Delta Hedging

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Overview

This notebook implements delta hedging strategies for European options, with detailed profit & loss (P&L) attribution and hedging error analysis.

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Features

1. Dynamic Delta Hedging Simulation

• Implements self-financing delta-neutral hedging strategy
• Simulates realistic stock price paths using Geometric Brownian Motion (GBM)
• Periodic rebalancing with configurable frequency (daily, weekly, etc.)
• Incorporates transaction costs to reflect real-world trading conditions

2. Detailed P&L Attribution

Decomposes hedging performance into four key components:

Component	Description	Impact
Delta P&L	First-order hedging effect	Linear with stock price changes
Gamma P&L	Hedging error from convexity	Captures second-order effects
Theta P&L	Time decay contribution	Reflects option’s time value erosion
Transaction Costs	Rebalancing friction	Real-world implementation cost

3. Hedging Error Analysis

• Quantifies relationship between rebalancing frequency and hedging accuracy
• Demonstrates the trade-off between hedging precision and transaction costs
• Statistical analysis across multiple market scenarios (100+ simulations)

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Technical Implementation

Core Components

1. Black-Scholes Framework: Option valuation and Greek calculations
2. Monte Carlo Engine: Stochastic stock price simulation
3. Portfolio Tracking: Real-time hedge ratio adjustment
4. Statistical Analysis: Multi-scenario performance assessment

Visualization Suite

• Stock price path and delta evolution
• Cumulative P&L attribution by component
• Hedging error vs. rebalancing frequency
• Distribution analysis of hedging outcomes

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Configuration

Key Parameters

Parameter	Type	Default	Description
S0	float	100.0	Initial stock price
K	float	100.0	Strike price
r	float	0.05	Risk-free interest rate (annual)
d	float	0.02	Dividend yield (annual)
sigma	float	0.2	Volatility (annual)
T	float	1.0	Time to maturity (years)
num_steps	int	50	Number of rebalancing steps
option_type	str	‘call’	‘call’ or ‘put’
transaction_cost	float	0.001	Transaction cost (fraction)
seed	int	None	Random seed for reproducibility

Customization

# Custom parameter set for high volatility scenario
high_vol_params = {
    'S0': 100.0,
    'K': 100.0,
    'r': 0.05,
    'd': 0.02,
    'sigma': 0.4,      # 40% volatility
    'T': 0.5,          # 6 months
    'num_steps': 100,  # Daily rebalancing
    'option_type': 'call',
    'transaction_cost': 0.002,  # 0.2% costs
    'seed': 123
}

results = delta_hedging_simulation(**high_vol_params)

Theory

Black-Scholes Model

The framework uses the Black-Scholes-Merton model for option pricing:

\[C(S,t) = S e^{-dT} N(d_1) - K e^{-rT} N(d_2)\]

Where:

\[d_1 = \frac{\ln(S/K) + (r - d + \sigma^2/2)T}{\sigma\sqrt{T}}\] \[d_2 = d_1 - \sigma\sqrt{T}\]

Delta Hedging Strategy

Objective: Construct a portfolio that replicates the option payoff

Portfolio Construction:

• Short 1 option (liability)
• Long Δ shares (hedge)
• Cash account (financing)

Rebalancing Rule:

\[\Delta_t = \frac{\partial C}{\partial S} = e^{-dT} N(d_1)\]

P&L Attribution

Total P&L Decomposition:

\[\text{PnL} = \Delta \cdot dS + \frac{1}{2}\Gamma \cdot (dS)^2 + \Theta \cdot dt - \text{Costs}\]

• Delta P&L: Linear hedge performance
• Gamma P&L: Convexity error (main source of hedging risk)
• Theta P&L: Time value decay
• Transaction Costs: Friction from rebalancing

Hedging Error

Discrete rebalancing introduces error that scales as:

\[\text{Error} \sim \sigma \sqrt{\Delta t}\]

Implication: More frequent rebalancing reduces error but increases costs.