Foreign exchange and covered interest parity, financial engineering for quants lesson

A company that keeps its accounts in domestic currency must pay $1{,}000{,}000$ units of a foreign currency in one year. It can buy the foreign currency today and deposit it abroad until the payment date, or it can agree now on an exchange rate for that date and set aside domestic cash to pay at that rate. When both routes are available on ideal terms, they deliver the same foreign amount on the same date, so they must cost the same today, and that equality fixes the forward exchange rate. This lesson writes every exchange rate with its units, derives covered interest parity from the two routes, reads the forward as an expectation under the domestic pricing measure, checks the reciprocal quotation and says which funding leg a cross-currency basis adjusts.

Key terms

Exchange-rate quotation

Definition Fix a domestic currency, in which values are measured, and a foreign currency. The spot exchange rate $S_t > 0$ is the number of domestic currency units that buy one foreign currency unit at time $t$ , so its unit is domestic currency per foreign currency. Without transaction costs, an amount of $X_f$ foreign units is worth

X_f \, S_t \ \text{domestic units},

and an amount of $X_d$ domestic units buys $X_d / S_t$ foreign units. A rise in $S_t$ means that the foreign currency has become more expensive in domestic terms, that is, it has appreciated against the domestic currency. Every formula on this page uses this quotation unless it says otherwise. Quotes in data sources are written in both directions, depending on the currency pair, so the unit of a quote has to be checked before the quote enters a formula.

In plain words An exchange rate is a price, and like every price it has a unit. Multiply a foreign amount by the rate to get domestic currency, because the foreign unit cancels, and divide a domestic amount by the rate to get foreign currency. Whether a larger number means a stronger or a weaker domestic currency depends on which way the quote is written.

Example With the constructed spot rate $S_0 = 1.2$ domestic units per foreign unit, $500$ foreign units are worth $500 \cdot 1.2 = 600$ domestic units, and $600$ domestic units buy $600 / 1.2 = 500$ foreign units. If the rate rises to $1.25$ , the same $500$ foreign units are worth $625$ domestic units, so the foreign currency has become more expensive. Multiplying the $600$ domestic units by $1.2$ instead would give a number whose unit is domestic squared per foreign, a sure sign that the quote has been inverted.

FX forward

Definition An FX forward agreed at time $0$ with maturity $T$ and forward rate $K$ , in domestic units per foreign unit, obliges the long party to pay $K$ domestic units at $T$ and to receive one foreign unit at $T$ , for each foreign unit of notional. Nothing is paid when the contract is agreed. Valued in domestic currency at the spot rate of the maturity date, the long position pays

S_T - K

domestic units at $T$ , and the short position pays $K - S_T$ . The forward price $F_{0,T}$ is the rate $K$ that gives a newly agreed contract zero value. An existing contract keeps its $K$ while spot and interest rates move. Suppose default-free zero-coupon bonds trade in both currencies, with time- $t$ prices $P_d(t,T)$ in domestic units for one domestic unit paid at $T$ and $P_f(t,T)$ in foreign units for one foreign unit paid at $T$ , and that there are no frictions. Valuing the two legs separately, the law of one price gives the value of the long position at a time $t < T$ in domestic units,

\begin{aligned} V_t &= S_t P_f(t,T) \\[0.4em] &\quad - K P_d(t,T) \\[0.4em] &= P_d(t,T)\bigl(F_{t,T} - K\bigr), \end{aligned}

where $F_{t,T}$ is the forward price of a contract agreed at $t$ for the same maturity. The forward fixes the conversion rate. It does not provide funding in either currency and does not remove the risk that the counterparty fails to pay.

In plain words An FX forward is a currency exchange arranged today for a later date at a rate fixed today. Agreeing to it at the fair rate costs nothing. Afterwards it gains or loses value as the spot rate and the two interest rates move, while its own rate stays where it was set.

Example A long forward on $1{,}000$ foreign units at $K = 1.23$ pays $1{,}000 \cdot (1.26 - 1.23) = 30$ domestic units if the spot rate at maturity is $1.26$ , and $-30$ domestic units if it is $1.20$ . In the constructed market of the next card, with constant continuously compounded rates $r_d = 0.05$ and $r_f = 0.02$ , a one-year forward agreed at the fair rate $K \approx 1.2365454$ is worth nothing at first. Half a year later the spot rate is $1.25$ and the rates are unchanged. A new contract for the remaining half year has the forward price $F_{0.5,1} = 1.25\,e^{0.015} \approx 1.2688913$ , so the existing contract is worth

\begin{aligned} V_{0.5} &= e^{-0.025}\,(F_{0.5,1} - K) \\[0.4em] &\approx e^{-0.025} \cdot 0.0323459 \\[0.4em] &\approx 0.0315473 \end{aligned}

domestic units per foreign unit, about $31{,}547$ domestic units on a notional of $1{,}000{,}000$ foreign units.

Covered interest parity

Definition Assume that default-free zero-coupon bonds maturing at $T$ trade at time $0$ in both currencies, at the positive prices $P_d(0,T)$ and $P_f(0,T)$ , and that anyone can buy and sell them at these prices, which means borrowing and lending in each currency at the same rate. Assume also that currency can be exchanged at the spot rate $S_0$ and through the forward without transaction costs or restrictions, that no counterparty or bond issuer defaults, and that both routes below settle on the same dates. Then the forward price is

F_{0,T} = S_0 \, \frac{P_f(0,T)}{P_d(0,T)} .

Two strategies deliver exactly one foreign unit at $T$ and nothing else. The first buys $P_f(0,T)$ foreign units at spot, for $S_0 P_f(0,T)$ domestic units, and invests them in the foreign bond. The second agrees to a long forward at $K$ and buys $K$ domestic bonds for $K P_d(0,T)$ domestic units, which pay the $K$ owed at $T$ . The law of one price equates the two costs, and solving for $K$ gives the formula. If a quoted $K$ were larger, selling the forward and running the first strategy with borrowed domestic money would cost nothing today and pay $K - F_{0,T} > 0$ domestic units per foreign unit at $T$ in every state, an arbitrage, and reversing every trade does the same when $K$ is smaller. With constant continuously compounded rates $r_d$ and $r_f$ , the bond prices are $e^{-r_d T}$ and $e^{-r_f T}$ , and

F_{0,T} = S_0 \, e^{(r_d - r_f)T}.

With simple rates over one accrual period, whose day-count fractions are $\alpha_d$ and $\alpha_f$ under the conventions of the two currencies, the same bond-price formula gives $F_{0,T} = S_0 (1 + r_d \alpha_d)/(1 + r_f \alpha_f)$ . Because the argument uses only today’s bond prices, the bond-price form also holds when interest rates are random. The forward lies above spot exactly when $P_f(0,T) > P_d(0,T)$ , which for constant rates means $r_d > r_f$ . Covered parity is a replication identity. Uncovered interest parity, a hypothesis about the physical expectation of $S_T$ , is a different statement that needs economic assumptions and does not follow from replication.

In plain words Locking in foreign currency for a future date can be done by buying it now and earning the foreign rate on it, or by fixing the rate now and earning the domestic rate on the money set aside to pay. Both routes end with the same foreign unit, so in an ideal market they cost the same today. The currency with the higher interest rate trades at a forward discount, because the forward rate has to offset the difference in interest earned.

Example Take the constructed values $S_0 = 1.2$ , $r_d = 0.05$ , $r_f = 0.02$ and $T = 1$ , with continuous compounding. The forward price is

\begin{aligned} F_{0,1} &= 1.2\,e^{0.03} \\[0.4em] &\approx 1.2365454 \end{aligned}

domestic units per foreign unit, so the forward points $F_{0,1} - S_0$ are about $0.0365454$ . The first strategy buys $e^{-0.02} \approx 0.9801987$ foreign units for $1.2 \cdot 0.9801987 \approx 1.1762384$ domestic units. The second sets aside $1.2365454\,e^{-0.05} \approx 1.1762384$ domestic units, the same cost. Positive forward points do not mean that the foreign currency is expected to appreciate. They only reflect that the domestic rate is higher.

Domestic pricing measure

Definition Let $(\mathcal{F}_t)$ be the market filtration, and let the domestic and foreign money-market accounts be

\begin{aligned} B_d(t) &= e^{\int_0^t r_d(s)\,ds}, \\[0.4em] B_f(t) &= e^{\int_0^t r_f(s)\,ds}, \end{aligned}

where $r_d$ and $r_f$ are the adapted short rates of the two currencies, $B_d$ is measured in domestic units and $B_f$ in foreign units. A domestic pricing measure $\mathbb{Q}_d$ is a probability measure equivalent to $\mathbb{P}$ under which the domestic price of every traded asset without intermediate payments, divided by $B_d(t)$ , is a martingale with respect to $(\mathcal{F}_t)$ . Such a measure need not be unique. One unit of the foreign account is a traded asset worth $S_t B_f(t)$ domestic units, so the process

\frac{S_t B_f(t)}{B_d(t)}, \qquad 0 \le t \le T,

is a $\mathbb{Q}_d$ -martingale. When both short rates are deterministic, the bond prices are $P_d(t,T) = B_d(t)/B_d(T)$ and $P_f(t,T) = B_f(t)/B_f(T)$ , the account values come out of the conditional expectation, and

\begin{aligned} &\mathbb{E}^{\mathbb{Q}_d}[S_T \mid \mathcal{F}_t] \\[0.4em] &\quad = S_t \, \frac{P_f(t,T)}{P_d(t,T)} = F_{t,T}, \end{aligned}

which is $S_t e^{(r_d - r_f)(T - t)}$ for constant rates. When rates are random, the forward is instead the ratio

F_{0,T} = \frac{\mathbb{E}^{\mathbb{Q}_d}[S_T / B_d(T)]}{\mathbb{E}^{\mathbb{Q}_d}[1 / B_d(T)]},

the mean of $S_T$ under the domestic $T$ -forward measure, which in general differs from $\mathbb{E}^{\mathbb{Q}_d}[S_T]$ . The foreign pricing measure $\mathbb{Q}_f$ uses the numeraire $B_f$ and foreign units, and it is a different measure. Each domestic pricing measure corresponds to a foreign one through the density

\frac{d\mathbb{Q}_f}{d\mathbb{Q}_d}\bigg\vert_{\mathcal{F}_T} = \frac{S_T B_f(T)}{S_0 B_d(T)} ,

which is positive and has $\mathbb{Q}_d$ -mean $1$ by the martingale property above.

