Lesson 6 of 6 6–9% of the exam

You are reading the 2026 curriculum (exams up to November 2026). Sitting in February 2027 or later? Go to the 2027 Quantitative Methods guide.

By Daniil Tarakanov Updated Oct 16, 20262026 curriculum

Forward exchange rates and option values by no arbitrage

Investing at home must match converting, investing abroad and converting back at the forward rate, so F = S × (1 + r_price) ÷ (1 + r_base). For an option, a mix of shares and a short option that pays the same in every outcome is risk-free, so it must earn the risk-free rate; that pins down the option's price.

6 min cram, 32 min full read. Based on the 2026 CFA® Program Level I curriculum, Quantitative Methods, Learning Module 2. This is the 2026 version of the lesson. Sitting in February 2027 or later? Read the updated 2027 version.

Tailor this to you: take the self-assessment

Before this: Cash flow additivity and implied forward rates

On the exam

Explain the cash flow additivity principle, its importance for the no-arbitrage condition, and its use in calculating implied forward interest rates, forward exchange rates, and option values

Explain how cash flow additivity gives forward exchange rates and option values, and calculate them in simple cases. Expect one forward-rate calculation with the quote convention as the trap, and a one-period option by hedge ratio.

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.

How does no arbitrage fix a forward exchange rate?

A forward exchange rate is a rate agreed today for exchanging currencies at a future date. The same principle as the forward interest rate fixes it: two ways to turn one unit of money today into money at the future date must give the same result. Otherwise there is an arbitrage.

An exchange rate is quoted as price currency per unit of base currency. Get this right first.

Reading an exchange-rate quote

INR/USD 83.00read it as INR 83.00 per USD 1

Price currency (INR) Memorize the currency you pay, the one in the answer
- Its interest rate goes on top F = S × (1 + r_price) ÷ (1 + r_base)
Base currency (USD) Memorize the currency being priced, one unit
- Its interest rate goes underneath

The two roads are:

Invest in the base currency at its rate.
Convert at spot into the price currency, invest at the price-currency rate, and convert back at the forward rate.

Both roads must end with the same amount of base currency.

Forward exchange rate (annual compounding) Memorize

F = S \times \frac{(1 + r_P)^{T}}{(1 + r_B)^{T}}

Check it by running both roads with USD 1,000.

Rearrange itForward exchange rate: solve for the forward, spot or interest rate

The question asks for

The spot rate is 83 units of the price currency per unit of the base currency. The 1-year interest rates are 6.50% (price currency) and 4.50% (base currency). The no-arbitrage forward rate is closest to:

Spot (price currency per 1 base)Price-currency interest rate, r_P%Base-currency interest rate, r_B%Years, T

F = S \frac{( 1 + r _{P} ) ^{T}}{( 1 + r _{B} ) ^{T}}

83 × 1.065^1 ÷ 1.045^1 = 84.5885

Forward (price currency per 1 base)84.5885

On your calculator

83×(1.065y^x1)÷(1.045y^x1)=

y^x power

1.065ENTER1y^x1.045ENTER1y^x÷83×

ENTER enter (on the HP, puts the number on the stack) · y^x power

With simple interest or continuous compounding the same logic applies and only the growth factors change.

Forward exchange rate (simple rates) Use it

F = S \times \frac{1 + r_P\,T}{1 + r_B\,T}

Forward exchange rate (continuous compounding) Use it

F = S\,e^{(r_P - r_B)\,T}

Exam tip Say the quote aloud: “INR per USD.” The currency after “per” is the base, and its rate goes underneath.

How does no arbitrage fix an option value?

A call option gives the right to buy a share at the exercise price. A put option gives the right to sell. Their payoffs depend on the share, so a portfolio of the share and the option can have a payoff that does not depend on which way the share moves.

Take one period with two outcomes. Hold some shares and sell one call. Choose the number of shares so that the portfolio has the same value in both outcomes. That number is the hedge ratio . Because the value is certain, the risk-free portfolio must earn the risk-free rate, and that fixes the option’s price.

Hedge ratio Memorize

h = \frac{c^{+} - c^{-}}{S^{+} - S^{-}}

Worked example A call on a USD 80 share

The share is USD 80 today and in a year is USD 100 or USD 64. A call with exercise price USD 90 pays 10 if the share is 100 and 0 if it is 64. The risk-free rate is 4%.

Hedge ratio: h = (10 − 0) ÷ (100 − 64) = 0.2778 shares per call.
Portfolio: buy 0.2778 shares, sell 1 call.
Value in a year: 0.2778 × 100 − 10 = 17.78, or 0.2778 × 64 − 0 = 17.78. The same either way.
Present value at the risk-free rate: 17.78 ÷ 1.04 = 17.09.
Today’s cost of the portfolio equals that: 0.2778 × 80 − c₀ = 17.09, so c₀ = 22.22 − 17.09 = USD 5.13.

