Greek Hedging Code Explanation

The provided Python code conducts a comprehensive analysis of delta hedging strategies for options trading. It begins by importing necessary libraries and fetching historical stock price data using Yahoo Finance API. After pre-processing the data, including calculating returns and volatility, the code defines functions for implied volatility calculation and delta hedging simulation. Implied volatility is computed using the Black-Scholes model, providing insight into the market's expectation of future volatility. Next, the code simulates stock price movements over time using geometric Brownian motion, considering a specified number of simulations, and realized volatility. It then performs dynamic delta hedging simulation for each scenario. For each simulation, the code calculates delta values, determines adjustment actions, computes P&L components, and accumulates interest on adjustments. These steps allow for a comprehensive assessment of the effectiveness of dynamic delta hedging in managing risk associated with options trading. The results of the simulations are visualized through histograms, showing the distribution of total cashflows for dynamic delta hedging, static delta hedging, and no hedging strategies. The mean, 5th percentile, and 95th percentile values are marked on the plots to provide insights into the performance of each strategy. In summary, the code provides a robust framework for analyzing and optimizing options trading strategies. By employing quantitative techniques and simulation-based approaches, it offers valuable insights into the impact of delta hedging on option P&L and overall risk management. It serves as a powerful tool for traders and analysts seeking to enhance their understanding of options trading dynamics and optimize their hedging strategies in dynamic market conditions.

Delta Hedging

Delta hedging is a risk management strategy commonly used in options trading to reduce or eliminate the exposure to changes in the price of the underlying asset. The strategy involves adjusting the portfolio's delta, which represents the sensitivity of the option's price to changes in the underlying asset's price. By continuously adjusting the portfolio's delta to offset changes in the option's price due to changes in the underlying asset's price, delta hedging aims to maintain a neutral position, thereby minimizing the impact of price movements on the portfolio's value. Theory: Delta, denoted by Δ, measures the rate of change of the option price with respect to changes in the price of the underlying asset. It represents the slope of the option price curve relative to changes in the underlying asset's price. Delta can be positive or negative, depending on whether the option is a call or put and whether the position is long or short. The Black-Scholes model provides a formula for calculating the delta of a European call or put option: For a European call option: Δ=N(d1)Δ=N(d1) For a European put option: Δ=N(d1)−1Δ=N(d1)−1 Where: • N(⋅)N(⋅) is the cumulative distribution function of the standard normal distribution. • d1d1 is the standardized risk-adjusted term in the Black-Scholes formula, given by: d1=ln⁡(SK)+(r+σ22)TσTd1=σTln(KS)+(r+2σ2)T Mathematical Equations:

1. Black-Scholes Delta Formula: o For a European call option: Δcall=N(d1)Δcall=N(d1) o For a European put option: Δput=N(d1)−1Δput=N(d1)−1 Where N(⋅)N(⋅) is the cumulative distribution function of the standard normal distribution, and d1d1 is the standardized risk-adjusted term.
2. Standardized Risk-Adjusted Term d1d1: d1=ln⁡(SK)+(r+σ22)TσTd1=σTln(KS)+(r+2σ2)T Where: o SS is the current price of the underlying asset. o KK is the strike price of the option. o rr is the risk-free interest rate. o σσ is the volatility of the underlying asset. o TT is the time to expiration of the option. References:
3. Black, F., & Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81(3), 637–654. DOI: 10.1086/260062
4. Hull, J. C. (2018). Options, Futures, and Other Derivatives (10th Edition). Pearson.
5. McDonald, R. (2014). Derivatives Markets (3rd Edition). Pearson. Delta hedging involves adjusting the delta of the option position by buying or selling the underlying asset to maintain a delta-neutral position. This strategy helps traders manage the risk associated with changes in the underlying asset's price and is a fundamental aspect of options trading.

Code Methodology

The methodology employed in this code revolves around simulating and analyzing delta hedging strategies for options trading. Here's a breakdown of the methodology:

1. Data Preparation and Importing: o The code starts by importing necessary libraries such as datetime, pandas, numpy, yfinance, and matplotlib. o It downloads historical stock price data using Yahoo Finance API.
2. Implied Volatility Calculation: o Implied volatility is calculated for European options using the Black-Scholes model. o The implied volatility function iteratively calculates the volatility needed to match the market price of the option.
3. Simulating Stock Price Movements: o Geometric Brownian motion is employed to simulate stock price movements over time. o Multiple simulations are performed to generate different paths for the stock price.
4. Delta Hedging Simulation: o Delta hedging is simulated for each path of the stock price. o Delta of the option is calculated at each time step using the Black-Scholes model. o Total delta position and adjustments are computed to manage the delta risk. o Adjustment actions (buy or sell) and cash flows are determined based on the changes in delta. o Interest accrued on adjustments is also calculated.
5. Analysis of P&L Components: o Profit and loss (P&L) components are computed for the option and the underlying stock. o Components include original option P&L, original stock P&L, adjustment P&L, carry (interest) on options, carry (interest) on stock, and interest on adjustments.
6. Visualizing Results: o Histograms are plotted to visualize the distribution of total cashflows for different hedging strategies: dynamic delta hedging, static delta hedging, and no delta hedging. o Key statistics such as mean, 5th percentile, and 95th percentile are annotated on the plots to provide insights into the risk and profitability of each strategy.
7. Summary Statistics and Interpretation: o Summary statistics are calculated to analyze the overall performance of each hedging strategy. o The effectiveness of dynamic delta hedging compared to static delta hedging and no hedging strategies is evaluated based on the distribution of total cashflows. Overall, the methodology involves utilizing mathematical models (such as Black-Scholes and geometric Brownian motion) to simulate option pricing and stock price movements. It focuses on managing delta risk through dynamic hedging strategies and evaluates the impact of different hedging approaches on profitability and risk management.