Binomial option-pricing model: formula sheet

Key formulas

Long call payoff at expiry

max(S_T − K, 0)

S_T is the share price at expiry. Never negative for the holder.

Long put payoff at expiry

max(K − S_T, 0)

Pays when the share price ends below the strike.

Short positions

Short call payoff = −max(S_T − K, 0); short put payoff = −max(K − S_T, 0)

The writer's payoff is the negative of the holder's. The writer's payoff is never positive; the writer's maximum profit is the premium received.

Profit at expiry (ignoring interest)

Profit = payoff − premium (long); profit = premium − payoff (short)

If asked to allow for interest, accumulate the premium to expiry at the risk-free rate first.

Breakeven share price

Call: K + premium; Put: K − premium

Holder's breakeven, ignoring interest on the premium.

Moneyness

Call in the money if S > K; put in the money if S < K

Intrinsic value now = max(S − K, 0) for a call, max(K − S, 0) for a put.

Effect of factors on a European option (other factors fixed)

S up: call up, put down. K up: call down, put up. Volatility up: both up. Rate up: call up, put down. Dividends up: call down, put up.

Time to expiry: more time raises American options; for European options the effect is not always upward (e.g. deep in-the-money European puts).

Put-call parity (no dividends, continuous compounding)

c + K e^(−rT) = p + S₀

European options, same strike K and expiry T, same underlying. r is the continuously compounded risk-free rate.

Put-call parity (discrete compounding)

c + K (1 + i)^(−T) = p + S₀

Use when the question gives an annual effective rate i.

Put-call parity with known dividends

c + K e^(−rT) + D = p + S₀, where D is the present value of dividends paid before T

The share holder receives dividends, so the share portfolio is worth more by D at the end. Move D to the call side.

European call bounds (no dividends)

max(S₀ − K e^(−rT), 0) ≤ c ≤ S₀

Lower bound comes from parity and p ≥ 0.

European put bounds (no dividends)

max(K e^(−rT) − S₀, 0) ≤ p ≤ K e^(−rT)

Upper bound: the put can pay at most K at T.

American options and parity

S₀ − K ≤ C − P ≤ S₀ − K e^(−rT) (no dividends)

Parity becomes an inequality for American options. Early exercise is the reason.

Arbitrage rule

If two portfolios have equal payoffs, buy the cheaper and sell the dearer

The profit today is the price difference, invested at the risk-free rate if needed.

Share price after one period

S_up = S0 × u; S_down = S0 × d

Requires d < u. No-arbitrage needs d < (1 + r) < u.

Option payoffs

Call: max(S_T − K, 0); Put: max(K − S_T, 0)

Compute the payoff in the up state (Cu) and the down state (Cd) separately.

Delta (hedge ratio)

Δ = (Cu − Cd) ÷ (S0·u − S0·d)

Change in option payoff divided by change in share price between the two states.

Cash position

B = (Cu − Δ·S0·u) ÷ (1 + r) = (Cd − Δ·S0·d) ÷ (1 + r)

Using discrete interest r for the period. With continuous rate, replace (1 + r) by e^(rT). Negative B means borrowing.

Option price

C0 = Δ·S0 + B

Cost today of the replicating portfolio.

No-arbitrage condition

d < 1 + r < u

If it fails, you can make a risk-free profit from the shares and cash alone.

Risk-neutral probability (continuous compounding)

q = (e^(rΔt) − d) ÷ (u − d)

Here u and d are the up and down factors per step, and Δt is the step length in years. Probability of a down move is 1 − q.

Risk-neutral probability (annual compounding rate i)

q = ((1 + i)^Δt − d) ÷ (u − d)

Use when the question gives an effective annual rate. For one yearly step, q = (1 + i − d) ÷ (u − d).

One-period option value

V0 = e^(−rΔt) × [q × Vu + (1 − q) × Vd]

Vu and Vd are the option values after an up and a down move.

Martingale check on the share

S0 = e^(−rΔt) × [q × S0u + (1 − q) × S0d]

Use it to check your q. The discounted expected share price must equal today's price.

No-arbitrage condition

d < e^(rΔt) < u

Ensures 0 < q < 1.

Multi-period value

V0 = e^(−rnΔt) × Σ P(j ups) × V(n, j), with P(j ups) = C(n, j) q^j (1 − q)^(n−j)

Valid for European options on a recombining tree with constant u, d and r.

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.

Risk-neutral probability of an up move

q = (e^(rΔt) − d) ÷ (u − d)

Uses a continuously compounded rate r and a step length Δt. If the question gives a simple annual rate per step, use (1 + i) in place of e^(rΔt). Needs d < e^(rΔt) < u for no arbitrage.

