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Michael Brenndoerfer

The Greeks and Option Risk Management › Delta Hedging and Delta-Neutral Portfolios

Michael BrenndoerferNovember 29, 20255 min read

Quantitative FinanceAll chapters

Part 5 of 15Stochastic Calculus and Pricing Theory

Chapter 31The Greeks and Option Risk Management › Delta Hedging and Delta-Neutral Portfolios

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Delta hedging is the practice of trading the underlying asset to offset the delta of an option position, creating a portfolio that is (to first order) immune to small price movements. This technique is fundamental to options market making and forms the basis of more sophisticated hedging strategies.

Constructing a Delta-Neutral PositionLink Copied

If you own a call option with $\Delta = 0.6$ , your portfolio gains $0.60 for every $1 increase in the stock. To neutralize this exposure, you short 0.6 shares. The combined position has:

\Delta_{\text{portfolio}} = \Delta_{\text{option}} - 0.6 \times \Delta_{\text{stock}} = 0.6 - 0.6 \times 1 = 0

where:

$\Delta_{\text{portfolio}}$ : net delta of the position
$\Delta_{\text{option}}$ : delta of the option
$\Delta_{\text{stock}}$ : delta of the stock (always 1)

A delta-neutral portfolio does not profit or lose from small movements in the underlying price. This allows you to isolate other sources of profit or risk, such as time decay or volatility changes. You typically maintain delta-neutral books so you can focus on earning the bid-ask spread without taking directional risk.

Dynamic HedgingLink Copied

The hedge is only valid at a point in time. As the stock price moves, delta changes (that's gamma), so you must continuously rebalance. In practice, you rehedge at discrete intervals, perhaps hourly or daily, or when delta drifts beyond a threshold. The frequency of rebalancing involves a trade-off: more frequent hedging keeps the portfolio closer to delta-neutral but incurs more transaction costs.

The following simulation demonstrates dynamic hedging. We generate a geometric Brownian motion price path and rebalance the hedge at each time step, tracking the cash flow and shares held to verify if the portfolio value tracks the option price:

In[22]:

Code

import numpy as np
import pandas as pd


def simulate_delta_hedge(S0, K, T, r, sigma, n_steps, seed=42):
    """Simulate delta hedging over the option's life."""
    np.random.seed(seed)

    dt = T / n_steps
    sqrt_dt = np.sqrt(dt)

    # Generate price path
    Z = np.random.standard_normal(n_steps)
    S = np.zeros(n_steps + 1)
    S[0] = S0

    for i in range(n_steps):
        S[i + 1] = S[i] * np.exp(
            (r - 0.5 * sigma**2) * dt + sigma * sqrt_dt * Z[i]
        )

    # Track hedge
    results = []
    shares_held = 0
    cash = 0

    for i in range(n_steps + 1):
        t_remaining = T - i * dt
        if t_remaining <= 0:
            t_remaining = 0.0001  # Avoid division by zero

        delta = black_scholes_delta(
            S[i], K, max(t_remaining, 0.0001), r, sigma, "call"
        )
        option_price = black_scholes_price(
            S[i], K, max(t_remaining, 0.0001), r, sigma, "call"
        )

        # Rebalance hedge
        trade = delta - shares_held
        shares_held = delta
        cash -= trade * S[i]  # Buy shares (cash outflow)
        cash *= np.exp(r * dt)  # Earn interest on cash

        hedge_portfolio_value = shares_held * S[i] + cash

        results.append(
            {
                "step": i,
                "time": i * dt,
                "stock_price": S[i],
                "delta": delta,
                "option_price": option_price,
                "shares_held": shares_held,
                "hedge_value": hedge_portfolio_value,
            }
        )

    return pd.DataFrame(results)

In[23]:

Code

# Simulation parameters
S0 = 100
K = 100
T = 0.25
r = 0.05
sigma = 0.20
n_steps = 63

# Run simulation
hedge_df = simulate_delta_hedge(S0, K, T, r, sigma, n_steps)

Out[24]:

Visualization

Line chart of one simulated stock-price path over a quarter, with a horizontal reference at the $100 strike. — Simulated delta hedging stock price path. The underlying asset follows a geometric Brownian motion, illustrating the stochastic nature of the price process against which the option is hedged.

Out[25]:

Visualization

Line chart of call delta over a quarter, fluctuating with the simulated stock path and falling toward zero below strike. — Evolution of call delta (shares held) during the simulation. Delta rises while the stock is above strike and falls toward zero as the simulated price moves below strike, requiring continuous hedge rebalancing.

Hedging Costs and GammaLink Copied

Delta hedging is not free. Each rebalancing trade incurs transaction costs (bid-ask spread, commissions). The hedging process also has an expected cost related to gamma and realized volatility.

When you're short gamma (short options), you systematically buy high and sell low: you buy more shares after the stock rises and sell shares after the stock falls. This "buying high, selling low" pattern loses money on average when volatility is positive, which is the theta you earn for being short options. You earn theta precisely because you must engage in this unfavorable trading pattern to maintain your hedges.

Conversely, when you're long gamma, you sell high and buy low during rebalancing, capturing the realized volatility. This is the core mechanism behind volatility arbitrage strategies. If you buy an option at an implied volatility of 20% and the stock realizes 25% volatility, your gains from delta hedging will exceed your losses from theta, generating a profit.