Multi-Period Binomial Tree Option Pricing with Backward Induction

Understand Multi-Period Binomial Trees

Key rules to remember

Risk-neutral probability

q = (e^(rΔt) − d) ÷ (u − d)

Use (1 + i − d) ÷ (u − d) if the rate i is per step and not continuously compounded. Here r is the continuously compounded risk-free rate and Δt is the step length.

No-arbitrage condition

d < e^(rΔt) < u

This is the same as 0 < q < 1. If it fails, there is an arbitrage and the model is invalid.

Node value (backward induction)

V = e^(−rΔt) × [q × V_up + (1 − q) × V_down]

Applied at every node. The discount is for one step only.

Terminal share prices

S(n, j) = S0 × u^j × d^(n − j)

j is the number of up moves out of n steps. This holds for a recombining tree.

European option, n-step closed form

V0 = e^(−rnΔt) × Σ [n! ÷ (j!(n − j)!)] × q^j × (1 − q)^(n − j) × payoff(S0 × u^j × d^(n − j))

Sum over j = 0 to n. Discount once over the whole term. Valid for European options only.

Replicating shares at a node

Δ = (V_up − V_down) ÷ (S_up − S_down)

Number of shares held over the next step. It changes from node to node.

How to solve Multi-Period Binomial Trees questions

1. 1Write down S0, u, d, the strike K, the number of steps n, and the risk-free rate per step. Check whether the rate is continuous or effective, because this changes the formula for q.
2. 2Calculate q. Check that 0 < q < 1. If not, say there is an arbitrage.
3. 3Draw the tree and label each node with its share price. Use S0·u^j·d^(n−j) to avoid errors.
4. 4Write the option payoff at each terminal node, for example max(S − K, 0) for a call or max(K − S, 0) for a put.
5. 5Step back one period at a time. At each node use V = discount × [q × V_up + (1 − q) × V_down]. Keep at least four decimal places in the working.
6. 6Continue until you reach the single node at time 0. That value is the option price.
7. 7Check your answer. Use the closed form with binomial weights, or put-call parity if you priced both a call and a put. For a European call the price must be at least max(S0 − K·discount factor, 0).
8. 8If asked, find the hedge ratio at a node using (V_up − V_down) ÷ (S_up − S_down) and state the borrowing or lending needed.

Quickest way: Collapse the tree with binomial weights

Common mistakes in Multi-Period Binomial Trees

• Using the real-world probability of an up move instead of q

  The question gives you a probability of an up move and it seems natural to use it.

  Fix: Always price with the risk-neutral q from the no-arbitrage formula. Real-world probabilities do not affect the price.

• Using the wrong formula for q

  Mixing the continuous form e^(rΔt) with an effective rate, or forgetting that Δt is the step length and not the whole term.

  Fix: Check the rate type first. For a rate per step i, use 1 + i. For continuous rate r, use e^(rΔt). Compute Δt as total time divided by steps.

• Discounting for the whole term at every node

  Students forget that each backward step discounts for one period only.

  Fix: Discount one step at each node. Only in the closed form do you discount once for the total n steps.

• Counting 2^n separate final prices

  Students draw the tree as if up-down and down-up were different outcomes.

  Fix: In a recombining tree ud = du. There are n + 1 final prices. Remember the weight 2q(1 − q) for the middle node in two steps, since two paths lead to it.

• Not checking 0 < q < 1

  Students go straight to the calculation.

  Fix: Always check d < growth factor < u. A q outside (0, 1) signals an arbitrage or an input error.

• Using the European backward induction for an American option

  The same tree is used, so students forget the extra test.

  Fix: For American options compare the continuation value with the exercise value at every node and keep the larger. For European options there is no such test.

Worked examples