Delta and delta hedging
1. The idea
Level 4 found an option’s delta on a tree: the number of shares that hedges one option over the next step. The formula gives delta at once, as the slope of the option’s price against the share price:
\[ \Delta_{\text{call}} = N(d_1), \qquad \Delta_{\text{put}} = N(d_1) - 1, \]
where \(d_1\) is that of the formula and \(N\) the cumulative standard normal distribution. A call’s delta is between 0 and 1, and a put’s between −1 and 0. With a dividend yield \(q\), both are multiplied by \(e^{-qT}\).
A position’s delta is the sum of its parts: each option counts with its delta, with a plus sign if it was bought and a minus sign if it was sold, and each share with a delta of 1. A position with a delta of zero is delta-neutral: a small move in the share price leaves its value about the same. A trader who has sold \(n\) calls makes the position delta-neutral by buying \(n\,\Delta\) shares; after selling puts, whose delta is negative, the trader sells shares short.
An example. For the six-month call of the formula’s lesson, with the share at $50 and a strike price of $52, \(d_1 = 0.0155\) and \(\Delta = N(0.0155) = 0.5062\). A trader who has sold 1,000 of these calls has a delta of \(-1{,}000 \times 0.5062 = -506.2\), and buys 506 shares.
Delta is only a tangent. It describes small moves; as the share price moves and time passes, delta changes, so the hedge has to be rebalanced. With options sold, a rise in the share price on its own makes the hedge buy shares and a fall makes it sell: rebalancing buys high and sells low, which costs money. Under the formula’s assumptions, with the hedge rebalanced all the time, the formula’s price is exactly what the hedge costs, financing included: that is why it is the option’s price. Rebalanced less often, the cost depends on the path of the share price and on how often the hedge is rebalanced.
2. Try it
If an answer is wrong, we will see what went wrong and how to fix it.
3. Delta on a chart
First, the delta of the call in the worked example below.
Now move the share price. The tangent at $80 has delta as its slope: near $80 it follows the call’s price, and further away it leaves it, which is why the hedge needs rebalancing.
4. One worked example, then your turn
In the worked example, a trader who has sold calls hedges with shares.
Now a hedge rebalanced a week later, with hints, one step at a time.
And one on your own: a hedge of puts bought.
5. Practice
Each question has new numbers. You master the skill when you get three right in a row, with at least one right answer for an option’s delta, one for a hedge and one for rebalancing a hedge. You can always open the solution, but then that question does not count.
6. Coming back to it
When you have mastered a skill, it comes back on the start page for a short review after 1 day. After each right answer the next review comes later: after 3, 7, 14 and then 30 days. After a wrong one, the reviews start again at 1 day.
This skill: