Understanding Big O Notation and Its Role in Algorithm Efficiency

Table of Links

Abstract and 1. Introduction

1.1 Option Pricing

1.2 Asymptotic Notation (Big O)

1.3 Finite Difference

1.4 The Black-Schole Model

1.5 Monte Carlo Simulation and Variance Reduction Techniques

1.6 Our Contribution

2. Literature Review
3. Methodology

3.1 Model Assumption

3.2 Theorems and Model Discussion

4. Result Analysis
5. Conclusion and References

1.2 ASYMPTOTIC NOTATION (𝐵𝑖𝑔 𝑂)

Big 𝑂 notation, denoted as 𝑂(𝑓(𝑛)), is a mathematical representation widely used in computer science to describe the upper bound or worst-case behavior of algorithms and functions as the input size, denoted as n, approaches infinity. In essence, it characterizes a function's growth rate or an algorithm's time complexity [3].

Formally, for a given function 𝑔(𝑛),𝑂(𝑔(𝑛)) , represents the set of functions for which there exists positive constants c and n₀ such that for all n greater than or equal to 𝑛0 , the function 𝑔(𝑛) is bounded above by 𝑐 times 𝑓(𝑛). Mathematically, it can be expressed as:

𝑂(𝑓(𝑛)) = { 𝑔(𝑛) ∶ ∃𝑐 > 0, ∃𝑛0 > 0, 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡

0 ≤ 𝑔(𝑛) ≤ 𝑐𝑓(𝑛) ∀ 𝑛 ≥ 𝑛0}

In simpler terms, if a function 𝑔(𝑛) can be bounded by a constant multiple of 𝑓(𝑛) for sufficiently large values of n, then 𝑔(𝑛) belongs to the set 𝑂(𝑓(𝑛)).

Big 𝑂 notation provides a concise way to analyze and compare the efficiency of algorithms, focusing on their scalability and performance characteristics without getting bogged down in specific implementation details. By understanding the asymptotic behavior of algorithms, developers can make informed decisions about algorithm selection and optimization strategies, crucial for designing efficient and scalable software systems.