On your calculator

10÷36=STO1 hedge ratio 0.2778
RCL1×64÷1.04=STO2 17.09, present value of the riskless payoff
RCL1×80−RCL2= 5.13

STO store · RCL recall a stored number

10ENTER36÷STO1 hedge ratio 0.2778
64×1.04÷STO2 17.09, present value of the riskless payoff
RCL180×RCL2− 5.13

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

Call value by replication Use it

c_0 = h\,S_0 - \frac{h\,S^{-} - c^{-}}{1 + r}

Read the same result another way: 0.2778 shares, partly paid for by borrowing 17.09 against the sure payoff, behave exactly like the call. That mix is a replicating portfolio . The call must cost what it costs.

Rearrange itOne-period call by replication: hedge ratio and value

The question asks for

A share trades at 80.00 and will be 100.00 or 64.00 in one year. A call has an exercise price of 90.00. The number of shares to hold per call written to form a risk-free portfolio is closest to:

Share price if up, S⁺Share price if down, S⁻Exercise price, X

h = \frac{c ^{+} - c ^{-}}{S ^{+} - S ^{-}}

(10.00 − 0.00) ÷ (100.00 − 64.00) = 0.2778

Hedge ratio, h (shares per call)0.2778

On your calculator

10−0=call payoffs, up less down
÷(100−64)=share prices, up less down

10ENTER0−call payoffs, up less down
100ENTER64−share prices, up less down
÷

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

InteractiveReplicate the option: find the share mix that cannot win or lose

Try this 0 of 3

Cut the exercise price to 80, then slide h until both states match again
Switch to the put and find its hedge
Raise the risk-free rate to 6% or more and watch the call

Option

Option type

Shares held per option, h0.2778

Share and rates

Share price today, S₀80.00

Up-state price, Su100.00

Down-state price, Sd64.00

Exercise price, X90.00

Risk-free rate, r4.00%

Hedge ratio, h*0.2778

Riskless value in 1 year17.78

PV at 4.00%17.09

Call value, c₀5.13

Show working

Payoffs: cu = 10.00, cd = 0.00

h* = |cu − cd| ÷ (Su − Sd) = 10.00 ÷ 36.00 = 0.2778

Riskless value: 0.2778 × 100.00 − 10.00 = 0.2778 × 64.00 − 0.00 = 17.78

PV = 17.78 ÷ 1.04 = 17.09

c₀ = h* × S₀ − PV = 22.22 − 17.09 = 5.13

Parity check: c₀ − p₀ = 5.13 − 11.67 = −6.54 = S₀ − X ÷ (1 + r) = 80.00 − 86.54

Call value on your calculator

10−0=÷(100−64)=STO1h* = 0.2778
RCL1×64−0=STO2riskless value 17.78
RCL1×80−(RCL2÷1.04)=c₀ = 5.13

STO store · RCL recall a stored number

10ENTER0−100ENTER64−÷STO1h* = 0.2778
RCL164×0−STO2riskless value 17.78
RCL180×RCL21.04÷−c₀ = 5.13

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

If a portfolio can't lose or win, it must earn the risk-free rate.

Same cash flows, same price: three uses

No-arbitrage pricing

Forward interest rate Memorize long loan against two short loans
Forward exchange rate Memorize invest at home against convert, invest abroad, convert back
Option value Use it option against a mix of shares and borrowing that pays the same
- Hedge ratio shares per option that makes the portfolio risk-free
- Risk-free portfolio must earn the risk-free rate

Go deeper The put, and the risk-neutral shortcut

The put. With the same tree, a put with exercise price 90 pays 0 if the share is 100 and 26 if it is 64. To hedge, hold 0.7222 shares long and the put: the portfolio is worth 72.22 in both outcomes. Today’s value is 72.22 ÷ 1.04 = 69.44, so p₀ = 69.44 − 0.7222 × 80 = USD 11.67. Put-call parity checks it: c₀ − p₀ = 5.13 − 11.67 = −6.54, and S₀ − X ÷ (1 + r) = 80 − 90 ÷ 1.04 = −6.54.

The put

26÷36=STO1 0.7222 shares
RCL1×100÷1.04=STO2 69.44, present value of the riskless payoff
RCL1×80=STO3 57.78, the shares today
RCL2−RCL3= 11.67

STO store · RCL recall a stored number

26ENTER36÷STO1 0.7222 shares
100×1.04÷STO2 69.44, present value of the riskless payoff
RCL2RCL180×− 11.67

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

Risk-neutral probabilities. The same price comes from an average of the two payoffs, discounted. The weight on the up outcome is q = (1.04 × 80 − 64) ÷ (100 − 64) = 0.5333, and c₀ = 0.5333 × 10 ÷ 1.04 = 5.13. The weight q is not a forecast of the share. It is the number that makes the share earn the risk-free rate.

Why real-world odds never enter. The hedged portfolio has the same value in both outcomes, so it does not matter how likely each one is. The full treatment is the binomial model in Derivatives, and foreign exchange (FX) forwards return in Economics.

Memorize this

MemorizeFX forward in three forms

Rates	Forward
Annual compounding	$S(1 + r_P)^T / (1 + r_B)^T$
Simple	$S(1 + r_P T) / (1 + r_B T)$
Continuous	$S\,e^{(r_P - r_B)T}$

Memory hook Price-currency rate on top, base-currency rate underneath. The higher-rate currency is worth less forward: more of it per unit of the other.

Know it cold Quote A/B is A per 1 B: A is the price currency. Match each rate to its currency.

MemorizeOption by replication, four steps

Memory hook Hedge, check, discount, solve.

Know it cold 1) $h = (c^+ - c^-)/(S^+ - S^-)$ . 2) Long $h$ shares, short 1 call: same value in both outcomes. 3) Discount that value at $r$ . 4) $c_0 = hS_0 -$ that present value.

Reading an exchange-rate quote

INR/USD 83.00read it as INR 83.00 per USD 1

Price currency (INR) Memorize the currency you pay, the one in the answer
- Its interest rate goes on top F = S × (1 + r_price) ÷ (1 + r_base)
Base currency (USD) Memorize the currency being priced, one unit
- Its interest rate goes underneath

Same cash flows, same price: three uses

No-arbitrage pricing

Forward interest rate Memorize long loan against two short loans
Forward exchange rate Memorize invest at home against convert, invest abroad, convert back
Option value Use it option against a mix of shares and borrowing that pays the same
- Hedge ratio shares per option that makes the portfolio risk-free
- Risk-free portfolio must earn the risk-free rate

Forward exchange rate (annual compounding)

F = S \times \dfrac{(1 + r_P)^{T}}{(1 + r_B)^{T}}

Quick example

Spot 0.9400 CHF per EUR, one-year rates CHF 1.5%, EUR 3.0%. 0.9400 × 1.015 ÷ 1.03 = 0.9263 CHF per EUR.

On your calculator

0.94×1.015÷1.03= 0.9263

0.94ENTER1.015×1.03÷ 0.9263

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

Forward exchange rate (simple rates)

F = S \times \dfrac{1 + r_P\,T}{1 + r_B\,T}

Quick example

Spot USD/GBP 1.2700, six-month rates USD 5%, GBP 4% (annual, simple). 1.27 × 1.025 ÷ 1.02 = 1.2762.

On your calculator

1.27×1.025÷1.02= 1.2762

1.27ENTER1.025×1.02÷ 1.2762

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

Forward exchange rate (continuous compounding)

F = S\,e^{(r_P - r_B)\,T}

Quick example

Spot INR/USD 83.00, continuous rates INR 6.5%, USD 4.5%, one year. 83 × e^0.02 = 84.68.

On your calculator

0.022NDe^x×83= 84.68

2ND second function (the shifted label above a key) · e^x e to the power x

0.02ge^x83× 84.68

g blue shift key · e^x e to the power x

Hedge ratio

h = \dfrac{c^{+} - c^{-}}{S^{+} - S^{-}}

Quick example

Share 80, then 100 or 64. Call, exercise price 90, pays 10 or 0. h = 10 ÷ 36 = 0.2778 shares per call.

On your calculator

10÷36= 0.2778

10ENTER36÷ 0.2778

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

Call value by replication

c_0 = h\,S_0 - \dfrac{h\,S^{-} - c^{-}}{1 + r}

Quick example

Same tree, risk-free rate 4%. 0.2778 × 80 − (0.2778 × 64 − 0) ÷ 1.04 = 22.22 − 17.09 = 5.13.

On your calculator

10÷36=STO1 0.2778
RCL1×64÷1.04=STO2 17.09
RCL1×80−RCL2= 5.13

STO store · RCL recall a stored number

10ENTER36÷STO1 0.2778
64×1.04÷STO2 17.09
RCL180×RCL2− 5.13

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

How it is asked

Forward exchange rate

USD/GBP spot 1.2700; USD 5%, GBP 4%. One-year forward?

