Let's explore one of the most fundamental concepts of option risk management.

Delta hedging.

Delta hedging is the technique traders use to minimize directional exposure to the underlying price.

The idea is bring the option portfolio's delta to zero or to any other desired targets by trading the underlying or a close proxy, like a future or ETF directly.

This is standard practice among market makers because it quickly eliminates a significant source of market risk.

Let's look at an example.

Imagine you've bought 100 call options each with a delta of 0.5.

That means the total position has a delta of plus 50.

To offset this, you'd need to trade 50 units of the underlying, in this case, 50 shares of the stock.

Assuming each option controls one share.

Similarly, if each option was a 100 share contract, you'd need to trade 5,000 shares.

The only question left is whether to buy or sell.

A long call gives you long exposure to the assets.

So to neutralize that long delta, you need to sell the stock short.

Now, let's see how this works. In practice.

Suppose you've put on the hedge and then the stock price drops by $1.

As the calls lose value, the option premium falls by 50 cents per contract across 100 calls.

That's a loss of $50 or $5,000 worth of 100 share contracts.

But the 50 shares you sold short, now trade $1 lower, so your hedge gains $50.

Ignoring costs and noting that Greeks are local measures, meaning they describe sensitivity for only small immediate moves, the net effect is approximately zero for that small price change.

After the move, delta will change and you'd typically need to reh.

Of course, in reality, there are some practical considerations.

For example, it's important to ensure that hedge amounts do not exceed a tradable size.

If instead of 100 calls you had 100,000 calls, the hedge would require selling 50,000 shares.

That raises the question, can you execute this trade quickly without moving the market? The liquidity is always a key consideration in Delta hedging, and there's one more point.

Sometimes both counterparties to the option trade want to hedge their delta.

For example, in our long call scenario, the call seller has a short delta position and may want to hedge this by buying stock.

In such cases, both sides can agree to a delta exchange as part of the trade.

Instead of hedging in the market and dealing with market impact slippage and costs, they provide the hedge directly to each other.

This option is then executed delta neutral.

So far we've talked about Delta in terms of a single option, but in practice, traders rarely just look at one contract.

They think in terms of positions and portfolios.

That's why it helps to distinguish between three different levels of delta.

First option delta.

This is the sensitivity of a single options premium to a change in the underlying asset price.

Second option position delta.

This shows the profit or loss impact of the total position in a single option and is calculated by multiplying the option delta by the number of contracts and if applicable by the contract size.

For example, if each contract represents 100 shares.

And third, the total position delta.

This is the delta of the whole portfolio portfolio.

It includes all option positions and any delta hedges, for example, stock or futures.

So it tells us the net sensitivity to moves in the underlying.

Let's go through the example shown here.

A trader is short 1000 at the money call options on a hundred shares each.

The delta is approximately plus 0.5 per option, as is typically the case for at the money options.

A short call has negative position delta.

So option position delta equals 1000 options times by 100 shares per contract times by the option delta of minus 0.5, giving minus 50,000 share equivalent to hedge part of that exposure, the trader has bought 45,000 shares.

Adding this hedge to the option position delta gives a total position delta of minus 50,000, plus 45,000 equals minus 5,000.

So what does this mean? If the stock price goes up by one, the overall portfolio would lose about IE minus 5,000 share equivalents times by $1.

This illustrates the key points when managing an options book.

You don't just look at the delta of one option in isolation.

You roll everything up into position deltas, and ultimately into the total delta of the portfolio.

That's the number that really drives risk management decisions.