Gamma, theta and vega
1. The idea
Delta is one of the Greeks: each says how an option’s price changes when one input changes and the others stay as they are. For a European option on a share without dividends:
- Gamma, \(\Gamma\), the change in delta when the share price rises by $1 (Level 4), the same for a call and a put: \(\Gamma = N'(d_1)/(S_0\sigma\sqrt{T})\).
- Theta, \(\Theta\), the change in the price as time passes, with nothing else changing; for a call, \(\Theta = -S_0N'(d_1)\sigma/(2\sqrt{T}) - rKe^{-rT}N(d_2)\) a year, divided by 365 for a day. An option loses time value as expiry comes closer, so theta is usually negative.
- Vega, the change in the price when the volatility rises by one percentage point: \(S_0\sqrt{T}\,N'(d_1)/100\), the same for a call and a put. A higher volatility makes every option worth more.
- Rho, the change in the price when the risk-free rate rises by one percentage point: for a call, \(KTe^{-rT}N(d_2)/100\).
Here \(N'(x) = e^{-x^2/2}/\sqrt{2\pi}\) is the height of the standard normal curve at \(x\), and the other symbols are those of the formula. For the six-month call of that lesson, with the share at $50 and a strike price of $52, gamma is 0.0376, theta −$0.0139 a day, vega $0.141 and rho $0.108.
The change in value. Over a short time \(\Delta t\) in which the share price changes by \(\Delta S\), a position’s value changes by about
\[ \Theta\,\Delta t + \Delta\,\Delta S + \tfrac{1}{2}\,\Gamma\,(\Delta S)^2, \]
where \(\Theta\), \(\Delta\) and \(\Gamma\) are the position’s theta, delta and gamma: \(\Delta\) is the change in its value, and \(\Gamma\) the change in \(\Delta\), when the share price rises by $1. \(\Delta S\) is in dollars, and \(\Delta t\) in the unit of theta, here days. Delta alone is the tangent of delta hedging; gamma bends it into a curve. In a delta-neutral position with a negative gamma, such as one of sold options, the gamma term, \(\tfrac{1}{2}\Gamma(\Delta S)^2\), is negative for a move either way. Sold options usually have a positive theta, so a day without a move adds to their value, though not always: a deep in-the-money European put can have a positive theta when bought, and so a negative one when sold.
Gamma-neutral. Shares have a delta of 1 and no gamma, so only options change a position’s gamma. To make a position with gamma \(\Gamma_P\) gamma-neutral, trade \(-\Gamma_P/\Gamma_T\) of a traded option with gamma \(\Gamma_T\): a positive number is bought, a negative one sold. That changes the delta too, so then trade shares to make the position delta-neutral as well. Options and shares are traded in whole numbers, so the position ends up gamma-neutral and delta-neutral as nearly as whole numbers allow.
2. Try it
If an answer is wrong, we will see what went wrong and how to fix it.
3. Delta and gamma on a chart
First, the call’s delta.
Now move the share price. The curve of delta and gamma follows the call’s price much further than delta’s tangent would.
4. One worked example, then your turn
In the worked example, we approximate a position’s change in value from its Greeks.
Now a position made gamma-neutral and then delta-neutral, with hints, one step at a time.
And one on your own: the gamma of a put.
5. Practice
Each question has new numbers. You master the skill when you get three right in a row, with at least one right answer for gamma, one for a change in value and one for a gamma-neutral hedge. You can always open the solution, but then that question does not count.
6. Coming back to it
When you have mastered a skill, it comes back on the start page for a short review after 1 day. After each right answer the next review comes later: after 3, 7, 14 and then 30 days. After a wrong one, the reviews start again at 1 day.
This skill: