Lesson 1 of 3 6–9% of the exam

By Daniil Tarakanov Updated Oct 20, 20262027 curriculum

The one-period binomial tree and the hedge ratio

Q: How do you calculate the hedge ratio in a binomial model?

Divide the difference between the option's two payoffs by the difference between the share's two prices: h* = (c₁ᵘ − c₁ᵈ) ÷ (S₁ᵘ − S₁ᵈ). With payoffs of 18 and 0 and share prices of 91 and 56, h* = 18 ÷ 35 = 0.5143.

Q: What is a one-period binomial option pricing model?

A model that lets the share move to just two prices at the end of one period, which is the option's whole life. With two outcomes you can hedge the option exactly with the share, and the hedge gives the option's no-arbitrage price.

Q: Why is the hedged portfolio risk-free?

The hedge ratio is chosen so the portfolio is worth the same amount whether the share goes up or down. A payoff that does not depend on the outcome has no risk, so it must grow at the risk-free rate.

If the share ends at one of two prices, combine shares and a sold call so the portfolio is worth the same in both states. That portfolio is risk-free, so it must earn the risk-free rate, which pins down the call's price. The share-to-option mix is the hedge ratio.

8 min cram, 32 min full read. Based on the 2027 CFA® Program Level I curriculum, Derivatives, Learning Module 10. Also part of the 2026 curriculum, so it applies to the November 2026 exam too.

Tailor this to you: take the self-assessment

Before this: Pricing and Valuing Options, Put–Call Parity and Option Replication, Probability Trees and Conditional Expectations

On the exam

Explain how to value a derivative using a one-period binomial model

Explain how to value a derivative with a one-period binomial model. Expect the hedge ratio h*, the value of the hedged portfolio, and the option price by replication. For a put, h* is negative.

How deeply to learn each part

Know the idea: Understand it in general. It frames other topics, but you will not calculate with it.
Use it: Understand it well enough to solve a question with it. Recognize when it applies.
Memorize: Learn it by heart. The exam will not give it to you.

You know this one. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still do.

You tested strong here. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still know it.

By building a portfolio from the share and the option that is worth the same in both outcomes. A forward can be priced without any view on where the share goes. An option cannot, because its payoff is lopsided, so we need a model of the share’s future price. The simplest is the binomial model : over one period, which is the whole life of the option, the share either goes up or goes down. Nothing else can happen.

The size of each move is set by the up and down gross returns , R_u and R_d. A gross return is the ending price divided by the starting price, so 30% up is R_u = 1.30 and 20% down is R_d = 0.80. The spread between them is the volatility of the share.

Share prices at the end of the period Use it

S_1^u = R_u S_0 \qquad S_1^d = R_d S_0

What sits on a one-period binomial tree

One-period binomial treeone period = the whole life of the option

Today (t = 0) the only unknown is the option price
- Share S₀ known
- Option c₀ or p₀ what we solve for
Up state Use it chance q, which the price never uses
- Share S₁ᵘ = R_u S₀ R_u greater than 1
- Option c₁ᵘ max(0, S₁ᵘ − X) for a call
Down state Use it chance 1 − q
- Share S₁ᵈ = R_d S₀ R_d less than 1
- Option c₁ᵈ max(0, S₁ᵈ − X) for a call

Notice what is missing. The chance of an up move, q, is on the tree but never enters the price. We will see why in the third lesson.

InteractiveFind the share mix that makes both outcomes equal

Try this 0 of 3

Slide h up to 0.60 and compare the two leaves
Switch to the put and slide h until both leaves match
Keep S₀ 70, X 73 and r 6%, widen the tree to R_u 1.50 and R_d 0.70, then re-hedge the call

Hedged portfolio

Option

Try another h: shares held per option0.5143

Share and option

Share price today, S₀ (EUR)70.00

Up gross return, R_u1.30

Down gross return, R_d0.80

Exercise price, X (EUR)73.00

Risk-free rate, r6%

Hedge ratio, h*0.5143

Portfolio in 1 year, both states28.80

PV at 6%27.17

Call value, c₀8.83

Show working

Tree: S₁ᵘ = 70.00 × 1.30 = 91.00; S₁ᵈ = 70.00 × 0.80 = 56.00

Payoffs: c₁ᵘ = 18.00, c₁ᵈ = 0.00

h* = (c₁ᵘ − c₁ᵈ) ÷ (S₁ᵘ − S₁ᵈ) = (18.00 − 0.00) ÷ (91.00 − 56.00) = 0.5143

Hedged portfolio: 0.5143 × 91.00 − 18.00 = 0.5143 × 56.00 − 0.00 = 28.80

PV = 28.80 ÷ 1.06 = 27.17

c₀ = h* × S₀ − PV = 36.00 − 27.17 = 8.83

Call value on your calculator

18−0=÷(91−56)=STO1|h*| = 0.5143
RCL1×56−0=STO2hedged portfolio 28.80
RCL1×70−(RCL2÷1.06)=c₀ = 8.83

STO store · RCL recall a stored number

18ENTER0−91ENTER56−÷STO1|h*| = 0.5143
RCL156×0−STO2hedged portfolio 28.80
RCL170×RCL21.06÷−c₀ = 8.83

ENTER enter (on the HP, puts the number on the stack) · STO store · RCL recall a stored number

Pick the share-to-option mix that makes both outcomes equal; the rest is discounting.

What is the option worth at each end of the tree?

Its payoff at expiry, which you know for certain once you know the share price. The only unknown is the price today.

What is the hedge ratio, and why does it make the portfolio risk-free?

The hedge ratio is the number of shares to hold for each option sold so that the portfolio ends at the same value in both states. Sell one call and buy h shares. At expiry the portfolio is worth h × S₁ − c₁ in each state. Setting the up value equal to the down value and solving for h gives:

Hedge ratio Memorize

h^* = \frac{c_1^u - c_1^d}{S_1^u - S_1^d}

The top is how much the option payoff changes between the states. The bottom is how much the share price changes. One share moves the portfolio by its price change, so h* shares cancel the option’s change. This is the risk-free hedged portfolio : its end value does not depend on which state occurs.

Hedged portfolio value at expiry Use it

V_1 = h^* S_1^u - c_1^u = h^* S_1^d - c_1^d

Rearrange itHedge ratio: solve for any input

The question asks for

A share priced at 91.00 or 56.00 at the end of one period supports an option that pays 18.00 if the share is up and 0.00 if it is down. The hedge ratio is closest to:

Option payoff if up, c₁ᵘOption payoff if down, c₁ᵈShare price if up, S₁ᵘShare price if down, S₁ᵈ

h^{*} = \frac{c _{1}^{u} - c _{1}^{d}}{S _{1}^{u} - S _{1}^{d}}

(18.00 − 0.00) ÷ (91.00 − 56.00) = 0.5143

Hedge ratio, h* (shares per option)0.5143

On your calculator

18−0=÷(91−56)=0.5143

18ENTER0−91ENTER56−÷0.5143

ENTER enter (on the HP, puts the number on the stack)

Go deeper Why not just buy one share per call?

One share per call would be right only if the option moved one-for-one with the share. A call that is out of the money in the down state moves less than the share, so you need a fraction. With payoffs of 18 and 0 against share prices of 91 and 56, the call moves 18 while the share moves 35, so you hold 18/35 of a share. Because option contracts cover many shares, you keep the same ratio in whole units: 5143 shares for 10,000 calls, for example.

How does the risk-free portfolio give the call price?

Because a portfolio with a certain payoff must grow at the risk-free rate. Otherwise you could borrow or lend at r and lock in a profit. Discount V₁ at r to get the portfolio’s value today, V₀. That value equals the cost of the shares minus the call premium you collected: V₀ = h*S₀ − c₀. Rearranging gives the call price. This is replication made exact: the share and a loan reproduce the call’s payoff, so the call must cost what that package costs.

Call price by replication Memorize

c_0 = h^* S_0 - \frac{V_1}{1 + r}

The result is the no-arbitrage option price . Any other price lets a trader lock in a riskless profit.

Rearrange itCall price by replication: solve for any input

The question asks for

A call is hedged with 0.5143 shares per call sold. The share trades at 70.00, the hedged portfolio is worth 28.80 at expiry in both states, and the risk-free rate for the period is 6.00%. The call price is closest to:

Hedge ratio, h*Share price today, S₀Hedged portfolio value at expiry, V₁Risk-free rate for the period, r%

c_{0} = h^{*} S_{0} - \frac{V _{1}}{1 + r}

0.5143 × 70.00 − 28.80 ÷ 1.06 = 36.00 − 27.17 = 8.83

Call price today, c₀8.83

On your calculator

0.514286×70−(28.8÷1.06)=8.83

0.514286ENTER70×28.8ENTER1.06÷−8.83

ENTER enter (on the HP, puts the number on the stack)

Exam tip Three moves, in order: payoffs at the leaves, h* from the spreads, then discount V₁ and subtract from h*S₀. Skipping the discount is the most common wrong answer.

