Delta Hedging from zero
1/5Why hedge at all?
You sold a call. If spot goes up, you owe money. Delta tells you exactly how much exposure you have. Hedging means buying delta shares of the underlying to neutralize that exposure.
Say you sell a call with delta 0.50. For every $1 the underlying rises, your short call loses $0.50. So you buy 0.50 shares of the underlying. Now when spot moves up, your hedge gains roughly what your option loses. You are delta neutral.
The widget below shows this in action. Drag spot away from $100 and compare the naked short call P&L to the hedged P&L. The hedge is not perfect — the residual you see is the gamma error, and it is the subject of the rest of this lesson.
Notice: the hedged P&L is much smaller than the unhedged P&L, but it is not zero. For small moves the hedge works well. For large moves the residual grows — because delta itself has changed.
The gamma problem
Delta changes as spot moves. The rate of change is gamma. After a big move, your hedge is stale and you need to rebalance.
Think of delta as a speedometer and gamma as acceleration. When gamma is high (at-the-money, near expiry), your speedometer swings wildly with small price changes. When gamma is low (deep ITM/OTM), the speedometer barely moves.
Every time delta shifts, you trade to get back to neutral. This is the rehedge. Step through the simulator below to see how it works in practice: spot moves, delta shifts, you trade, and fees accumulate.
Watch the gamma column. When gamma is large, even a $5 move forces a meaningful rehedge. Each trade has a fee. Over the life of a 30-day option you might rehedge 20–40 times. Those fees add up.
The cost of hedging: ½Γ(ΔS)²
Each rebalance has a gamma P&L of ½Γ(ΔS)². But theta decays your short option position in your favor. The tradeoff between gamma cost and theta income is the heart of options market making.
When you are short gamma (sold options), every rehedge costs you money: you buy high and sell low as the market whipsaws. The formula for this cost per move is ½ × Γ × ΔS².
In exchange, your short option bleeds value every day via theta. That is your income. If the daily theta income exceeds the gamma cost from actual market moves, you profit.
(rebalance loss)
(daily decay)
Drag the move size slider. Small moves = theta wins. Large moves = gamma wins. The breakeven move is where the two bars are equal. This is why option sellers love quiet markets and fear volatile ones.
Hedge frequency matters
More frequent hedging keeps delta tight but racks up fees. Less frequent hedging saves on fees but lets the hedge drift, blowing up P&L variance.
There is no free lunch. Hedging every $1 move keeps you close to neutral, but you pay a fee on every trade. Hedging every $10 move is cheap, but between adjustments your exposure can swing violently.
The simulator below runs the same random price path with three different hedge thresholds. Click “New price path” multiple times to see how outcomes vary.
After a few paths, the pattern is clear: tight hedging has consistent (small) P&L. Wide hedging has wild swings — sometimes very profitable, sometimes a disaster. Most MMs hedge somewhere in between, balancing cost against variance.
Realized vs implied
This is the fundamental bet. If realized vol exceeds the implied vol you sold at, gamma cost exceeds theta income and you lose. If realized vol is lower, you win. Everything else is detail.
When you sell an option at 60% implied vol, you are betting that the underlying will actually move less than 60% annualized. Your theta income is calibrated to 60% vol. Your gamma costs depend on what actually happens.
Drag both sliders below. When implied vol exceeds realized vol, the green theta line stays above the red gamma line — the seller profits. Flip them and the seller bleeds.
This is why the volatility risk premium exists. Implied vol tends to exceed realized vol, compensating sellers for fees, slippage, gap risk, and the asymmetric pain of short gamma. The premium is not free money — it is compensation for real risk.
Where to go next:
Delta Hedging reference — the full page with all the details
Gamma — the Greek driving the need to rebalance
Vol regimes — when the volatility risk premium is wide vs thin