USD is the price currency: 1.2700 × 1.05 ÷ 1.04 = 1.2822. Inverting the rates (1.2579) is the trap.
One-period call value

Share 90, then 108 or 72. Call, exercise 96, r = 3%. Value?

h = 12 ÷ 36 = 0.3333; value = 0.3333 × 90 − 0.3333 × 72 ÷ 1.03 = 6.70. Forgetting to discount (6.00) is the trap.
Define the hedge ratio

In one-period replication the hedge ratio is the number of what?

Shares per option that makes the portfolio risk-free.

Exam traps

Inverting the quote: putting the base-currency rate on top.
Forgetting to discount the risk-free portfolio value before solving for the option.
Valuing the option with real-world probabilities instead of replication.

Checkpoint

3 exam-style questions. About 90 seconds each.

1The spot USD/GBP rate is 1.2700 (USD per GBP). One-year interest rates are 5% in USD and 4% in GBP (annual compounding). The no-arbitrage one-year forward rate is closest to:
2A share is at EUR 90 and will be EUR 108 or EUR 72 in one year. A call with an exercise price of EUR 96 is valued with a 3% risk-free rate. Its value is closest to:
3In a one-period replication, the hedge ratio is best described as the number of:

Key takeaways

Forward FX: F = S × (1 + r_price) ÷ (1 + r_base).
The higher-interest currency trades at a forward discount.
Replicate the option with shares; the hedged portfolio earns the risk-free rate.

Forward exchange rate (annual compounding)

F = S \times \dfrac{(1 + r_P)^{T}}{(1 + r_B)^{T}}

Quick example

Spot 0.9400 CHF per EUR, one-year rates CHF 1.5%, EUR 3.0%. 0.9400 × 1.015 ÷ 1.03 = 0.9263 CHF per EUR.

On your calculator

0.94×1.015÷1.03= 0.9263

0.94ENTER1.015×1.03÷ 0.9263

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

Forward exchange rate (simple rates)

F = S \times \dfrac{1 + r_P\,T}{1 + r_B\,T}

Quick example

Spot USD/GBP 1.2700, six-month rates USD 5%, GBP 4% (annual, simple). 1.27 × 1.025 ÷ 1.02 = 1.2762.

On your calculator

1.27×1.025÷1.02= 1.2762

1.27ENTER1.025×1.02÷ 1.2762

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

Forward exchange rate (continuous compounding)

F = S\,e^{(r_P - r_B)\,T}

Quick example

Spot INR/USD 83.00, continuous rates INR 6.5%, USD 4.5%, one year. 83 × e^0.02 = 84.68.

On your calculator

0.022NDe^x×83= 84.68

2ND second function (the shifted label above a key) · e^x e to the power x

0.02ge^x83× 84.68

g blue shift key · e^x e to the power x

Hedge ratio

h = \dfrac{c^{+} - c^{-}}{S^{+} - S^{-}}

Quick example

Share 80, then 100 or 64. Call, exercise price 90, pays 10 or 0. h = 10 ÷ 36 = 0.2778 shares per call.

On your calculator

10÷36= 0.2778

10ENTER36÷ 0.2778

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

Call value by replication

c_0 = h\,S_0 - \dfrac{h\,S^{-} - c^{-}}{1 + r}

Quick example

Same tree, risk-free rate 4%. 0.2778 × 80 − (0.2778 × 64 − 0) ÷ 1.04 = 22.22 − 17.09 = 5.13.

On your calculator

10÷36=STO1 0.2778
RCL1×64÷1.04=STO2 17.09
RCL1×80−RCL2= 5.13

STO store · RCL recall a stored number

10ENTER36÷STO1 0.2778
64×1.04÷STO2 17.09
RCL180×RCL2− 5.13

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

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Next module: Statistical Measures of Asset Returns Mean, median, mode and what outliers do to them

Questions about this lesson?

Where this comes back

Exchange Rate Calculations Economics
The One-Period Binomial Model Derivatives

Questions candidates ask

How do you calculate a forward exchange rate?

Multiply the spot rate by (1 + price-currency rate) ÷ (1 + base-currency rate). For INR/USD at 83.00 with rates of 6.5% in INR and 4.5% in USD, the one-year forward is 83.00 × 1.065 ÷ 1.045 = 84.59.

What is a hedge ratio in option pricing?

It is the number of shares to hold per option written so that the portfolio has the same value whichever way the share moves. It equals the spread of option payoffs divided by the spread of share prices, (c⁺ − c⁻) ÷ (S⁺ − S⁻).

Why are probabilities not needed to price an option?

The hedged portfolio is worth the same in every outcome, so it earns the risk-free rate whatever the odds. Its price today follows from discounting that value, and the option's price is what is left. Real-world probabilities never enter.

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