A one-step binomial tree
1. The idea
A binomial tree is a simple picture of how a share price moves: over one step, it goes from \(S_0\) today either up to \(S_u\) or down to \(S_d\). An option’s value today, \(V\), then follows in three steps.
The option’s payoff after each move: \(V_u\) after the up move and \(V_d\) after the down move. With \(S_T\) the share price at expiry and \(K\) the strike price, a call’s payoff is \(\max(S_T - K, 0)\) and a put’s \(\max(K - S_T, 0)\), as in Level 1. A cash-or-nothing call’s payoff is a fixed amount if \(S_T > K\), and a cash-or-nothing put’s if \(S_T < K\); otherwise it is zero. The options here are European: they can be exercised only at expiry.
The risk-neutral probability of an up move, for a share that pays no dividends, as here: the probability under which the share grows, on average, at the risk-free rate:
\[ p = \frac{e^{r\Delta t} - d}{u - d}, \]
with the up factor \(u = S_u/S_0\), the down factor \(d = S_d/S_0\), the risk-free rate \(r\) with continuous compounding, as a decimal, and the time step \(\Delta t\), the length of the step in years. On a one-step tree the step is the option’s whole life, so \(\Delta t = T\), the time to expiry.
The value today: the present value of the two payoffs averaged with \(p\) and \(1 - p\):
\[ V = e^{-r\Delta t}\,\big[\,p\,V_u + (1-p)\,V_d\,\big]. \]
An example. A share costs $75 today and will be $84 or $66 in four months; the risk-free rate is 3%. A call with a strike price of $78 has the payoff \(V_u = 6\) or \(V_d = 0\). With \(u = 84/75 = 1.12\), \(d = 66/75 = 0.88\) and \(\Delta t = 4/12\),
\[ p = \frac{e^{0.03 \times 4/12} - 0.88}{1.12 - 0.88} = 0.5419 \]
and \(V = e^{-0.03 \times 4/12} \times (0.5419 \times 6 + 0.4581 \times 0) = \$3.22\).
The hedge behind the price. Whoever sells an option can remove its risk by holding
\[ \Delta = \frac{V_u - V_d}{S_u - S_d} \]
shares per option, the option’s delta: here \((6 - 0)/(84 - 66) = \frac{1}{3}\) of a share. The position is then worth the same after either move, \(\frac{1}{3} \times 84 - 6 = \frac{1}{3} \times 66 = 22\): it carries no risk, so the real chance of an up move does not enter the price. For a put, \(\Delta\) is negative: the seller sells shares short.
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
First, a quick check of the put’s payoff.
Here is the question’s tree. Work out the payoffs, then today’s value, or step through it.
Now move the strike price. The higher it is, the larger the put’s payoff after a fall, and the more it is worth today.
4. One worked example, then your turn
In the worked example, we find the hedge for a call.
Now one with hints, one step at a time.
And one on your own: a cash-or-nothing call, which pays a fixed amount or nothing.
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 the hedge, for a call’s or a put’s value and for a cash-or-nothing option. 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: