Delta Hedging: How Market Makers Cover Directional Risk

Before this article: Delta Explained: The Directional Sensitivity of an Option.

The dealer's problem

When a dealer sells an option, they're left with directional exposure they don't want. If they sold calls, they lose money if the market rises. If they sold puts, they lose money if the market falls. Since their business isn't betting on direction, they neutralize that exposure by buying or selling the underlying asset. That process is called Delta Hedging.

The formula

The amount of the underlying a dealer needs to buy or sell to stay neutral is:

$$ Q = \Delta \times N \times M $$

Where:

$Q$ = quantity of the underlying to hedge (shares, index units, equivalent futures contracts).
$\Delta$ = the option's Delta.
$N$ = number of contracts sold.
$M$ = the contract multiplier (100 for standard stock/ETF/index options).

Key idea: this isn't a discretionary decision by the dealer. It's a mechanical formula. That's why the aggregate behavior of all dealers is, to a meaningful degree, predictable: it's the foundation of this entire method.

What the dealer buys or sells, by position

Dealer's position	What happens if price rises	How they hedge
Sold Calls (short calls)	Loses money on that position	Buys the underlying
Sold Puts (short puts)	Gains on that position (negative Delta shrinks)	Sells the underlying
Bought Calls (long calls)	Gains money	Sells the underlying to stay neutral
Bought Puts (long puts)	Loses money	Buys the underlying to stay neutral

The hedging loop

This isn't a one-time event: the dealer continuously rebalances their hedge as the price moves and Delta changes with it (you'll see exactly why in the next article, Gamma).

sequenceDiagram
    participant T as Trader
    participant D as Dealer
    participant M as Market (underlying/future)
    T->>D: Buys Calls
    D->>D: Calculates total Delta to hedge
    D->>M: Buys the underlying (Delta Hedging)
    M-->>D: Price moves
    D->>D: Delta changes with price
    D->>M: Rebalances the hedge
    Note over D,M: The cycle repeats<br/>all day long

Numeric example

Example: dealers, in aggregate, sold 10,000 option call contracts on the underlying with an average Delta of 0.40. Using the standard index option contract multiplier ($M = 100$):

$$Q = 0.40 \times 10{,}000 \times 100 = 400{,}000 \text{ index units}$$

That's the magnitude of directional exposure dealers need to neutralize by buying the equivalent underlying. Illustrative figure to show the mechanics.

Next step

You've seen that the dealer continuously rebalances their hedge. What's missing is why: how fast Delta changes as price moves is exactly what Gamma measures.

Continue with: Gamma Explained: The Rate of Change of Delta