Before pricing an option, do we need to predict whether the stock will rise?

Start with the question

Do you need the real-world probability of an up move to price the option?

Do not reach for a formula yet. Put your own up-move probability into the expectation and see what different analysts obtain.

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The two states one period from now

Start with the payoff; Δ, B, and p* remain hidden for now.

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Mechanism, not prediction

Find a portfolio that pays exactly the same in both states

Adjust Δ shares of stock and the initial risk-free position B. When both states match, the cost of building the portfolio today pins down the option price.

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Your replicating portfolio

Stock position Δ The number of shares held per unit of the option.

Initial risk-free position B (NTD) B<0 means borrowing today; the terminal position is B × e^(rT).

Adjust both Δ and B until the residual is zero in each state.

Adjust one-period market parameters

S₀ (NTD)

K (NTD)

r (annual rate, %)

T (years)

u (up factor)

d (down factor)

State-by-state check

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State	Stock price	Option payoff	Portfolio value at maturity	Residual
Up	—	—	—	—
Down	—	—	—	—

Discovery moment

Change π again: which quantities actually move?

The three cards represent different concepts. Move π from low to high and identify the only card that changes.

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Real-world up-move probability π

65%

Your forecast

— NTD

e^(−rT)[πfᵤ+(1−π)fᴅ]

Changes with your probability judgment

Replication cost

— NTD

ΔS₀+B

Determined by state-by-state replication

Risk-neutral price

— NTD

e^(−rT)[p*fᵤ+(1−p*)fᴅ]

Equal to the replication cost

Risk-neutral weight p*—

Difference between the two prices—

Put-call parity residual—

p* = [e^(rT) − d] / (u − d). It is a pricing weight that makes the discounted stock a fair game, not a forecast of the stock’s real-world up-move probability.

What you should notice Changing the up-move probability you believe changes your forecast, but it does not change the replicable no-arbitrage price. Pricing comes from state-by-state replication; p* is the weight that makes the discounted stock a fair game.

Concept check: if you become more optimistic about an up move, which quantity must change? The subjective discounted expectation computed with real-world probability π Risk-neutral weight p* The Δ and B that replicate the payoff state by state The no-arbitrage price

Choose an answer, then check your understanding.

Bridge the two representations

First reproduce the one-period price, then extend the tree

The default CRR case is calibrated to the same one-period up and down factors. Its price must match exactly. Verify the bridge first, then load the three-period example.

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Δt = T/N—

u / d—

p*—

f(node) = e^(−rΔt) × [p* f(up) + (1−p*) f(down)]

European only: At intermediate nodes, only continuation values are discounted; immediate exercise is not compared, so there is no early-exercise decision.

Backward-induction workspace

Matched settings are ready. Roll back one layer to verify the exact price match.

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Time-zero priceNot yet rolled back

One-period benchmark—

CRR minus one-periodRoll back to compare

Call / Put with the same parameters—

Parity residual—

Black–Scholes–Merton continuous-time benchmark

The same S₀, K, r, σ, and T are used. BSM does not depend on the tree depth N, so it stays fixed while you refine the CRR tree.

BSM option price—

d₁ / d₂—

CRR − BSMRoll back to compare

Complete the CRR rollback to compare the discrete-tree price with the continuous-time benchmark.

Show the BSM formula used here

d₁ = [ln(S₀/K) + (r + σ²/2)T] / (σ√T)
d₂ = d₁ − σ√T
C = S₀Φ(d₁) − Ke^(−rT)Φ(d₂)
P = Ke^(−rT)Φ(−d₂) − S₀Φ(−d₁)

Assumptions for this benchmark: European exercise, no dividends, constant r and σ, and the standard lognormal BSM model. This panel is a comparison benchmark; the lesson’s main pricing mechanism remains replication and CRR backward induction.

Four takeaways ① Forecasting uses π; no-arbitrage pricing uses p*.
② Replication cost and the risk-neutral price are two representations of the same price.
③ Backward induction repeats the one-period problem from right to left.
④ With S₀, K, r, σ, and T fixed, BSM stays fixed while finite-N CRR prices can change as the tree is refined.

Before pricing an option, do we need to predict whether the stock will rise?

Do you need the real-world probability of an up move to price the option?

The two states one period from now

Find a portfolio that pays exactly the same in both states

State-by-state check

Exact replication solution

Change π again: which quantities actually move?

First reproduce the one-period price, then extend the tree

Backward-induction workspace

Black–Scholes–Merton continuous-time benchmark

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