Options and Derivatives · Lesson 5
The Greeks
How much an option's price moves when one input moves a little. Delta, gamma, theta and vega, each worked out by formula and checked by repricing, and what a hedged owner gains and pays each day.
Know the basics? Read the lesson notesOne euro up
The Black–Scholes lesson priced a made-up call: the right, but not the duty, to buy a share for €105 (the strike) on one day a year from now. The share costs €100 today, interest is 5% a year, and the share's volatility, how widely its yearly returns typically spread, is 17.42%. The Black–Scholes formula, which prices such a call from those five inputs, gives €6.94. Every number in this lesson is made up.
Now the share rises by €1, to €101, and nothing else changes. What is the call worth?
The formula gives €7.4850, about €0.55 more: the answer is B.
How much the call's price moves per euro of the share's move is its delta. Delta and its relatives each measure how much the price moves when one input moves a little and the others stay put. They are called the Greeks, after the Greek letters most of them are named with.
Delta, two ways
The first way needs no new formula, only the price. Bump and reprice: move the share a little each way, price the call again at each, and divide the change in price by the change in share price.
At €99 the formula gives €6.4157; at €101, €7.4850. How much does the price change per euro, to four decimals?
Hint: from €99 to €101 the share moves €2. Divide the change in price by 2.
The second way is a formula. Recall N(d1) from the Black–Scholes lesson: the bell curve's area to the left of d1, a number between 0 and 1. Delta = N(d1), the part of the Black–Scholes formula that the last lesson called the shares in the copy (the shares and loan that reproduce the call's payoff): 0.5347. The two agree, as they should. The copy has to move like the call, so it holds as many shares as the call moves per euro.
The call's price against the share's price (made-up inputs)
Dots: the prices at €100 minus and plus the bump. Dashed: the line through them, whose steepness is the bump's estimate of delta.
Drag the bump smaller. At €20 the line's steepness is 0.5178; from €0.50 down it reads 0.5347. Delta is the slope of the price curve at today's price: the steepness of the line that just touches the curve there.
Gamma: how fast delta moves
Delta is not fixed. Far below the strike the call is almost sure to pay nothing, and a euro on the share barely moves it: delta near 0. Far above, it moves almost euro for euro with the share: delta near 1. In between, delta climbs. How fast it climbs, the change in delta per euro of share price, is gamma.
Where is gamma largest?
Price and delta as the share's price moves (made-up inputs)
Top: the call's price, with the line that just touches it. Bottom: delta. Gamma is how steeply the bottom curve climbs.
Gamma peaks a little below the strike (here €95.6). At €100 it is 0.0228: one euro up should take delta to about 0.5347 + 0.0228 = 0.5575, and the formula's delta at €101 is 0.5573. Bump and reprice checks gamma the same way as delta, with delta in place of the price; the ledger shows both.
How far a slope reaches
Delta alone predicts a straight line: after a move of m euros, the price changes by about delta × m. But delta changes along the way, by gamma × m by the end of the move, so on average delta is off by half that. The correction is ½ × gamma × m².
For a €10 rise, the line gives 6.94 + 0.5347 × 10 = €12.29, and the correction is ½ × 0.0228 × 10² = 1.14. What is the estimate with both, in euros?
Hint: add the two: 12.29 + 1.14.
The formula gives €13.35 at €110: the line alone is €1.06 short, the line plus the correction €0.08 over.
The call's price after a move, and two estimates from today's Greeks (made-up inputs)
Orange: the formula's price. Dashed: delta's line. Grey: the line plus ½ × gamma × move².
Drag to a €20 fall. The line predicts 6.94 − 0.5347 × 20 = −€3.75, a price below 0, which no call can have, since the worst it can pay is nothing. With the correction, €0.81; the formula, €0.74. The Greeks are slopes at one point: good for small moves, worse for large ones, delta alone worst.
Theta: the cost of waiting
Now leave the share at €100 and let one day pass. The call has a day less in which to end above €105, and the formula gives €6.9232, against €6.9389 today (shown to four decimals): 0.0157 euros less. The change in price as time passes is theta. Here it is −€0.0157 a day, the same by the theta formula and by repricing a day later (to four decimals); per year, × 365, it is −€5.73.
Same share price, but a month left instead of a year. Does the call lose more or less a day?
The call's price as its last day comes closer (made-up inputs)
Orange: the price by share price. Dashed: what the call pays on its last day. Dot: the share at €100.