Continuation value at a node

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

V_up and V_down are the option values at the next two nodes. For an American option these are already the larger-of values.

American put value at a node

V = max( K − S, C )

K − S is the exercise value when positive. If K − S is negative, exercise value is zero, so V is at least C.

American call value at a node

V = max( S − K, C )

For a share with no dividends, C is never below S − K, so early exercise is not optimal.

Value at expiry

Put: max(K − S, 0). Call: max(S − K, 0)

Same starting point as for European options.

Ordering of values

American value ≥ European value

Equality holds for a non-dividend call. For a put the American value can be strictly higher.

CRR up factor

u = e^(σ√Δt)

σ is annual volatility. Δt is the step length in years, so Δt = T ÷ n.

CRR down factor

d = 1/u = e^(−σ√Δt)

This makes the tree recombine, so an up then a down move returns the price to S₀.

Risk-neutral probability of an up move

p = (e^(rΔt) − d) ÷ (u − d)

r is the continuously compounded risk-free rate. The probability of a down move is 1 − p.

No-arbitrage condition

d < e^(rΔt) < u

Equivalent to 0 < p < 1. Check it when Δt is large.

Option value by backward induction

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

Apply at each node, working back from the final payoffs.

Step length

Δt = T ÷ n

T is the option term in years and n is the number of steps.

Convergence statement

binomial price → Black-Scholes price as n → ∞ (Δt = T ÷ n → 0)

Holds for European options with u and d calibrated to σ as above.

Quick revision

• Call payoff at expiry = max(S − K, 0); put payoff = max(K − S, 0).
• No arbitrage means two portfolios with the same payoff must have the same price.
• Put-call parity for European options on a non-dividend share: C − P = S₀ − K·e^(−rT).
• One-period replication: hold Δ = (Vu − Vd) ÷ (S₀u − S₀d) shares, financed by cash.
• Option value = S₀·Δ + cash, where the cash amount may be negative (borrowing).
• Risk-neutral probability of an up-move: q = (e^(rh) − d) ÷ (u − d) for step length h.
• Option value = e^(−rh) × [q·Vu + (1 − q)·Vd].
• The real-world probability of an up-move does not enter the price.
• No-arbitrage requires d < e^(rh) < u; otherwise q falls outside 0 to 1.
• In multi-period trees, work backwards from expiry, applying the one-step formula at each node.
• American option value at a node = max(exercise value, continuation value).
• In a Cox-Ross-Rubinstein style tree, u = e^(σ√h) and d = 1/u; the price tends to the Black-Scholes value as steps increase.

Common mistakes

• Drawing a call payoff that goes negative below K. Fix: The holder can walk away, so the payoff is max(S_T − K, 0). It is flat at zero below K.
• Confusing payoff with profit. Fix: Read the question for the word profit. Then always subtract the premium (long) or add it (short).
• Discounting the share price instead of the strike Fix: Only the strike is paid in the future, so only K is discounted. S₀ is already a price today.
• Applying parity to American options as an equality Fix: Parity as an equality needs European options. For American options use the inequality.
• Using the real-world probability of an up move to price the option. Fix: In replication, probabilities never appear. Solve the two payoff equations for Δ and B.
• Forgetting to discount the cash amount B. Fix: B is invested or borrowed for the period. It grows by (1 + r) or e^(rT), so include that factor in both equations.
• Using the real-world probability p in the pricing formula. Fix: Real-world p never enters the price. Always compute q from u, d and r. Use p only if asked for a real-world expected payoff.
• Writing q = (u − e^(rΔt)) ÷ (u − d), which is the down probability. Fix: The up probability has e^(rΔt) − d on top. Check that q × u + (1 − q) × d = e^(rΔt).
• Using the real-world probability of an up move instead of q 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 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.

Exam tips

• Sketch first. A quick labelled diagram with K, the premium and the breakeven often earns marks even if arithmetic slips.
• Read whether the question asks for payoff or profit, and whether to allow for interest on the premium. State your assumption.
• For factor questions, change one variable at a time and give direction plus a reason. Mention European or American if it matters.
• In MCQs, check the sign: for a short position, flip the diagram about the horizontal axis.
• In Paper B, a payoff function is a one-line formula. In R use pmax(S − K, 0) for a call to handle many prices at once.
• Write the parity equation first and name each term. Examiners award marks for the correct setup.
• State clearly that parity needs European options with the same strike and expiry.
• In arbitrage questions, show the cash flows at time 0 and at expiry in both cases, S_T > K and S_T ≤ K.