Position delta is the profit or loss of the entire option position relative to a $1 change in the price of the underlying asset.
It is usually expressed in dollar form: a position delta (“PD”) of 20 on a stock option spread, for example, would mean that a $1 change in the underlying stock would result in a total loss of $20.
More formally it is the sum of the deltas of each leg of a spread, each multiplied by the number of contracts held, all multiplied by 100.
Let’s look at this in more detail.
What Is Delta?
Before considering position delta, let’s review the concept of delta.
This is one of the options greeks used to risk manage option positions and describes the effect on option prices of changes on the underlying asset price. Here’s a bit more on these option greeks:
Delta, then, is the change in option price resulting from a $1 change in the underlying stock or other asset.
Calculating Position Delta | Examples With Single Option Positions
So how is it calculated? Let’s build this up by way of a series of examples.
1 Long Call Option Over 100 Shares
Let’s look first, for simplicity, at 1 long call option – an option to buy 100 shares at a strike price of $80/share some time in the future when the underlying stock price is $100.
This is an in the money option with a positive delta of, say, 0.7. So a $1 increase on the underlying asset, to $101, would result in an increase of $0.70 per share in the option’s price.
Our option contract covers 100 shares and so the total dollar impact of a $1 change in the value of the stock would result in a 100x $0.70 = $70 profit.
So this option spread has a PD of 70.
Note that you can perform a similar calculation by considering the profit/loss if the stock price decreases by $1 to $99. In this case the resulting loss would be $70 as the short position has negative delta.
Therefore a profit results from an increase in stock price, and a loss from a decrease. As the profit and loss is ‘in the same direction’ as a rise and fall of the stock price the PD is positive.
10 Long Call Options
Let’s look at the same in the money options position, but this time where 10 contracts are held.
Using the same logic as above a $1 increase/decrease in stock price would result in a profit/loss of $70 per contract.
But because we now own 10 contracts, the total profit/loss would be $700.
Therefore this option position’s PD is 700.
1 Short Call Option Over 100 Shares:
This is an example of negative PD as the short call option has negative delta.
From the above calculations, and assuming we are still considering the same $80 strike price in the money option, a $1 increase/decrease in stock price would result in a loss/profit of $70 (ie an increase causes a loss and a decrease causes a profit).
Hence there is a negative position delta of -$70
(Using similar logic the PD of 10 of these short in the money call options is -$700).
So we can see that the position delta of a single option position is given by the following formula:
Calculating Position Delta | Multiple Option Legs
The basic principle behind calculating the net position delta of an option spread with multiple legs is to calculate the position delta of each leg and add them together.
Again, let’s look at some examples:
1 Out Of The Money Bull Call Spread
Suppose we purchase an out of the money bull call spread.
This consists of one long call with a lower strike price and one short call with a higher strike price. Both calls have the same underlying stock, the same expiration date and, in this case, are both out of the money.
So, with the stock price at $100, we may purchase a $110 call option (with delta of 0.4) and sell a $120 call option (with a delta of -0.3) – for a debit.
The position delta of the $110 long call option = 1 contract x 0.4 x 100 = $40
The position delta of the $120 short call option = 1 contract x -0.3 x 100 = -$30
Therefore the trader’s position delta = $40 – $30 = $10.
In other words for every $1 increase/decrease in stock price a trader makes a profit/loss of $10.
Covered Call Over 100 Shares
A covered call is a popular option strategy, particularly for owners of a share portfolio wishing to earn some additional income by selling call options against this underlying stock.
The spread usually involves a sold ‘at the money’ or ‘out of the money ‘ call option for every 100 shares owned.
So let’s say we own 100 shares (with a delta of 1). Hence
Position Delta of Shares = 1 * 100 = $100
And let’s say we sell an out of the money call option contract, with a delta of -0.4.
Hence the option’s position delta = 1 x -0.4 x 100 = -$40
So the net position delta = $100 – $40 = $60
Conclusion
Position delta is a useful measure that provides the dollar profit or loss of an options strategy resulting from a change in underlying stock price.
It can be easily calculated from the deltas of a spread’s constituent legs.
