Lesson 11 of 20 · 6 min
Delta as a rate of change
Delta is how much the option's price moves for a one-dollar move in the underlying, right now.
Delta is the first and most used of the Greeks, and its primary meaning is the simple one: it is the slope of the option's price against the underlying's price at this instant.
A delta of 0.533 means that if the underlying rises one dollar, the option's price rises about 53 cents. Calls have deltas between 0 and 1; puts have deltas between −1 and 0, because they move against the underlying. Multiply by the contract multiplier to get dollars: a 0.533 delta on one contract of 100 shares is about $53 per dollar of underlying move.
The word instant is doing work. Delta is a derivative, a slope measured at a point, and it is only accurate for a small move. In the base case, delta is 0.533 at $100, so a one-dollar rise predicts $0.53 of option gain; the actual gain is $0.56. The 3-cent gap is the curvature that lesson 13 measures. Over a five-dollar move the prediction would be off by far more.
Delta's second use is as an exposure measure. A 0.533 delta on one contract behaves, for small moves, like holding 53 shares. That is what makes it the natural unit for hedging, and it is why delta and the replicating package's share quantity are literally the same number — as lesson 8 showed, delta is N(d₁), the weight on the stock leg.
Delta varies systematically with moneyness. In the base case: the 95 call has delta 0.787, the 100 call 0.533, the 105 call 0.275 and the 110 call 0.106. Deep in the money it approaches 1; far out of the money it approaches 0.
Key terms
- Delta
- The change in an option's price per one-dollar change in the underlying. Between 0 and 1 for calls, −1 and 0 for puts.
- Share equivalent
- Delta times the contract multiplier. It converts an option position into the number of shares it currently behaves like.
- Local accuracy
- The property that delta predicts well for small moves and progressively worse for large ones, because the price relationship is curved.
Worked example
Delta's prediction against the actual price
Base case: underlying 100, strike 100, 30 days, rate 4%, volatility 25%. Call $3.02, delta 0.533.
- Delta's prediction for a $1 rise
- +$0.53
- Actual call price at $101
- $3.58
- Error over $1
- $0.03
- Delta's prediction for a $5 rise
- +$2.66
- Actual call price at $105
- $6.33
- Error over $5
- $0.65
A gain of $0.56.
A gain of $3.31.
Twenty times the one-dollar error.
Delta is nearly exact over a dollar and materially wrong over five. The error grows faster than the move does.
Check yourself
You hold three contracts of the base-case call with a delta of 0.533. The underlying rises $1. What is the approximate gain, and why only approximate?
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Educational only. Nothing in this lesson is investment, legal or tax advice, or a recommendation to enter into any transaction. Figures are illustrative and computed from the inputs stated in the lesson.