Introduction

Pseudo-random number generation is a crucial concept in simulation and bootstrapping methods. In this chapter, we will explore the principles behind pseudo-random number generation, its importance in generating simulated data, and the methods used to achieve randomness within computational simulations.

Principles of Pseudo-Random Number Generation

Pseudo-random numbers are sequences of numbers that appear to be random but are generated through deterministic algorithms. Unlike true random numbers, which are derived from unpredictable physical processes, pseudo-random numbers are computed using mathematical formulas. These formulas generate numbers that mimic randomness but are reproducible given the same initial conditions. Pseudo-random number generation forms the basis for creating simulated data in various financial modeling scenarios.

Methods for Pseudo-Random Number Generation

There are several methods for generating pseudo-random numbers. One commonly used method is the Linear Congruential Generator (LCG). The LCG generates numbers using the formula:

$$X_{n+1}=\left(a X_n+c\right) \bmod m$$ Where:

• $X_{n+1}$ is the next pseudo-random number.
• $a$ is the multiplier.
• $c$ is the increment.
• $m$ is the modulus. While pseudo-random number generators provide a useful approximation of randomness, they have limitations. The sequence of numbers they generate eventually repeats, and they are sensitive to the initial seed value. This sensitivity can lead to patterns if not managed properly.

Importance in Simulation

Pseudo-random number generation plays a critical role in simulation and bootstrapping. Simulated financial scenarios often require generating large amounts of random data to represent uncertain variables, such as market prices, interest rates, or economic indicators. Pseudo-random numbers provide a practical way to generate these data points and simulate potential outcomes under various conditions.

Example: Consider a simulation that models the potential price movement of a stock over time. To simulate this, pseudo-random numbers can be used to generate daily price changes based on historical volatility. By repeatedly applying the generated changes to the initial stock price, a simulated price trajectory can be constructed.