What changes for a put?

The hedge ratio formula is the same, but its sign is negative, because a put pays more when the share falls. A negative h* does not mean short the shares. It means the option and the shares are held in the same direction: buy the put and buy |h*| shares. The shares gain when the put loses, and the other way round.

Worked example Price the put with the same tree

Same share and tree, put with X = EUR 73. Payoffs: p₁ᵘ = max(0, 73 − 91) = 0; p₁ᵈ = max(0, 73 − 56) = 17.

h* = (0 − 17) ÷ (91 − 56) = −0.4857. Buy the put and buy 0.4857 shares.
Up: V₁ = 0.4857 × 91 + 0 = 44.20. Down: V₁ = 0.4857 × 56 + 17 = 44.20.
V₀ = 44.20 ÷ 1.06 = 41.70. This is the cost of the shares plus the put, so p₀ = 41.70 − 0.4857 × 70 = 41.70 − 34.00 = EUR 7.70.

On your calculator

17÷35=STO1 0.4857
×91=÷1.06=STO2 41.7
70×RCL1=+|-+RCL2= 7.7

STO store · RCL recall a stored number · +|- change sign

17ENTER35÷STO1 0.4857
91×1.06÷STO2 41.7
RCL270RCL1×− 7.7

ENTER enter (on the HP, puts the number on the stack) · STO store · RCL recall a stored number

What if the market price is not the model price?

An arbitrage. If the call trades above its model value, sell it, buy h* shares and borrow the difference at the risk-free rate. The hedge is risk-free, so you owe less at expiry than the hedge is worth.

If the call is below the model value, do the reverse: buy the call, sell h* shares short and lend the proceeds.

Go deeper Why the model needs R_d < 1 + r < R_u

If both moves were above 1 + r, the share would beat the risk-free asset in every state. You would borrow at r and buy the share, with no risk and no cost. If both were below 1 + r, you would short the share and lend. Either way arbitrage exists before any option is priced. So no arbitrage requires R_d < 1 + r < R_u, which is the same as saying the risk-neutral probability in the next lesson lies strictly between 0 and 1.

Go deeper More than one period

Longer trees repeat this one step backward: price every node at the last step as above, treat the results as the payoffs for the step before, and keep going. That is a Level II topic. The one-period version is the whole Level I method.

Memorize this

MemorizeThe binomial hedge, step by step

Step	Call (sold)	Put (bought)
Payoff at expiry	$\max(0,\ S_1 - X)$	$\max(0,\ X - S_1)$
Hedge ratio	$h^* = \frac{c_1^u - c_1^d}{S_1^u - S_1^d} > 0$	$h^* = \frac{p_1^u - p_1^d}{S_1^u - S_1^d} < 0$
Position	sell 1 call, buy $h^*$ shares	buy 1 put, buy $\|h^*\|$ shares
Value at expiry	$V_1 = h^* S_1 - c_1$	$V_1 = \|h^*\| S_1 + p_1$
Price today	$c_0 = h^* S_0 - V_1/(1+r)$	$p_0 = V_1/(1+r) - \|h^*\| S_0$

Memory hook Spread of option payoffs over spread of share prices. Equalize the two end values, discount at r, then take away the cost of the shares.

Know it cold $V_1$ is the same in both states, so it grows at the risk-free rate: $V_0 = V_1/(1+r)$ . Always discount.

MemorizeNo arbitrage and the tree

Memory hook The risk-free rate must sit between the two moves, or one state would dominate and you could borrow to buy the share for free.

Know it cold $R_d < 1 + r < R_u$ . Gross returns, not net: 1.30 and 0.80, not 30% and −20%.

