DP
Binomial Treesadvanced

Delta Hedging

Delta hedging is the practice of holding Δ\Delta shares of the underlying for every option written, so that small movements in the stock cancel out at the portfolio level. Because Δ\Delta itself moves with SS, TT, and σ\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.

Why it matters

Picture a market maker who has just sold 100 ASX SPI 200 index calls. Each call has Δ=0.60\Delta = 0.60. To stay neutral the maker buys $100 \times 0.60 = 60$ 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.

Formulas

Delta-neutral portfolio
Π=−1 option+Δ⋅shares\Pi = -1 \text{ option} + \Delta \cdot \text{shares}
P&L decomposition (continuous time)
dΠ=Θ dt+12Γ (dS)2d\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
ϵ≈12Γ[(ΔS)2−σ2S2Δt]\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 Δ=0.6\Delta = 0.6, gamma Γ=0.04\Gamma = 0.04 per A$1.

Solution

Initial hedge: buy $100 \times 0.6 = 60$ shares. After a A$1 rise in the stock, delta climbs by about $0.04$ to $0.64$. The desk must buy $100 \times 0.04 = 4$ 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 12Γ[(ΔS)2−σ2S2Δt]\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 Δ\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 Δ\Delta shares per option to be delta-neutral
  • •Rebalance as delta drifts with SS, TT, σ\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 ΔMin Var. HR

In learning paths

Pricing Models Path

Sources

  1. 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.
  2. 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.
  3. 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
← Back to the atlasSee in the network →