Drag the days down. The curve sinks toward the dashed payoff.
Vega: the price of uncertainty
The last input to move is the volatility. The change in price per percentage point of volatility is vega (not a Greek letter, despite the family name). By its formula vega is €0.3974 a point, and repricing at 17.43% and 17.41% agrees to four decimals. More volatility raises the price because the call gains from big rises and loses nothing extra from big falls.
If the volatility rose two points, to 19.42%, what would vega predict for the call's price, in euros?
Hint: today's €6.94 plus 2 × 0.3974.
The formula gives €7.73 too: near these inputs, the price moves almost in a straight line with the volatility.
For the precise: rho, the fifth Greek
Rho is the change in price per point of the interest rate. For this call it is about €0.4431 a point: higher interest shrinks PV(K), the strike's value today, which helps a call. With the rate written as in this course, PV(K) = K ÷ (1 + r)T, rho per 1.00 of r is T·PV(K)·N(d2) ÷ (1 + r); textbooks that write PV(K) = Ke−rT drop the ÷ (1 + r). Checked by bump and reprice in the lesson's script.
The Greeks of a European call
Each Greek is a slope of the Black–Scholes price: its change per small move of one input, with the others held fixed (in calculus, a partial derivative). Together they estimate the change in price when the inputs move a little. Below, φ(x) is the bell curve's height at x, as N(x) is its area to the left of x, and PV(K) is the strike's value today.
- Δ, Γ, Θ
- delta, gamma and theta of a European call (one that can be used only on its last day): delta per euro of share price, gamma per euro, theta per year (÷ 365 for a day)
- vega
- per 1.00 of volatility (÷ 100 for a percentage point)
- φ(x)
- the bell curve's height at x: e−x²/2 ÷ √(2π)
- rc
- the yearly rate r written as a continuous rate, ln(1 + r): 0.0488 for 5%
- m, t, s
- small changes: the share's move in euros, the years that pass (t in years), the change in volatility (s as a decimal: 0.02 for two points)
- S, K, T, σ
- the share's price today, the strike, the years left, the volatility
- d1, d2, N, PV(K)
- as in the Black–Scholes formula: d1 = [ln(S ÷ PV(K)) + ½σ²T] ÷ (σ√T), d2 = d1 − σ√T, PV(K) = K ÷ (1 + r)T
d1 = 0.0871, φ(0.0871) = 0.3974, N(d1) = 0.5347, N(d2) = 0.4653, PV(K) = 100.
Δ = 0.5347. Γ = 0.3974 ÷ (100 × 0.1742 × 1) = 0.0228. Vega = 100 × 0.3974 × 1 = 39.74, so €0.3974 a point (the same digits as φ(d1) only because S = 100 and T = 1 here). Θ = −100 × 0.3974 × 0.1742 ÷ 2 − 0.0488 × 100 × 0.4653 = −3.46 − 2.27 = −€5.73 a year, ÷ 365 = −€0.0157 a day. Each matches bump and reprice to four decimals.
- the Black–Scholes model holds: fixed, known volatility; a log price moving as a random walk (a running total of random up-or-down steps) with no jumps; free trading at one interest rate;
- the changes are small, for the estimate on the third line.
Moves are large (a €20 fall: delta's line said −€3.75, the formula €0.74), or the model's price itself is wrong: in a world with fatter tails (more very large moves than the bell curve allows), the Black–Scholes lesson found the formula's price off, and its slopes are slopes of that price. The lesson "Implied volatility and the smile" picks this up.
What is the call's gamma, to four decimals?
Hint: φ(d1) ÷ (S × σ × √T), with S = 110, σ = 0.2, T = 1.
Gamma and theta: two sides of one trade
A hedge is a holding that cancels a risk. An owner of the call can hedge it by selling 0.5347 shares (borrowed and sold, a short sale). For a small move, the shares then lose what the call gains, or gain what it loses, at the rate of delta. What is left is the correction from earlier, ½ × gamma × m², which is positive whichever way the share moves.
Waiting costs theta, €0.0157 a day. The owner's cash, €53.47 from the shares sold minus €6.94 for the call, is €46.53. At the continuous rate 4.88% (the theory card's rc) it earns €0.0062 a day, which leaves €0.0095 a day to pay. So the owner gains on a day when ½ × 0.0228 × m² is more than €0.0095. Setting them equal gives m² = 0.83, m = €0.91.
That is one day's standard deviation (one typical day's move) in the model: 100 × 0.1742 × √(1 ÷ 365) = €0.91. The match is exact in this model, because there the gain ½ × gamma × m² equals the day's cost exactly when m is one day's standard deviation. A bell curve puts about 68.3% of days within one standard deviation of the middle, and the rest outside it.
Draw 10,000 days as the model says, and reprice the hedged owner each day with the formula. On what share of days does the owner gain?
One hedged day, 10,000 simulated times (made-up inputs; first 1,000 days drawn)
Each dot: one day's share move and the hedged position's gain or loss, repriced with the formula, before costs. Curve: ½ × gamma × move² minus €0.0095. Black = day with a loss; orange = day with a gain.
In this simulation the owner gained on 3,166 of 10,000 days, 31.7%, and lost on 6,834. The average was −€0.0001 a day: about 0.
For the precise: the average's error bar
Two standard errors either side of the average (a standard error is how far an average of that many days typically lands from the long-run value): −€0.0003 to €0.0002, which contains 0.
Press Seller. Whoever sold the call and holds the shares against it has the opposite: small gains on about 2 days in 3, larger losses on the rest. Gamma is what the owner buys; theta is what the owner pays for it. In the model, on average, the two balance, before costs. The next lesson, "Delta hedging: where the P&L comes from", follows the hedge, and its profit and loss, through a whole year.
What the Greeks assume
The call is worth €6.94. What is wrong with the report?
Two more assumptions sit inside every Greek on this page. Each is a slope of the model's price, so it carries the model's world: fixed volatility, no jumps, free trading. Where the model's price is wrong, as for the far strike in the Black–Scholes lesson's fat-tailed world (one with more very large moves than the bell curve allows), its slopes are not the real ones either. And each Greek holds the other inputs still. When two move at once, the estimate adds their parts, which works only for small changes.
A put's delta
If the share rises by a small amount, the put's price falls. By how much per euro, to four decimals?
Hint: per euro on the share, C moves by its delta, S by 1, and PV(K) not at all.
A small move on a new call
The share rises €2. Using delta and gamma, what is the call's new price, in euros?
Hint: 4.80 + delta × 2 + ½ × gamma × 2².
Vega on a new call
If the volatility rises by one percentage point, to 21%, by how much does vega say the call's price rises, in euros (to four decimals)?
Hint: vega = S × φ(d1) × √T per 1.00 of volatility; divide by 100 for one point.
Two days in three
Judging by this lesson's simulation, what does winning on about two days in three show on its own?
Lesson done
In the Black–Scholes model, each Greek is a slope of an option's price: its change for a small move in one input, the others held fixed. For this lesson's call, delta was 0.5347, gamma 0.0228, theta −€0.0157 a day and vega €0.3974 a point, and each agreed with bump and reprice. An owner hedged with delta (who sells delta shares against the call) gains from moves (gamma) and pays for waiting (theta); in this lesson's simulation of the model it gained on 3,166 of 10,000 days and averaged about 0, before costs. The Greeks describe small changes inside the model; a large move or a world with fatter tails is outside what they promise.
Key ideas
- The Greeks
- measures of how much an option's price moves when one input moves a little and the others stay put
- Delta (Δ)
- the change in an option's price per euro of the share's price; for a European call in the Black–Scholes model, N(d1), the shares in the copy
- Gamma (Γ)
- the change in delta per euro of the share's price; for a European call in the Black–Scholes model, largest with the share near the strike (for this lesson's call, a little below it, at €95.6)
- Theta (Θ)
- the change in an option's price as time passes, the others fixed; for a European call in the Black–Scholes model, negative
- Vega
- the change in an option's price per point of volatility; for a European call in the Black–Scholes model, positive
- Bump and reprice
- estimating a Greek by moving one input a little, pricing again, and dividing the change in price by the move
- Hedge
- a holding that cancels a risk, such as selling delta shares against a call you own
- φ(x)
- the bell curve's height at x: e−x²/2 ÷ √(2π)
Δ = N(d1) · Γ = φ(d1) ÷ (Sσ√T) · vega = Sφ(d1)√T
Θ = −Sφ(d1)σ ÷ (2√T) − rc·PV(K)·N(d2)
change in price ≈ Δ·m + ½Γ·m² + Θ·t + vega·s, for small changes