What sits on a one-period binomial tree

One-period binomial treeone period = the whole life of the option

Today (t = 0) the only unknown is the option price
- Share S₀ known
- Option c₀ or p₀ what we solve for
Up state Use it chance q, which the price never uses
- Share S₁ᵘ = R_u S₀ R_u greater than 1
- Option c₁ᵘ max(0, S₁ᵘ − X) for a call
Down state Use it chance 1 − q
- Share S₁ᵈ = R_d S₀ R_d less than 1
- Option c₁ᵈ max(0, S₁ᵈ − X) for a call

The risk-free hedge, call versus put

Risk-free hedged portfoliosame value in both states, so it earns r

Call Memorize h* is positive
- Sell 1 call short the option
- Buy h* shares long the share
- V₀ = h*S₀ − c₀ then c₀ = h*S₀ − V₁ ÷ (1 + r)
Put Memorize h* is negative
- Buy 1 put long the option
- Buy |h*| shares long the share, not short
- V₀ = |h*|S₀ + p₀ then p₀ = V₁ ÷ (1 + r) − |h*|S₀

Share prices at the end of the period

S_1^u = R_u S_0 \qquad S_1^d = R_d S_0

Quick example

Share 45, up gross return 1.20, down gross return 0.90. It ends at 45 × 1.20 = 54.00 or 45 × 0.90 = 40.50.

On your calculator

45×1.2= 54
45×0.9= 40.5

45ENTER1.2× 54
45ENTER0.9× 40.5

ENTER enter (on the HP, puts the number on the stack)

Hedge ratio

h^* = \dfrac{c_1^u - c_1^d}{S_1^u - S_1^d}

Quick example

Share 50 ends at 62 or 46. A call with X = 50 pays 12 or 0. h = (12 − 0) ÷ (62 − 46) = 0.75 share per call sold

On your calculator

12÷(62−46)= 0.75

12ENTER62ENTER46−÷ 0.75

ENTER enter (on the HP, puts the number on the stack)

Hedged portfolio value at expiry

V_1 = h^* S_1^u - c_1^u = h^* S_1^d - c_1^d

Quick example

Same share and call, h = 0.75. Up: 0.75 × 62 − 12 = 34.50. Down: 0.75 × 46 − 0 = 34.50. The same in both states

On your calculator

12÷(62−46)=×62−12= 34.5

12ENTER62ENTER46−÷62×12− 34.5

ENTER enter (on the HP, puts the number on the stack)

Call price by replication

c_0 = h^* S_0 - \dfrac{V_1}{1 + r}

Quick example

S₀ = 50, h = 0.75, V₁ = 34.50, r = 2%. c₀ = 0.75 × 50 − 34.50 ÷ 1.02 = 37.50 − 33.82 = 3.68*.

On your calculator

12÷(62−46)=×50−(34.5÷1.02)= 3.68

12ENTER62ENTER46−÷50×34.5ENTER1.02÷− 3.68

ENTER enter (on the HP, puts the number on the stack)

How it is asked

Hedge ratio from the tree

Share 36 goes to 45 or 30. A call with X = 36 pays 9 or 0. Hedge ratio?

(9 − 0) ÷ (45 − 30) = 0.60. Traps: 1.67 (upside down, 15 ÷ 9) and 0.25 (9 ÷ 36, divides by S₀).
Call price by replication

S₀ = 80, h* = 0.50, hedged portfolio worth 36.00 at expiry, r = 4%. Call?

0.50 × 80 − 36.00 ÷ 1.04 = 40.00 − 34.62 = 5.38. Trap: 4.00 (V₁ not discounted).
Which side of the hedge?

The hedge ratio for a put is −0.45. What do you do?

Buy the put and buy 0.45 shares. Trap: selling shares because of the minus sign.
Mispriced option

The market call price is above the model value. Arbitrage?

*Sell the call, buy h shares, borrow** the difference at r.

Exam traps

Hedge ratio upside down: it is the spread of option payoffs over the spread of share prices, (c₁ᵘ − c₁ᵈ) ÷ (S₁ᵘ − S₁ᵈ), never the reverse.
Forgetting to discount the hedged portfolio's value: V₁ is a value at expiry, so c₀ = h*S₀ − V₁ ÷ (1 + r).
Treating a put's hedge ratio as a short position: h* is negative, which means buy the put and buy shares.
Dividing the call payoff by S₀ instead of the spread S₁ᵘ − S₁ᵈ.

Checkpoint

4 exam-style questions. About 90 seconds each.

