Forward exchange rates and option values by no arbitrage

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.

Read every exchange rate in words: INR 83.00 per USD means one US dollar (the base currency ) costs 83.00 Indian rupees (the price currency ). The forward is quoted the same way.

Exam tip Watch the slash. Some sources write INR per USD as INR/USD (price currency first), others as USD/INR (base currency first, the way dealers usually quote it). Do not trust the order: find the currency there is exactly one unit of. That is the base, and its interest rate is the one subtracted.

INR 83.00 per USDone unit of the base currency costs 83.00 of the price currency

• Price currency (INR) Memorize the currency you pay; the forward is in it
  • Its rate enters with a plus sign F = S × e^((r_price − r_base)T)
• Base currency (USD) Memorize the currency there is one unit of
  • Its rate is subtracted

The two roads are:

1. Invest in the base currency at its rate.
2. 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. With continuous compounding , interest is added at every instant and the growth factor over T years is e^(rT), so the forward depends on the gap between the two rates.

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

Here S and F are price currency per unit of base, r_P and r_B are the continuously compounded rates of the price and base currencies, and T is in years.

If interest rates change while the spot rate stays put, the forward moves with the gap. If the rupee rate rises from 6.5% to 7.0% and the dollar rate stays at 4.5%, the forward becomes 83.00 × e^0.025 = 85.10: a wider gap means a bigger forward discount for the higher-rate currency.

If a question gives annually compounded rates, use (1 + r) instead. The two answers differ, so read how the rates compound. In this module’s questions the rates are usually continuously compounded, even when the stem only says “risk-free rates”.

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

Go deeper Other rate conventions

Money-market rates are often quoted as simple interest. The same logic applies and only the growth factors change.

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

Six months, USD 1.0800 per EUR, simple rates of 4% (USD) and 2% (EUR) a year: 1.0800 × (1 + 0.04 × 0.5) ÷ (1 + 0.02 × 0.5) = 1.0907.

On your calculator

TI BA II Plus HP 12C

1. 1.08×1.02÷1.01= 1.0907

1. 1.08ENTER1.02×1.01÷ 1.0907

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

How does no arbitrage fix an option value?

A call option gives the right, but not the obligation, to buy a share at the exercise price. A put option gives the right, but not the obligation, 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.

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

The hedge ratio says how many shares offset the moves of one option: here 0.2778 shares offset one call.

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

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 : a portfolio with the same value and payoffs as the option. So the call must cost what the mix costs.

A put by replication

Use it

A put gains when the share falls, so the hedge holds shares and buys the put.

$$p_0 = \frac{h\,S^{+} + p^{+}}{1 + r} - h\,S_0, \quad h = \frac{p^{-} - p^{+}}{S^{+} - S^{-}}$$

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 Put–call parity, and the risk-neutral shortcut

Put–call parity. The put value from replication agrees with the call: c₀ − p₀ = 5.13 − 11.67 = −6.54, and S₀ − X ÷ (1 + r) = 80 − 90 ÷ 1.04 = −6.54.

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.

Go deeper Real options: the same idea in a project

A real option is a choice built into a business project: the right, but not the obligation, to delay, abandon, expand or shrink it. Like a financial option it can only add value, because you use it only when it helps. Valuing one follows the same logic of comparing the project’s value with and without the choice; Corporate Issuers covers it.