Basic, lesson 1 of 3
The greeks working together
Delta tells you what to hedge, gamma tells you how fast that changes, theta is what you pay for gamma, and vega is your bet on implied vol. Know the sign of each for every position.
10 min, 3 checks
1Delta is a hedge ratio
Delta is the number of shares an option behaves like. A desk adds up the delta of every option in a book, times the contract size, and trades shares or futures against the total. The book is delta neutral when that sum is zero.
| Position | Delta | Gamma | Theta | Vega |
|---|---|---|---|---|
| Long call | + | + | - | + |
| Short call | - | - | + | - |
| Long put | - | + | - | + |
| Short put | + | - | + | - |
Read the table in pairs. Anyone long an option, call or put, is long gamma and long vega and pays theta. Anyone short an option collects theta and is short gamma and vega. Only delta depends on whether it is a call or a put.
Worked example
Netting a small book
- 1Long 20 calls, delta 0.55, 100 shares each: 20 x 100 x 0.55 = +1,100 shares.
- 2Short 15 puts, delta -0.35: short a negative delta is positive, so -15 x 100 x -0.35 = +525 shares.
- 3Net delta: 1,100 + 525 = +1,625 shares.
The book behaves like owning 1,625 shares. To be delta neutral, sell 1,625 shares.
Check 1
You own 30 put contracts (100 shares each) with delta -0.25. How many shares do you buy to be delta neutral?
2Gamma and theta are one trade off
Gamma is the change in delta for a $1 move. A long gamma position gets longer as the stock rises and shorter as it falls, which is the shape you want. Theta is the rent for that shape. An option with a lot of gamma, at the money and close to expiry, also loses value fastest each day. You cannot have one without the other.
New delta is about Old delta + Gamma x Move
Price change with gamma included: Delta x Move + 0.5 x Gamma x Move^2.
Worked example
A call after a $3 move
- 1Call delta 0.50, gamma 0.04, stock rises $3.
- 2New delta: 0.50 + 0.04 x 3 = 0.62.
- 3Price change: 0.50 x 3 + 0.5 x 0.04 x 9 = 1.50 + 0.18 = $1.68.
Delta alone said $1.50. Gamma adds $0.18, and it adds that whichever way the stock moves, which is why long gamma is called being long convexity.
Check 2
A call has delta 0.30 and gamma 0.05. The stock rises $2. What is the new delta?
3Vega by strike and expiry
Vega is dollars per point of implied vol. It is largest at the money and grows with the square root of time, so a one year option has about twice the vega of a three month option at the same strike. Gamma does the opposite: it is concentrated in short dated options.
| Option | Gamma | Vega | Theta per day |
|---|---|---|---|
| 1 week, at the money | Very high | Low | High |
| 3 months, at the money | Moderate | Moderate | Moderate |
| 1 year, at the money | Low | High | Low |
| 3 months, far out of the money | Low | Low | Low |
This is why desks speak of gamma books and vega books. Short dated options are a bet on how much the stock moves this week; long dated options are a bet on where implied vol goes.
Check 3
You own 500 contracts (100 shares each) with vega 0.15 per vol point. Implied vol falls 2 points. How much do you lose, in dollars?
Terms in this lesson: Delta, Gamma, Implied volatility, Futures contract, Convexity.
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