Maths in Motion: The Quant's Guide to Finance…

Here are the main mathematical tools 🧰 and techniques forming the backbone of quantitative analysis in finance, helping professionals in the field to model, analyze, and make informed decisions on complex financial products and strategies.

1. Calculus: 
- Example: Calculating the delta of an option, which indicates how the price of the option changes concerning a change in the underlying stock price. Delta is the first derivative of the option's price with respect to the stock price.

2. Linear Algebra:
- Example: Portfolio optimization using Markowitz's Efficient Frontier, which relies on matrices to compute portfolio variances and covariances.

3. Probability and Statistics:
- Example: Value at Risk (VaR) is a statistical technique used to measure and quantify the level of financial risk within a firm or investment portfolio over a specified time frame.

4. Stochastic Calculus:
- Example: The Black-Scholes-Merton model, which prices European options by using a stochastic differential equation.

5. Partial Differential Equations (PDEs):
- Example: The heat equation used in finance to model how the prices of a financial derivative evolves over time.

6. Time Series Analysis:
- Example: Using ARIMA (Autoregressive Integrated Moving Average) models to forecast stock prices based on past price datas.

7. Numerical Methods:
- Example: Using the Monte Carlo simulation to estimate the price of an American option, which cannot be priced analytically like European options.

8. Optimization:
- Example: Determining the optimal weights of assets in a portfolio to achieve the highest expected return for a giving level of risk using the Sharpe Ratio.

9. Graph Theory:
- Example: Analyzing interbank lending to identify potential system-wide financial risks, where each bank is a node and lending relationships are edges.

10. Machine Learning and Data Science:
- Example: Utilizing neural networks to predict stock price movements based on various input features, such as past prices, trading volume, and other financial indicators.

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