Delta hedging — concept
About this app
The writer neutralizes price risk: price path with rebalance points on top, below the P/L scissors — unhedged swings wildly, hedged hugs zero, tighter with every increase in rebalancing frequency. Debunks “option sellers bet against the client” and “hedging once is enough”.
The idea
Selling an option exposes the writer to the price risk of the underlying. With delta hedging the writer holds Δ shares against it, where Δ is the sensitivity of the option value to the price. For small moves, option and shares then offset each other. Because Δ changes with price and time, the share position has to be adjusted regularly (rebalancing), with the remaining money earning interest in a cash account. The more often you rebalance, the closer the portfolio replicates the option. A market maker earns the premium and the bid-ask spread without betting on direction.
What you see
A bank sells one call at its Black-Scholes price C₀ (rate 3 %, maturity one year) and hedges it. The top chart shows a random price path with the strike line and markers at each rebalancing point. Below are two P&L lines for the writer: the naked short position, which swings with the price, and the hedged position of Δ shares plus cash. The values panel lists premium, final price, both results, current Δ and the number of rebalancings.
Try it
Set the rebalancing frequency to N = 1: the hedged P&L swings noticeably again, because Δ drifts over the year. Raise N step by step to 50 and draw several paths with “new path”: the hedged line stays ever closer to zero while the unhedged one keeps fluctuating. Change strike and volatility and check whether the pattern holds.
Frequently asked questions
- What is delta hedging?
- Delta hedging protects an option position against small moves in the underlying by holding Δ units of the underlying in the opposite direction. Δ is the derivative of the option value with respect to the price. Since Δ changes all the time, the hedge has to be adjusted regularly.
- Do option sellers bet against their clients?
- A market maker who sells options usually hedges the price risk with delta hedging. Its result then depends very little on which way the price moves. It earns the premium and the bid-ask spread, not the client’s loss.
- Why is hedging once not enough?
- An option’s delta depends on the price and the time to expiry, so it keeps changing. A hedge set up once no longer fits after larger price moves. Only frequent rebalancing keeps the position close to risk-free.
- What can I change in this app?
- You set the rebalancing frequency N from 1 to 50, the strike K and the volatility σ, and generate a fresh random path with “new path”. The app compares the P&L of the unhedged and the hedged short position.
Subject: Financial derivatives
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