The Greeks

Why this matters

The last three lessons each found that an option's price responds to something: the share price, the calendar, expected volatility. An options screen puts a number on each of those responses, labelled with a Greek letter. They look like jargon and they're really a compact answer to the practical question, "what happens to this option if...?" Bond investors already have the idea. Duration says how far a bond's price moves when yields change, and convexity says how duration itself changes. The Greeks do the same job for options, for more than one kind of change.

The Options Industry Council, the education arm of the options clearing house, calls delta, gamma, theta, vega, and rho the most commonly used Greeks, and describes all of them as theoretical estimates.¹ Keep both words in mind.

Delta: the share price

Delta is a theoretical estimate of how much an option's premium may change given a $1 move in the underlying share. An option with a delta of 0.50 can be expected to move about $0.50 for a $1 move, up or down. For an option that's been bought, delta is between 0 and 1 for a call and between 0 and −1 for a put.¹

Strike	Call premium	Call delta	Put premium	Put delta
$40	$10.80	0.91	$0.45	−0.09
$45	$6.85	0.76	$1.50	−0.24
$50	$3.90	0.56	$3.40	−0.44
$55	$2.00	0.36	$6.50	−0.64
$60	$0.95	0.20	$10.35	−0.80

All figures in this lesson come from a standard pricing model, rounded. A put's delta is negative because a put gains when the share falls.

Three readings of the same number are all useful.

• A rate of change. If STU rises $1, the $50 call should gain about $0.56, or $56 a contract. The $60 call gains about $20 and the $40 call about $91.
• An equivalent number of shares. One $50 call behaves, for small moves, like 56 shares of STU. Three $55 calls, at 0.36 each, behave like about 108 shares. This is the honest measure of how much exposure an options position carries.
• A rough guide to the odds. Delta approaches 1 as a call goes deeper in the money, because it's increasingly likely to finish there.¹ Delta is sometimes read as a rough stand-in for the chance of finishing in the money, so that 0.20 suggests about one in five. That's a rule of thumb, and it isn't what delta is defined as.

Gamma: how delta changes

Delta isn't fixed. Intrinsic Value, Extrinsic Value, and Moneyness showed a call gaining $3.30 on the first $5 rise in the share and $4.05 on the next. Gamma is how much delta is expected to change for a $1 move in the share.¹ The $50 call has a gamma of 0.043, so if STU rises from $50 to $51, its delta goes from 0.56 to about 0.60.

Move in STU	Delta alone	Delta and gamma	Model's actual change
Up $1	+$0.56	+$0.58	+$0.58
Up $5	+$2.79	+$3.33	+$3.28
Down $5	−$2.79	−$2.25	−$2.24

The gamma correction is half of gamma times the move squared: ½ × 0.043 × 5 × 5 = $0.54. It's added in both directions.

This is convexity again, and it works in the option buyer's favour as it did for the bondholder in Duration and Convexity. The call gains more than delta predicts when the share rises and loses less than delta predicts when it falls. Options that have been bought, calls or puts, always have positive gamma, and options that have been sold have negative gamma.¹ For a writer, the same curve bends the wrong way: losses speed up as the share moves against them.

The rest of this lesson is for members

Still to come in this lesson: Theta: the calendar, Vega: volatility, Rho: interest rates, How the Greeks change, The trade behind the letters, What the Greeks can't do, the key takeaways, and 2 quiz questions with explanations.

Membership opens every lesson in the curriculum, the capstone projects, the interactive tools, and the members-only channels in the Hyperion Entities Discord. The first lesson of every module stays free.

Become a member Sign in