Delta and gamma from a tree
1. The idea
A tree tells more than a price. On a share that pays no dividends, the number of shares that hedges the option over the next step, at any node, is its delta, as in the one-step tree:
\[ \Delta = \frac{V_u - V_d}{S_u - S_d}, \]
where \(V_u\) and \(V_d\) are the option’s values at the two nodes one step later, and \(S_u\) and \(S_d\) the share prices there. Someone who has sold the option holds \(\Delta\) shares per option: a call’s delta is between 0 and 1, so the hedge is a number of shares bought; a put’s is between −1 and 0, so the hedge is a number of shares sold short.
Delta is different at each node, so the hedge has to be changed after every step. How fast it changes is the option’s gamma, \(\Gamma\): the change in delta per $1 of share price. From a two-step tree,
\[ \Gamma = \frac{\Delta_u - \Delta_d}{\tfrac12\,(S_{uu} - S_{dd})}, \]
where \(\Delta_u\) and \(\Delta_d\) are the deltas at the nodes after one up and one down move, and \(S_{uu}\) and \(S_{dd}\) the highest and lowest share prices two steps ahead. Half the distance between them is the distance between the middles of the two price ranges over which \(\Delta_u\) and \(\Delta_d\) are measured.
Two facts connect this to what comes next. For European options on a share without dividends, a call’s delta minus the put’s delta is exactly 1 at every node, because a call less a put has the payoff \(S_T - K\), like a forward. And gamma is largest near the strike price close to expiry, where a small move decides whether the option ends in the money: that is where a hedge needs changing most often.
2. Try it
If an answer is wrong, we will see what went wrong and how to fix it.
3. The tree, node by node
Delta today needs the values after one step. What is the call worth after an up move?
Step through the tree, then click the two nodes after one step: their values over the change in the share price give delta.
Now move today’s share price. The call’s delta rises from 0 toward 1 as the share rises, the put’s from −1 toward 0, and the gap between them stays 1.
4. One worked example, then your turn
In the worked example, we find gamma for a call on a tree built from a volatility.
Now one with hints: delta of an American put, which uses the values that include early exercise.
And one on your own: 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 delta and one for gamma. 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: