Delta: the slope
How much an option moves when the stock moves $1, and why traders also read it as rough odds.
6 minute read
Delta tells you how much an option’s price moves when the stock moves $1.
Think of it as a speedometer. It doesn’t tell you where the option’s price is. It tells you how fast that price is moving compared with the stock.
Our $100 call, with the stock at $100, has a delta of 0.53. If the stock rises to $101, the call should gain about $0.53. The model prices it at $3.58, up $0.56 from $3.02. The few extra cents come from gamma, the subject of chapter 4.
Why it is called the slope
Draw the call’s value against the stock price and you get a smooth curve. Delta is the steepness of that curve at the current stock price.
The diagram shows it as a tangent: a straight line that touches the curve at one point and tilts exactly the way the curve tilts there. The little triangle scales the slope up to a $10 move so you can see it. Ten times 0.53 is $5.33.
Delta
0.53Call value
$3.02
- Value of the call today
- Tangent line (its slope is delta)
- Value at expiry
Drag across the chart, or focus it and use the arrow keys, to change the stock price.
If the stock rises $1 from $100.00, the call gains about $0.53 a share, or $53 a contract. One contract moves like owning 53 shares.
Delta and the odds
Close, but not the same. For a call, delta runs a little above the odds. The gap grows with more time and more volatility.
What the numbers mean
A call’s delta runs from 0 to 1. Far out of the money, the curve is almost flat. At $90 the delta is 0.08, so a $1 move in the stock barely registers. Deep in the money, the curve runs parallel to the stock. At $110 the delta is 0.92, and the call moves almost dollar for dollar. Near the strike, it sits close to one half.
A put gains when the stock falls, so its delta is negative, from 0 to −1. The $100 put has a delta of −0.47. A $1 rise in the stock takes about $0.47 off its price.
Delta also tells you how much stock a contract acts like. One contract covers 100 shares, so a delta of 0.53 moves like 53 shares. Traders use this to hedge, which means taking an opposite position to cancel out risk. Selling 53 shares against one of these calls leaves you roughly flat for small moves.
Delta as rough odds
Traders also read delta as the chance that the option finishes in the money. A call with a delta of 0.26 gets called a one-in-four shot. It is a handy shortcut, and it is close.
The model computes the actual chance separately. For that 0.26 delta call, the chance is 24%. For our at-the-money call it is 50% against a delta of 0.53. The two differ because delta also gives weight to how far past the strike the stock might go.
Delta does not stay fixed. Drag the stock price and the tangent tilts. How quickly it tilts is the next Greek, gamma.
Try it on real data
Compare the delta of a SPY strike near the stock price with one far out of the money. One moves almost dollar for dollar; the other barely moves.
See delta on SPY