Australian National University · FACULTY OF BUSINESS & ECONOMICS

FINM6041 · Applied Derivatives

Q: How many steps can a binomial tree have in the FINM6041 exam?

A maximum of 4 steps — the lecturer states this explicitly in the exam brief, so you will never have to grind out a huge tree by hand. The skill being tested is the method (compute p, build the tree, roll back, check early exercise), not arithmetic stamina.

Q: Is the risk-neutral probability p a real probability?

No. It is a pricing construct — the probability under which the stock earns the risk-free rate — sometimes called a pseudo, risk-adjusted or martingale probability. The actual real-world chance of an up-move does not enter the option price at all, because the option is valued relative to the stock through a riskless hedge.

Q: When do I need to check for early exercise?

For American options, at every non-terminal node — including today's root. Compare the rollback value with the immediate-exercise payoff and keep the larger. For European options you never check; you just roll back. A common trap is a deep in-the-money American put whose early-exercise value beats continuation at a down node.

Q: How do I choose u and d?

If the question gives volatility σ per year, use the Cox-Ross-Rubinstein formulas u = e^{σ√Δt} and d = 1/u. If instead it says the stock goes up or down by a stated percentage, use those factors directly. For a dividend-yield underlying use p = (e^{(r−q)Δt} − d)/(u − d), and for a futures underlying p = (1 − d)/(u − d).

Q: How does the binomial model relate to Black-Scholes?

They are the same idea at different resolutions. As you add steps, the binomial distribution of the terminal price approaches the lognormal, and the binomial price converges to the Black-Scholes-Merton value. Binomial is the tool of choice when no closed form exists — American and path-dependent options.

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Chapter 5 of 11 · FINM6041

The Binomial Option-Pricing Model

The Binomial Option-Pricing Model (Lecture 5) is the course's first real pricing engine and the intuition-builder for everything after it. The key idea is a delta-hedged riskless portfolio: combine the option with Δ = (f_u − f_d)/(S_u − S_d) shares so the position is riskless, and it must therefore earn the risk-free rate — which lets you back out the option's value with no reference to anyone's expected return. That logic collapses into risk-neutral valuation: price the option as f = e^{−rΔt}[p·f_u + (1 − p)·f_d] using the risk-neutral probability p = (e^{rΔt} − d)/(u − d). You extend it to multi-step trees by building the price tree forward and rolling back node by node, choose u and d from volatility via u = e^{σ√Δt} and d = 1/u, and handle American options by taking max(rollback value, immediate exercise) at every node. In the exam, binomial trees are capped at 4 steps.

In this chapter

What this chapter covers

01The delta-hedged riskless portfolio: Δ = (f_u − f_d)/(S_u − S_d)
02Risk-neutral valuation: f = e^{−rΔt}[p·f_u + (1 − p)·f_d]
03The risk-neutral probability p = (e^{rΔt} − d)/(u − d) — and why it is not a real-world probability
04Building the price tree forward, then rolling back node by node
05American options: max(rollback value, immediate-exercise payoff) at every node, including the root
06Choosing u and d: u = e^{σ√Δt}, d = 1/u when σ is given; use stated up/down moves otherwise
07Adjustments for other underlyings: dividend yield p = (e^{(r−q)Δt} − d)/(u − d), futures p = (1 − d)/(u − d)
08The 4-step exam cap and convergence to Black-Scholes as steps increase

Worked example · free

Two-step American put — checking early exercise at every node

Q [12 marks]. A stock trades at S₀ = $45. Over each 1-year step it either rises by a factor u = 1.25 or falls by a factor d = 0.80; the continuously compounded rate is r = 4% p.a. Value a 2-year American put with strike X = $46 using a two-step binomial tree, and compare it with the otherwise-identical European put.

2 marksRisk-neutral probability: p = (e^{rΔt} − d)/(u − d) = (e^{0.04} − 0.80)/(1.25 − 0.80) = (1.040811 − 0.80)/0.45 = 0.5351. Build the price tree: after step 1, 56.25 (up) or 36 (down); terminal nodes 70.3125 (uu), 45 (ud) and 28.80 (dd).
2 marksTerminal put payoffs, max(X − S_T, 0): f_uu = max(46 − 70.3125, 0) = 0; f_ud = max(46 − 45, 0) = 1; f_dd = max(46 − 28.80, 0) = 17.20.
3 marksUp node: rollback = e^{−0.04}[0.5351·0 + 0.4649·1] = 0.4466; early exercise = max(46 − 56.25, 0) = 0. Take the larger → 0.4466.
3 marksDown node: rollback = e^{−0.04}[0.5351·1 + 0.4649·17.20] = 8.1963; early exercise = max(46 − 36, 0) = 10. Since 10 > 8.1963, exercise early → value 10.
2 marksRoot: e^{−0.04}[0.5351·0.4466 + 0.4649·10] = $4.70; early exercise at the root = max(46 − 45, 0) = 1 < 4.70, so hold. The European put (rollback only) is worth $3.89, so the early-exercise right is worth about $0.81 — nearly all of it created at the deep in-the-money down node.