1A share at USD 40 will be USD 48 or USD 36 in one period. A call with an exercise price of USD 40 pays USD 8 or zero. The hedge ratio is closest to:
2To create a risk-free portfolio with a sold call, an investor should:
3A share at EUR 60 will be EUR 75 or EUR 48 in one year. A call with an exercise price of EUR 62 pays EUR 13 or zero, and the risk-free rate is 5%. The hedge ratio is 0.4815 and the hedged portfolio is worth EUR 23.11 at expiry in both states. The call price is closest to:
4In a one-period binomial model the hedge ratio for a put option is −0.33. To form a risk-free portfolio, the investor should:

Key takeaways

Two outcomes plus the right mix of shares and options give a risk-free portfolio.
h* = (c₁ᵘ − c₁ᵈ) ÷ (S₁ᵘ − S₁ᵈ). Sell the call, buy h* shares.
The hedged portfolio earns the risk-free rate, so c₀ = h*S₀ − V₁ ÷ (1 + r).
A put's h* is negative: buy the put and buy |h*| shares.

Share prices at the end of the period

S_1^u = R_u S_0 \qquad S_1^d = R_d S_0

Quick example

Share 45, up gross return 1.20, down gross return 0.90. It ends at 45 × 1.20 = 54.00 or 45 × 0.90 = 40.50.

On your calculator

45×1.2= 54
45×0.9= 40.5

45ENTER1.2× 54
45ENTER0.9× 40.5

ENTER enter (on the HP, puts the number on the stack)

Hedge ratio

h^* = \dfrac{c_1^u - c_1^d}{S_1^u - S_1^d}

Quick example

Share 50 ends at 62 or 46. A call with X = 50 pays 12 or 0. h = (12 − 0) ÷ (62 − 46) = 0.75 share per call sold

On your calculator

12÷(62−46)= 0.75

12ENTER62ENTER46−÷ 0.75

ENTER enter (on the HP, puts the number on the stack)

Hedged portfolio value at expiry

V_1 = h^* S_1^u - c_1^u = h^* S_1^d - c_1^d

Quick example

Same share and call, h = 0.75. Up: 0.75 × 62 − 12 = 34.50. Down: 0.75 × 46 − 0 = 34.50. The same in both states

On your calculator

12÷(62−46)=×62−12= 34.5

12ENTER62ENTER46−÷62×12− 34.5

ENTER enter (on the HP, puts the number on the stack)

Call price by replication

c_0 = h^* S_0 - \dfrac{V_1}{1 + r}

Quick example

S₀ = 50, h = 0.75, V₁ = 34.50, r = 2%. c₀ = 0.75 × 50 − 34.50 ÷ 1.02 = 37.50 − 33.82 = 3.68*.

On your calculator

12÷(62−46)=×50−(34.5÷1.02)= 3.68

12ENTER62ENTER46−÷50×34.5ENTER1.02÷− 3.68

ENTER enter (on the HP, puts the number on the stack)

All Derivatives formulas, every lesson

Practice exam-style questions on The One-Period Binomial Model

Next lesson Pricing calls and puts with risk-neutral probabilities

Questions about this lesson?

Where this comes back

Pricing calls and puts with risk-neutral probabilities The One-Period Binomial Model

Questions candidates ask

How do you calculate the hedge ratio in a binomial model?

Divide the difference between the option's two payoffs by the difference between the share's two prices: h* = (c₁ᵘ − c₁ᵈ) ÷ (S₁ᵘ − S₁ᵈ). With payoffs of 18 and 0 and share prices of 91 and 56, h* = 18 ÷ 35 = 0.5143.

What is a one-period binomial option pricing model?

A model that lets the share move to just two prices at the end of one period, which is the option's whole life. With two outcomes you can hedge the option exactly with the share, and the hedge gives the option's no-arbitrage price.

Why is the hedged portfolio risk-free?

The hedge ratio is chosen so the portfolio is worth the same amount whether the share goes up or down. A payoff that does not depend on the outcome has no risk, so it must grow at the risk-free rate.

Spotted an error? Tell us and we will fix it.

The one-period binomial tree and the hedge ratio

On the exam

How can you price an option exactly if the share can only go to one of two prices?

What is the option worth at each end of the tree?

What is the hedge ratio, and why does it make the portfolio risk-free?

How does the risk-free portfolio give the call price?

What changes for a put?

What if the market price is not the model price?

Memorize this

How it is asked

Exam traps

Checkpoint

Key takeaways

Where this comes back

Questions candidates ask