Valuing a Derivative Using a One-Period Binomial Model: formula sheet

Key formulas

Up and down prices

S₁⁺ = S₀ × u ; S₁⁻ = S₀ × d

u and d are gross factors (1 + return), not percentage moves.

Factors from returns

u = 1 + up return ; d = 1 + down return

A 15% fall gives d = 0.85, not 0.15.

No-arbitrage condition on factors

d < 1 + r < u

If 1 + r is outside this range, one asset dominates the other and arbitrage exists.

Risk-free growth over the period

Bond value at end = B₀ × (1 + r)

Use the rate for the length of the single step, not an annual rate unless the step is one year.

Call payoffs at expiry

c₁⁺ = max(0, S₁⁺ − X) ; c₁⁻ = max(0, S₁⁻ − X)

Put payoffs: max(0, X − S₁). Shown here so you can read the tree outputs.

Hedge ratio (delta) for a call

h = (c⁺ − c⁻) ÷ (S⁺ − S⁻)

Positive for a call. Use put payoffs p⁺ and p⁻ for a put; the result is negative.

Up and down prices

S⁺ = S × u ; S⁻ = S × d

u and d are gross factors, for example u = 1.25 means a 25% rise.

Call payoffs at expiry

c⁺ = max(0, S⁺ − X) ; c⁻ = max(0, S⁻ − X)

X is the exercise price. Put payoffs are max(0, X − S).

Replicating portfolio value

c = h × S + B

B is the bond position today. B is negative when you borrow.

Bond position from the down state

B = [c⁻ − h × S⁻] ÷ (1 + r)

Same B results from the up state: [c⁺ − h × S⁺] ÷ (1 + r).

Hedged position

Long h units, short 1 call: value today = h × S − c

The payoff is riskless in both states; it earns the risk-free rate only if the option is fairly priced.

Up and down factors

u = S↑ ÷ S0; d = S↓ ÷ S0

Up factor is above down factor. Often d = 1 ÷ u in problems, but only if the question says so.

Risk-neutral probability of up move

π = (1 + r − d) ÷ (u − d)

r is the risk-free rate for the period. Assumes the underlying pays no income; with income, adjust as the question directs.

Risk-neutral probability of down move

1 − π = (u − 1 − r) ÷ (u − d)

Check that π and 1 − π add to 1.

Option value today

V0 = [π × Vu + (1 − π) × Vd] ÷ (1 + r)

Discount at the risk-free rate, not a required return on the stock.

European call payoffs

cu = max(0, Su − X); cd = max(0, Sd − X)

X is the exercise price.

European put payoffs

pu = max(0, X − Su); pd = max(0, X − Sd)

A put pays when the price at expiry is below X.

Up and down factors

u = S+ ÷ S0 ; d = S− ÷ S0

S+ and S− are the possible prices after one period.

Risk-neutral probability of an up move

π = (1 + r − d) ÷ (u − d)

Use for an asset with no income. r is the risk-free rate for the period. The down probability is 1 − π.

Option value (risk-neutral)

c0 = [π × c+ + (1 − π) × c−] ÷ (1 + r)

Works for puts too: use the put payoffs p+ and p−.

Hedge ratio

Calls: h = (c+ − c−) ÷ (S+ − S−). Puts: h = (p+ − p−) ÷ (S+ − S−)

h is the number of shares you hold to hedge one short option. A positive h means a long position in the shares. A negative h means a short position in the shares. For a short call, h is positive, so you hold h shares long. For a short put, the put payoff is larger in the down state, so h is negative, and the hedged position is a short put plus |h| shares sold short. If you are long the option, take the opposite share position.

Arbitrage profit

Profit today = |market price − model value|

Overpriced: sell the option, buy the replicating portfolio. Underpriced: buy the option, sell the replicating portfolio.

No-income forward price

F0 = S0 × (1 + r)

A forward priced at this level has zero value at initiation.

American option value (one period)

Value = max(immediate exercise value, European value)

Call exercise value = max(0, S0 − X). Put exercise value = max(0, X − S0).

Quick revision

• The one-period binomial model has one up move (u) and one down move (d) for the underlying.
• Up and down prices: Su = S0 × u and Sd = S0 × d.
• Hedge ratio h = (Vu − Vd) ÷ (Su − Sd).
• A call has a positive h. A long call is replicated by long h units of the underlying plus borrowing. A put has a negative h. A long put is replicated by short |h| units of the underlying plus lending. To hedge, take the opposite position in the derivative.
• Risk-neutral probability of an up move: π = (1 + r − d) ÷ (u − d), and the down probability is 1 − π.
• Derivative value = [π × Vu + (1 − π) × Vd] ÷ (1 + r).
• Discount at the risk-free rate, not at a required return on the underlying.
• The risk-neutral probability is not the real-world probability of an up move.
• Option value does not depend on investors' risk preferences.
• If the market price is above the model value, sell the derivative and buy the replicating portfolio; if below, buy the derivative and sell the replicating portfolio.
• Payoffs at expiry: call = max(S − X, 0); put = max(X − S, 0).
• No arbitrage requires d < 1 + r < u.

Common mistakes

• Using the percentage move instead of the factor, e.g. d = 0.20 for a 20% fall. Fix: Always write d = 1 − fall and u = 1 + rise before multiplying.
• Using an annual risk-free rate for a step shorter than a year. Fix: Match r to the step length. For a one-year step use r as given; for a different period, adjust it as the question states.
• Dividing by the wrong spread, such as using S instead of S⁺ − S⁻. Fix: Delta is payoff difference over price difference. Both differences are up minus down.
• Forgetting to discount the bond position by (1 + r). Fix: Divide the future bond amount by (1 + r) to get B. Check that h × S⁺ + B × (1 + r) equals the up payoff.
• Using the real-world probability of an up move instead of π. Fix: For valuation, always compute π from u, d and r. Real-world probabilities are a distractor.
• Discounting at the stock's expected return or a required return. Fix: Under risk-neutral pricing, discount the expected payoff at the risk-free rate.
• Using the real-world probability of an up move instead of π. Fix: Ignore the real-world probability. Always compute π from u, d and r.
• Buying the option when it is overpriced, or selling when it is underpriced. Fix: Say it aloud: market above value, so sell the option. Market below value, so buy it.

Exam tips

• Read whether the question gives returns or prices; convert to factors before anything else.
• Check that r is for the same period as the tree step.
• For assumption questions, a statement that the price can take many values or that borrowing costs more than r contradicts the model.
• With three options and no penalty, a quick u and d calculation usually removes the options built from the percentage moves.
• Questions are three-option MCQs with numbers in ascending order. Compute h first. A wrong delta usually gives a wrong value that still appears as an option.
• Do a sanity check: call value must be positive, below S, and below h × S in the usual case where you borrow.
• If the down payoff for a call is zero, the bond leg is just −h × S⁻ ÷ (1 + r). This saves time.
• Read whether the question asks for the number of shares, the borrowing amount or the option value. They are different numbers from the same working.