Enhancing Hedge Error Approximations in the Black-Scholes Model

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

3. METHODOLOGY

3.1 MODEL ASSUMPTION

Let us consider a set containing n number of securities 𝑋𝑖 , 𝑖 = 1,2,3, … , 𝑛 which follows the multivariate continuous time

In a generalized way, we can write equation (4) as

With 𝜑 = (𝜑1 ,𝜑2 , … ,𝜑𝑛)′. We are able to identify in closed form the trade-off between hedging idiosyncratic versus systematic option risk at the portfolio level by concentrating just on one systematic risk element. Now we can rewrite (5) for the ith security as

However, the normal Black-Scholes hedge is no longer perfect in discrete time; that is, the expected return of the hedge portfolio no longer vanishes nearly absolutely but merely in expectation. In this study, we demonstrate that a smaller hedge error variance in discrete time can be obtained with various hedge portfolios [17]. Instead of concentrating on the linear exposure to overall risk, that is, systematic plus idiosyncratic risk, these alternative hedge portfolios highlight the higher-order exposure to the systematic risk element.

Let us discuss another mathematical assumption based on Finite difference concept [18]. Finite difference methods approximate derivatives of functions by expressing them as weighted combinations of function values at nearby points. One way to derive finite difference formulas is through Taylor series expansion.

Consider a function 𝑓(𝑥) that is sufficiently smooth, such that it has continuous derivatives up to some order in a neighborhood of a point 𝑥 [19]. The Taylor series expansion of 𝑓(𝑥) about 𝑥 is:

Now, let's focus on approximating the first derivative 𝑓′(𝑥). Subtracting 𝑓(𝑥) from both sides of the Taylor series expansion gives:

If we solve this equation (9) for 𝑓′(𝑥), we get:

From the concept of asymptotic notation, we can write equation (11) as:

This is a finite difference approximation for the first derivative of 𝑓(𝑥). The error decreases as Δ𝑥 decreases, with higher-order terms becoming relatively less significant.

Similarly, higher-order derivatives can be approximated using finite differences derived from Taylor series expansions [20]. These finite difference formulas provide a way to numerically approximate derivatives of functions, which is fundamental in many areas of applied mathematics and computational science [21].