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Mathematics Fundamentals for Quantitative Finance

🎯 Learning Objectives

  • Review essential calculus concepts
  • Understand linear algebra basics
  • Master statistics and probability
  • Apply math to financial problems

Mathematics is the foundation of quantitative finance. This chapter reviews essential mathematical concepts you'll use throughout your quant journey. Don't worry if you're rusty - we'll explain everything clearly!

Calculus Basics

Derivatives

What it is: The rate of change of a function

In Finance: Used to measure sensitivity (Greeks in options), optimization

Key Concepts: - First Derivative: Rate of change - Second Derivative: Rate of change of the rate of change - Chain Rule: For composite functions

Example - Price Change: 

# If price P(t) = 100 + 5t (price increases by $5 per day)
# Derivative dP/dt = 5 (constant rate of change)

Integration

What it is: The reverse of differentiation, finding the area under a curve

In Finance: Used in continuous-time models, calculating expected values

Key Concepts: - Definite Integral: Area under curve between two points - Indefinite Integral: Antiderivative

Optimization

What it is: Finding maximum or minimum values

In Finance: Portfolio optimization, finding best parameters

Methods: - Gradient Descent: Iterative optimization - Lagrange Multipliers: Constrained optimization

Linear Algebra

Vectors and Matrices

Vectors: Arrays of numbers representing quantities with direction

In Finance: Portfolio weights, returns, prices

import numpy as np

# Portfolio weights
weights = np.array([0.4, 0.3, 0.3])  # 40% stock A, 30% stock B, 30% stock C

# Returns
returns = np.array([0.05, 0.03, 0.02])  # 5%, 3%, 2% returns

# Portfolio return = weighted sum
portfolio_return = np.dot(weights, returns)
print(f"Portfolio Return: {portfolio_return:.4f}")  # 0.035 = 3.5%

Matrix Operations

Matrix Multiplication: Used in portfolio calculations

Inverse Matrix: Solving systems of equations

Eigenvalues/Eigenvectors: Used in Principal Component Analysis (PCA)

Dot Product

Formula: a · b = Σ(aᵢ × bᵢ)

In Finance: Calculating portfolio returns, correlations

# Calculate portfolio return
weights = np.array([0.5, 0.3, 0.2])
returns = np.array([0.10, 0.05, 0.03])
portfolio_return = np.dot(weights, returns)

Statistics

Descriptive Statistics

Mean (Average): 

import numpy as np

returns = [0.05, 0.03, -0.02, 0.08, 0.01]
mean_return = np.mean(returns)
print(f"Mean Return: {mean_return:.4f}")

Median: Middle value when sorted

Standard Deviation: Measure of volatility 

std_dev = np.std(returns)
print(f"Standard Deviation: {std_dev:.4f}")

Variance: Square of standard deviation 

variance = np.var(returns)

Probability Distributions

Normal Distribution: Most common in finance - Bell-shaped curve - Defined by mean (μ) and standard deviation (σ)

from scipy import stats
import matplotlib.pyplot as plt

# Generate normal distribution
mu, sigma = 0.05, 0.15  # 5% mean return, 15% volatility
normal_dist = stats.norm(mu, sigma)

# Calculate probability of return > 10%
prob = 1 - normal_dist.cdf(0.10)
print(f"Probability of return > 10%: {prob:.4f}")

Other Distributions: - Log-normal: For stock prices (prices can't be negative) - Student's t: For small samples - Chi-square: For variance testing

Correlation and Covariance

Covariance: Measures how two variables move together 

# Covariance between two stocks
stock_a = [0.05, 0.03, -0.02, 0.08]
stock_b = [0.04, 0.02, -0.01, 0.07]
covariance = np.cov(stock_a, stock_b)[0, 1]

Correlation: Normalized covariance (-1 to +1) - +1: Perfect positive correlation - 0: No correlation - -1: Perfect negative correlation

correlation = np.corrcoef(stock_a, stock_b)[0, 1]
print(f"Correlation: {correlation:.4f}")

Probability

Basic Probability

Probability: Likelihood of an event (0 to 1)

In Finance: Probability of profit, probability of loss

Conditional Probability

P(A|B): Probability of A given B

In Finance: Probability of stock going up given market is up

Bayes' Theorem

Formula: P(A|B) = P(B|A) × P(A) / P(B)

