Delta-Hedging Simulation for European Options

Overview

This Python project simulates a delta-hedging strategy for European options, a risk management technique commonly used by options traders to reduce exposure to price movements of the underlying asset. The goal is to analyze how the frequency of rebalancing affects the portfolio’s Profit and Loss (PnL) distribution. Using Monte Carlo simulations of the underlying asset’s Brownian motion, the hedge is dynamically adjusted along each path. The results highlight the trade-off between more frequent hedging for better risk control and the potential impact of higher transaction costs.

Requirements

• Python3
• Jupyter notebook

Usage

1. Clone the repository to your local machine:

   git clone https://github.com/Sassouly/Delta-dynamics.git

2. Navigate to the project directory:

   cd Delta-dynamics

Simulation Details

Underlying Model

The asset price follows a stochastic differential equation under the Black–Scholes framework: The asset price follows the stochastic differential equation $dS_t = \mu S_t dt + \sigma S_t dW_t$.

Simulation Parameters

• Initial price of the underlying asset
• Strike price
• Time to maturity
• Risk-free interest rate
• Volatility of the underlying asset
• Number of time steps
• Number of simulated price paths

Option Pricing and Greeks

At each time step, the Black–Scholes formula is used to compute the option price and its Greeks (Delta, Gamma, Vega, Theta, Rho). The Delta is particularly important as it determines the hedge ratio: the number of shares to hold in order to maintain a delta-neutral position.

Dynamic Hedging Process

• The underlying price paths are simulated using a vectorized approach for computational efficiency.
• The portfolio is rebalanced at each step to remain delta-neutral.
• At maturity, the PnL is calculated as the difference between the option payoff and the hedging portfolio’s final value.

Objective

The simulation illustrates how the frequency of rebalancing and the volatility of the underlying asset impact the effectiveness of the hedge and the resulting PnL distribution. It highlights the trade-off between hedging accuracy and transaction costs in practice.

Results

The simulation provides a quantitative illustration of how delta-hedging reduces risk exposure in option portfolios. As the hedging frequency increases, the variance of the PnL decreases significantly, showing a convergence toward zero. This behavior is consistent with the theoretical results derived from the Black–Scholes model, where continuous hedging would lead to a perfectly risk-neutral portfolio. Empirically, the simulation shows that:

• The mean PnL across all simulated paths is close to zero, confirming that delta-hedging neutralizes the directional exposure of the option.
• The variance of the PnL decreases approximately with the inverse square root of the hedging frequency.
• With more frequent hedging, the distribution of PnL becomes narrower and more symmetric, illustrating improved replication accuracy of the option payoff.
• Conversely, lower rebalancing frequency leads to larger deviations caused by unhedged Gamma risk between adjustments.

Contact

If you have any questions or suggestions, feel free to contact me:

Name: Assouly Sacha
LinkedIn: www.linkedin.com/in/sacha-assouly