Delta Hedging

Q: A delta-hedged portfolio must be rebalanced because:

Delta changes as the stock price moves. Delta is a function of S, T, and σ, so any stock movement changes Δ. The rate of change is gamma Gamma = ∂ Δ / ∂ S. Without rebalancing, the portfolio drifts away from neutrality and accumulates direction risk (Hull 2022, §19.4).

Delta hedging is the practice of holding $\Delta$ shares of the underlying for every option written, so that small movements in the stock cancel out at the portfolio level. Because $\Delta$ itself moves with $S$ , $T$ , and $\sigma$ , the hedge must be rebalanced as the market moves. In the Black–Scholes limit of continuous rebalancing with no frictions, delta hedging perfectly replicates the option payoff and the hedging cost equals the option price.

Try it yourself

Greeks explorer

Move the inputs and watch the five Black-Scholes Greeks update. Each Greek is a sensitivity of the option price: delta to the spot, gamma to delta itself, vega to volatility, theta to the passage of time, and rho to the rate. They all build on d₁ and d₂.

Curve Δ DeltaAt spot 0.5840

Δ DeltaN(d₁)0.5840

Γ Gammaφ(d₁)/(S·σ·√T)0.02758

𝒱 Vegaper 1.00 vol · per 1% = 0.275827.582

Θ Thetaper year · per day = −0.0208−7.587

ρ Rhoper 1.00 rate · per 1% = 0.258925.886

d₁0.2121

d₂0.0707

Vega and gamma are identical for a call and a put; delta, theta and rho differ by side.

Spot S100

Strike K100

Volatility σ20%

Maturity T0.50 yr

Rate r4.0%

Why it matters

Picture a market maker who has just sold 100 ASX SPI 200 index calls. Each call has $\Delta = 0.60$ . To stay neutral the maker buys 100 × 0.60 = 60 index-equivalent units. If the market rises and delta drifts to 0.70, the maker buys 10 more units. If the market falls and delta drops to 0.45, the maker sells 15 units. Buying high and selling low is the cost of being short gamma. The premium received compensates the maker for that systematic bleed.

Before you read on — recall

A delta-hedged portfolio must be rebalanced because:

Formulas

Delta-neutral portfolio

\Pi = -1 \text{ option} + \Delta \cdot \text{shares}

P&L decomposition (continuous time)

d\Pi = \Theta\, dt + \tfrac{1}{2}\Gamma\, (dS)^2

Stock-price risk is hedged. Residual P&L is time decay plus gamma times realised variance (Hull 2022, §19.4).

Hedging error per rebalance

\epsilon \approx \tfrac{1}{2}\Gamma\bigl[(\Delta S)^2 - \sigma^2 S^2 \Delta t\bigr]

Worked examples

Scenario

Wholesale desk shorts 100 ASX-listed equity calls each with $\Delta = 0.6$ , gamma $\Gamma = 0.04$ per A$1.

Solution

Initial hedge: buy 100 × 0.6 = 60 shares. After a A$1 rise in the stock, delta climbs by about 0.04 to 0.64. The desk must buy 100 × 0.04 = 4 more shares to restore neutrality. After a A$1 fall, delta drops by 0.04 and the desk sells 4 shares. Each round trip of buy-high-sell-low is the gamma cost that the option premium must cover.

Common mistakes

✗Delta hedging eliminates all risk. It does not. The portfolio is locally riskless only against small stock moves. Gaps, jumps, stochastic volatility, and transaction costs all leave residual P&L. Hull (2022) §19.4 shows the daily P&L is approximately $\tfrac{1}{2}\Gamma[(\Delta S)^2 - \sigma^2 S^2 \Delta t]$ , which is zero in expectation only if realised volatility equals the volatility used to compute $\Delta$ .
✗You hedge once and forget. Real desks rebalance continuously to daily depending on gamma, liquidity, and transaction costs. The optimal rebalance frequency is a trade-off between hedging error and trading cost (Whalley & Wilmott 1997).
✗Delta hedging makes the option seller risk-free. No, it makes them volatility-neutral on average, not in every state. The seller still loses if realised vol exceeds implied vol used to price and hedge.

Revision bullets

•Hold $\Delta$ shares per option to be delta-neutral
•Rebalance as delta drifts with $S$ , $T$ , $\sigma$
•Continuous-time replication is the basis of Black–Scholes
•Gamma cost = buying high, selling low between rebalances
•Residual risk from gaps, jumps, and transaction costs

Quick check

A delta-hedged portfolio must be rebalanced because:

Connected topics

Delta Δ The Greeks Min Var. HR

More in Binomial Trees

1-Step Tree 2-Step Tree Risk-Neutral p Backward Ind.Early Exercise Div. Trees

In learning paths

Pricing Models Path

Sources

Hull (2022), §19.4
Hull, John C. Options, Futures, and Other Derivatives. 11th ed. Pearson, 2022. ISBN 978-0-13-693997-9.
Decomposes delta-hedged P&L into theta and gamma terms and discusses rebalancing in practice.
Black & Scholes (1973)
Black, Fischer and Myron Scholes. "The Pricing of Options and Corporate Liabilities." Journal of Political Economy 81(3), 1973, pp. 637–654.
Foundational paper that derives the option price as the cost of dynamically delta-hedging the option in continuous time.
Hull & White (2017)
Hull, John C. and Alan White. "Optimal Delta Hedging for Options." Journal of Banking & Finance 82, 2017, pp. 180–190.
Empirical work showing that practical delta hedges benefit from a correction for the empirical relationship between implied vol and the underlying.

How to cite this page

Dr. Phil's Quant Lab. (2026). Delta Hedging. Derivatives Atlas. https://phucnguyenvan.com/concept/delta-hedging