Delta Hedging
You sell someone a call. Now you have a problem: if the price goes up, you owe them money. How much money? That depends on delta.
Delta is the hedge ratio. A call with delta 0.5 means: for every dollar the underlying rises, the option gains 50 cents. So if you're short that call, you buy 0.5 units of the underlying. Price goes up, the option costs you more, but your hedge makes it back. You're flat.
That's the idea. In practice, it's much harder than it sounds.
Start simple: the one-time hedge
Imagine delta never changed. You'd sell the option, hedge once, and collect your premium at expiry. Drag the slider to see what this looks like:
At one specific price, the hedge is perfect. But watch what happens as you move the slider -- the required hedge changes dramatically, especially near the strike. A static hedge is only right for an instant.
The problem: delta moves
The moment the underlying moves, your hedge is wrong. The rate at which delta changes is gamma, and it's the central fact of delta hedging:
- Near the strike, gamma is high. A small move flips the hedge requirement. You're constantly adjusting.
- Far from the strike, gamma is low. The option is either deep in the money or out of the money, and delta barely budges. You can mostly leave it alone.
- Near expiry, gamma explodes. A 0DTE option near the strike might need re-hedging every few minutes. A 90-day option might go a whole day without needing an adjustment. See gamma and gamma exposure for more.
And price isn't the only thing that moves delta:
- Implied vol changes shift delta too. This is vanna. A sudden vol spike can force a hedge adjustment even if the underlying hasn't moved. In crypto, where IV swings 5-10 points in an hour, this matters.
- Time passing shifts delta. This is charm. Even in a perfectly flat market, your hedge drifts daily as the option decays. OTM options lose delta, ITM options gain it. See charm.
Every one of these shifts means the market maker has to trade: buy or sell perps to get back to neutral. Each adjustment is a real trade with real costs.
The cost of each hedge
Every re-hedge costs money in three ways:
- Trading fees -- 1-5 bps per trade. Sounds tiny. But over 30 hedge adjustments on a 30-day option, that's 30-150bps of cumulative fee drag on the notional.
- Spread crossing -- the theoretical hedge price is the mid-market. Your actual fill is on the bid or ask side. Every hedge trade gives up half the spread.
- Slippage -- if you're hedging size, you eat through the book. In fast markets, the price moves while you're trying to execute.
These costs are predictable and manageable in normal conditions. The real problem is what happens when conditions aren't normal.
How often to hedge
More frequent hedging keeps your delta tight but racks up fees. Less frequent hedging saves on fees but lets the hedge drift, blowing up your P&L variance:
Try the different market regimes. In a volatile market, hedging every 4 days is terrifying -- your delta drifts massively between adjustments. In a calm market, daily hedging is overkill and the fees destroy your edge.
The optimal frequency depends on the tradeoff:
Hedge more often
When the cost of NOT hedging is high
- High gamma (ATM, near expiry)
- High realized vol (big daily moves)
- Low trading fees (cheap to re-balance)
- Concentrated book (less natural netting)
Hedge less often
When the cost of hedging is high
- Low gamma (far OTM/ITM positions)
- Calm markets (small moves between adjustments)
- High fees (each trade eats into edge)
- Thin liquidity (your trades move the market)
Here's a concrete example of why this matters: an MM paying 2bps per hedge re-hedges 30 times on a 30-day option. That's 60bps of fee drag on the notional. If the option has 3% of edge (IV - realized), fees just ate 20% of the profit.
The book: why netting matters
Nobody hedges one option at a time. A real market maker runs a book -- hundreds of positions across different strikes and expiries. And in a book, deltas cancel:
Move the price around. Notice how individual positions have large deltas, but the net (what actually needs hedging) is much smaller. A short call and a short put at the same strike roughly cancel each other's delta. The MM only hedges the leftover.
This is why portfolio margin is so important. If the exchange recognizes that your book is internally hedged, you tie up less capital. If it margins each position in isolation, you're posting collateral against risk you don't actually have.
A well-diversified book might net 60-90% of gross delta internally. The hedge trades that hit the perp market are just the residual.
What actually kills the trade
Everything above is the normal cost of doing business. MMs can model fees, estimate gamma costs, and price their options accordingly. What they can't easily price is the tail:
Liquidity vanishes when you need it most. The biggest gamma-driven hedge adjustments happen during sharp moves -- exactly when order books thin out. Toggle between normal and crash conditions to see the difference:
Gap risk. Price can jump faster than any algorithm can hedge. Liquidation cascades in crypto can move price 10% in seconds with nothing tradeable in between. Your hedge was set at 69k and the next fill is at 62k. The October 2025 crash liquidated nineteen billion dollars in 24 hours -- many MMs couldn't re-hedge fast enough.
Basis blowout. The hedge instrument isn't the underlying -- it's a perp. In stressed markets, funding spikes, basis blows out, and the perp you're using as a hedge moves differently from what the option references. You're hedged on paper but exposed in practice.
Vol of vol. IV itself is volatile. A sudden repricing changes the option's value and its delta (via vanna) without any spot move. The vol of vol is the thing that makes options market making genuinely hard, not the basic gamma mechanics.
Why anyone does this
Given all of the above, why sell options at all? Because implied vol doesn't just equal realized vol. It exceeds it -- persistently. This is the volatility risk premium.
But the premium isn't free money. It's compensation for:
- Fee drag across dozens of trades
- Slippage on real execution
- The asymmetry of short gamma (small daily theta wins, occasional large losses on moves)
- Liquidity risk and gap risk in tail events
- All the second-order effects (vanna, basis, vol of vol) that make the real world messier than the model
An MM profits when IV exceeds realized vol by enough to cover all of these frictions. Not just IV > realized. IV > realized + fees + slippage + gap risk + everything else. Drag the sliders to see how the math works:
Try setting IV just slightly above realized vol and watch the P&L go negative from frictions alone. Then crank fees up to see how sensitive the trade is to execution costs. See vol regimes for more on when the premium is wide vs thin.
Simulate it
The simulator below runs a complete lifecycle: random price path, hedge adjustments on your schedule, fees on every trade, and a full P&L breakdown. Open the trade log to see every individual hedge.
Play with it:
- Crank fees to 5-10bps and watch them eat the entire edge
- Set hedge frequency to 5+ days and watch the P&L variance explode
- Compare 7-day vs 60-day DTE -- short-dated options have much more aggressive re-hedging
- Hit "New price path" a few times -- same parameters, wildly different outcomes. That's path dependency.
What this leaves out
This page uses simplified Black-Scholes math. In practice:
- MMs hedge to bands, not targets. They let delta drift within a tolerance, only re-hedging when it breaches a threshold. This is more realistic but makes P&L path-dependent and harder to model.
- Vega hedging creates additional flow not captured here. MMs hedge their vol exposure using other options or vol products.
- Skew and term structure affect hedging in ways a flat-vol model doesn't capture.
- Real P&L is path-dependent. Two paths with identical realized vol can produce very different hedging outcomes depending on the sequence of moves.
- This model doesn't simulate liquidity or gap risk -- the things that actually matter most in tail events.