The American put is worth $4.70, versus $3.89 for the European put. The extra $0.81 comes from exercising early at the down node, where the immediate payoff of $10 beats the $8.20 continuation value.

Sia tip — The whole point of the American question is the node-by-node max(rollback, immediate exercise) — and you must check the root too. Also keep every decimal in p and the intermediate node values; the exam says to retain all decimal places and awards no marks for a final number shown without working.

Glossary

Key terms

Risk-neutral probability (p): The 'pseudo-probability' p = (e^{rΔt} − d)/(u − d) under which the stock is expected to grow at the risk-free rate. It is a pricing device, not a forecast — the real-world probability of an up-move is irrelevant to the option's value.
Delta-hedged portfolio: A holding of Δ = (f_u − f_d)/(S_u − S_d) shares against one short option that has the same value in both future states, hence is riskless and must earn r. This is the no-arbitrage argument the whole model rests on.
Up and down factors (u, d): The multiplicative moves of the underlying over one step. When volatility σ is given, u = e^{σ√Δt} and d = 1/u; when a question states the stock 'moves up/down by x%', use those factors directly.
Roll-back (backward induction): Working from the terminal payoffs back to today, applying f = e^{−rΔt}[p·f_u + (1 − p)·f_d] at each node until you reach the root — the standard way to price any multi-step tree.
Early-exercise test: At every node of an American option, compare the rollback (continuation) value with the immediate-exercise payoff and keep the larger. A European option skips this test and uses rollback only.
CRR parametrisation: The Cox-Ross-Rubinstein choice u = e^{σ√Δt}, d = 1/u, which links the tree to the stock's volatility and makes the binomial price converge to Black-Scholes as the number of steps grows.

FAQ

The Binomial Option-Pricing Model FAQ

How many steps can a binomial tree have in the FINM6041 exam?

A maximum of 4 steps — the lecturer states this explicitly in the exam brief, so you will never have to grind out a huge tree by hand. The skill being tested is the method (compute p, build the tree, roll back, check early exercise), not arithmetic stamina.

Is the risk-neutral probability p a real probability?

No. It is a pricing construct — the probability under which the stock earns the risk-free rate — sometimes called a pseudo, risk-adjusted or martingale probability. The actual real-world chance of an up-move does not enter the option price at all, because the option is valued relative to the stock through a riskless hedge.

When do I need to check for early exercise?

For American options, at every non-terminal node — including today's root. Compare the rollback value with the immediate-exercise payoff and keep the larger. For European options you never check; you just roll back. A common trap is a deep in-the-money American put whose early-exercise value beats continuation at a down node.

How do I choose u and d?

If the question gives volatility σ per year, use the Cox-Ross-Rubinstein formulas u = e^{σ√Δt} and d = 1/u. If instead it says the stock goes up or down by a stated percentage, use those factors directly. For a dividend-yield underlying use p = (e^{(r−q)Δt} − d)/(u − d), and for a futures underlying p = (1 − d)/(u − d).

How does the binomial model relate to Black-Scholes?

They are the same idea at different resolutions. As you add steps, the binomial distribution of the terminal price approaches the lognormal, and the binomial price converges to the Black-Scholes-Merton value. Binomial is the tool of choice when no closed form exists — American and path-dependent options.

Studying with AI? Sia — free AI financial modeling tutor works through FINM6041 step by step.

Study strategy

Exam move

Anchor everything to the riskless-hedge story rather than memorising p in isolation: the option plus Δ shares is riskless, so it earns r, and rearranging that gives p = (e^{rΔt} − d)/(u − d). Then drill one clean routine — compute p, build the price tree forward, roll back with f = e^{−rΔt}[p·f_u + (1 − p)·f_d] — until it is automatic on a 2- to 4-step tree. For American options, write 'max(rollback, exercise)' at every node as a physical habit, and always re-check the root. Get the u,d decision right up front (σ given → CRR; percentage moves stated → use them), and know the two variations the exam loves: the dividend-yield and futures versions of p. Keep all decimals through the tree and show each node — a correct price with no working scores zero.

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