Two-step binomial trees
1. The idea
A two-step tree uses the one-step calculation twice: we value the option one step before expiry, then work back to today. Each step multiplies the share price by \(u\) or by \(d\), so after two steps it is \(S_0u^2\), \(S_0ud\) or \(S_0d^2\). The option is then valued backwards:
- At expiry, the value at each node is the option’s payoff there.
- One step earlier, the value at each node is the discounted risk-neutral average of the two values that follow it:
\[ V = e^{-r\Delta t}\,\big[\,p\, V_u + (1-p)\, V_d\,\big], \qquad p = \frac{e^{(r-q)\Delta t} - d}{u - d}, \]
where \(V\) is the option’s value at a node, \(V_u\) and \(V_d\) its values at the two nodes one step later, after an up and a down move, \(r\) the risk-free rate and \(q\) the dividend yield, both with continuous compounding, as decimals, and \(\Delta t\) the time step: the time to expiry \(T\) divided by the number of steps \(n\), \(\Delta t = T/n\), in years. A six-month option on a two-step tree has \(\Delta t = 0.5/2 = 0.25\) years, three months; on a twelve-step tree, \(\Delta t = 0.5/12\) years, half a month.
- Repeat, one step at a time, until today.
The same three steps value a tree with any number of steps. A five-step or a twelve-step tree has more nodes and a shorter time step, \(\Delta t = T/n\), but each node is valued exactly as here, from the two nodes after it; the lesson on American options works through three steps, and the option calculator values American options on a tree of a thousand steps. The method also stays the same when the up and down moves come from a volatility, when the share pays a dividend yield, and for other payoffs. Each of these comes with its own example in section 4.
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
Every tree starts with \(p\). It is the same at every node, since \(u\), \(d\) and \(\Delta t\) are.
Here is the question’s tree. Before each step back, try one node yourself, or let the tree show it. Click any filled node to see how its value comes from the two after it.
Now move the strike price. The gap between the two lines is the call’s time value: its value today less its payoff if it expired now.
4. One worked example, then your turn
In the worked example, two things change. The up and down moves follow from the share’s volatility \(\sigma\), the standard deviation of its yearly return with continuous compounding, as a decimal: \(u = e^{\sigma\sqrt{\Delta t}}\) and \(d = 1/u\), the Cox-Ross-Rubinstein tree. And the share pays a dividend yield \(q\), which enters \(p\) and nothing else: the share grows by \(r - q\) on average, but the option is still discounted at \(r\), since it receives no dividends. For a stock index, \(q\) is its dividend yield; for a currency, \(q\) is the foreign interest rate.
The same put on trees with more steps: each is valued backwards in the same three steps, with the time step \(\Delta t = T/n\) and \(u = e^{\sigma\sqrt{\Delta t}}\). Move the number of steps: the value swings around one number and settles at it, the value of the Black-Scholes-Merton formula, which Level 7 teaches.
Other payoffs change only the values at expiry: \(\min(S_T, K)\), a fixed amount if the share ends above \(K\) (cash-or-nothing), or the share itself if it ends above \(K\) (asset-or-nothing). Now one with hints: a derivative whose payoff is the share price, capped at $75.
And one on your own: a cash-or-nothing 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 with the factors given, one with the factors from a volatility, and one with another payoff. 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: