returns:
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    price *= (1 + r)
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    price_path.append(price)
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# Plot
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plt.plot(price_path)
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plt.title("Simulated Random Walk")
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plt.xlabel("Time Steps")
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plt.ylabel("Price")
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plt.show()

Explanation:

• We assume the daily log returns are normally distributed with mean and variance .
• The price evolves multiplicatively.

Fitting a Linear Model in Python#

Below is a simple example of fitting a linear regression to hypothetical dependent?(asset A returns) and independent?(asset B returns) data.

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import numpy as np
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import statsmodels.api as sm
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# Hypothetical returns
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np.random.seed(42)
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asset_A = np.random.normal(0.001, 0.02, 1000)  # Dependent variable
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asset_B = np.random.normal(0.0005, 0.02, 1000) # Independent variable
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# Add a constant to the independent variable (intercept)
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X = sm.add_constant(asset_B)
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model = sm.OLS(asset_A, X).fit()
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print(model.summary())

Explanation:

• OLS is Ordinary Least Squares regression from the Statsmodels library.
• Were regressing asset_A returns on asset_B returns, yielding estimates for (intercept) and (slope).

High-Frequency Trading Considerations#

In high-frequency trading (HFT), statistical and probabilistic analysis occur on extremely short timescales:

• Latency and Microstructure: Price changes can be modeled with point processes or reflected in state-based Markov models.
• Order Book Dynamics: Changes in order book depth, limit orders, and cancelations are analyzed in real-time.
• Ultra-Low Latency Systems: Need for efficient data structures and algorithms to handle massive incoming data.

Risk-Neutral Pricing and Derivatives#

In options and futures markets, a risk-neutral probability measure is used for pricing:

• BlackScholes: Prices European options assuming underlying price follows geometric Brownian motion.
• Risk-Neutral World: Expected return of the underlying asset is the risk-free rate, simplifying pricing models.

Monte Carlo Simulations at Scale#

Monte Carlo methods sample from probability distributions to simulate a wide range of scenarios:

• Pricing Complex Derivatives: Can handle path-dependent options.
• Stress Testing: Evaluate portfolio under extreme market conditions.
• Parallelization: Running thousands or millions of simulations requires scalable computing.