In Finance: Updating beliefs based on new information

Expected Value

Formula: E[X] = Σ(xᵢ × P(xᵢ))

In Finance: Expected return, expected profit

# Expected return
returns = [0.10, 0.05, -0.05]
probabilities = [0.5, 0.3, 0.2]
expected_return = sum(r * p for r, p in zip(returns, probabilities))
print(f"Expected Return: {expected_return:.4f}")

Time Series Mathematics

Returns

Simple Return: R = (P₁ - P₀) / P₀

Log Return: r = ln(P₁ / P₀)

import pandas as pd

# Calculate simple returns
prices = [100, 105, 103, 108]
returns = pd.Series(prices).pct_change().dropna()
print(returns)

# Calculate log returns
log_returns = np.log(pd.Series(prices) / pd.Series(prices).shift(1)).dropna()

Volatility

Historical Volatility: Standard deviation of returns

Annualized Volatility: σ_annual = σ_daily × √252

# Calculate annualized volatility
daily_returns = [0.01, -0.02, 0.03, 0.01, -0.01]
daily_std = np.std(daily_returns)
annual_vol = daily_std * np.sqrt(252)  # 252 trading days per year
print(f"Annualized Volatility: {annual_vol:.4f}")

Financial Mathematics

Compound Interest

Formula: A = P(1 + r)ⁿ

In Finance: Future value of investments

# Future value
principal = 1000
rate = 0.05  # 5% annual
years = 10
future_value = principal * (1 + rate) ** years
print(f"Future Value: ${future_value:.2f}")

Present Value

Formula: PV = FV / (1 + r)ⁿ

In Finance: Discounting future cash flows

Net Present Value (NPV)

Formula: NPV = Σ(CFₜ / (1 + r)ᵗ)

In Finance: Evaluating investment opportunities

Regression Analysis

Linear Regression

Formula: y = α + βx + ε

In Finance: Finding relationships (e.g., stock vs. market)

from sklearn.linear_model import LinearRegression

# Example: Stock return vs Market return
market_returns = [0.05, 0.03, -0.02, 0.08]
stock_returns = [0.06, 0.04, -0.01, 0.09]

model = LinearRegression()
model.fit(np.array(market_returns).reshape(-1, 1), stock_returns)
beta = model.coef_[0]
alpha = model.intercept_

print(f"Beta: {beta:.4f}")
print(f"Alpha: {alpha:.4f}")

R-squared

What it is: Proportion of variance explained

In Finance: How well model explains returns

Key Formulas for Quantitative Finance

Sharpe Ratio

Formula: SR = (Rₚ - Rₑ) / σₚ

  • Rₚ = Portfolio return
  • Rₑ = Risk-free rate
  • σₚ = Portfolio volatility

Value at Risk (VaR)

Formula: VaR = μ - z × σ

  • μ = Mean return
  • z = Z-score (e.g., 1.96 for 95% confidence)
  • σ = Standard deviation
# Calculate VaR
mean_return = 0.05
std_dev = 0.15
confidence = 0.95
z_score = stats.norm.ppf(1 - confidence)
var = mean_return - z_score * std_dev
print(f"95% VaR: {var:.4f}")

Practice Exercises

Exercise 1: Calculate Portfolio Return

Given: - Stock A: 40% weight, 10% return - Stock B: 35% weight, 5% return - Stock C: 25% weight, 8% return

Calculate portfolio return.

Exercise 2: Calculate Correlation

Calculate correlation between two stocks with returns: - Stock X: [0.05, 0.03, -0.02, 0.08] - Stock Y: [0.04, 0.02, -0.01, 0.07]

Exercise 3: Calculate Volatility

Calculate annualized volatility from daily returns: - Daily returns: [0.01, -0.02, 0.03, 0.01, -0.01, 0.02, -0.01]

Resources for Further Learning

  • Khan Academy: Free math courses
  • 3Blue1Brown: Visual math explanations
  • Coursera: Mathematics for Finance courses
  • Books: "Mathematics for Finance" by Capinski & Zastawniak

Key Takeaways: - Calculus: Derivatives measure sensitivity, integration for continuous models - Linear Algebra: Vectors and matrices for portfolio calculations - Statistics: Mean, variance, correlation for data analysis - Probability: Expected values and distributions - Time Series: Returns and volatility calculations

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