In plain words Measure every domestic price in units of the domestic bank account. Under the domestic pricing measure those relative prices have no drift, and the foreign deposit is one of them once it is converted into domestic currency. With deterministic rates this forces the pricing-measure mean of the future spot rate to equal the forward price. These are pricing weights, not forecasts. The physical mean of $S_T$ depends on the probabilities of the real world, and the forward price alone says nothing about it.

Example Keep the constructed values $S_0 = 1.2$ , $r_d = 0.05$ , $r_f = 0.02$ and $T = 1$ , and suppose that $S_T$ can take only the two values $F_{0,1} \pm 0.1$ , about $1.3365454$ and $1.1365454$ . The martingale condition requires $\mathbb{E}^{\mathbb{Q}_d}[S_T] = F_{0,1}$ , which forces the domestic pricing probability $\tfrac12$ on each value. Physical probabilities of $0.3$ on the higher value and $0.7$ on the lower one are equally consistent with no arbitrage, and they give a physical mean of about $1.1965454$ , below spot even though the forward is above it. Under $\mathbb{Q}_d$ the mean of the reciprocal rate is

\begin{aligned} &\tfrac12 \cdot \frac{1}{1.3365454} \\[0.4em] &\quad + \tfrac12 \cdot \frac{1}{1.1365454} \\[0.4em] &\approx 0.8140284, \end{aligned}

larger than $1/F_{0,1} \approx 0.8087046$ by Jensen’s inequality, since $x \mapsto 1/x$ is strictly convex. The density above moves the weights to about $0.5404352$ and $0.4595648$ under $\mathbb{Q}_f$ , and under these weights the mean of $1/S_T$ is exactly $1/F_{0,1}$ . A domestic-measure mean and a foreign-measure mean answer different questions.

Reciprocal quotation

Definition The reciprocal quotation writes the exchange rate as foreign units per domestic unit, $\tilde{S}_t = 1/S_t$ . A forward to receive one foreign unit for $K$ domestic units is, scaled by $1/K$ , an agreement to deliver one domestic unit for $1/K$ foreign units. So the fair forward in the reciprocal quotation is exactly the reciprocal of the original forward, and in its parity formula the two currencies exchange roles,

\begin{aligned} \tilde{F}_{0,T} &= \frac{1}{F_{0,T}} \\[0.4em] &= \tilde{S}_0 \, \frac{P_d(0,T)}{P_f(0,T)} \\[0.4em] &= \tilde{S}_0 \, e^{(r_f - r_d)T}, \end{aligned}

the last form for constant continuously compounded rates. An increase in $\tilde{S}_t$ means that the domestic currency has become more expensive. Forward points change sign under inversion, since $F_{0,T} > S_0$ exactly when $1/F_{0,T} < 1/S_0$ . With bid and ask quotes, inversion also exchanges the two sides, because the reciprocal of a bid in one quotation is an ask in the other.

In plain words Turning the quote upside down turns the forward upside down too, and the rate difference in the exponent changes sign with it. A rule that follows the units always works. The exponent is the rate of the currency in which the quote is counted minus the rate of the currency being priced.

Example In the constructed market the reciprocal spot rate is $1/1.2 \approx 0.8333333$ foreign units per domestic unit, and the reciprocal forward is $1/1.2365454 \approx 0.8087046$ , which equals $0.8333333\,e^{0.02 - 0.05}$ . The reciprocal forward points are about $-0.0246287$ foreign units per domestic unit, negative while the original points are positive. Applying the original exponent to the reciprocal spot would give $0.8333333\,e^{0.03} \approx 0.8587121$ , whose reciprocal $1.1645346$ is far from the forward $1.2365454$ in the original quotation.

Cross-currency basis

Definition A cross-currency basis is a spread $b$ , quoted under a stated convention, that is added to the interest rate of one named currency leg so that a parity-style formula reproduces a quoted forward or cross-currency swap. In the convention used here the spread is added to the foreign rate and the domestic rate is left unchanged. With constant continuously compounded rates the quoted forward is then written as

F^{\text{mkt}}_{0,T} = S_0 \, e^{(r_d - r_f - b)T},

and the basis implied by a quoted forward is

\begin{aligned} b &= r_d - r_f \\[0.4em] &\quad - \frac{1}{T}\ln\frac{F^{\text{mkt}}_{0,T}}{S_0} . \end{aligned}

The same forward corresponds to the spread $-b$ when the spread is added to the domestic rate instead, and market conventions also differ in the benchmark rates, the tenor and the compounding used. A nonzero basis says that the ideal hypotheses of covered parity fail for the benchmark rates chosen, for example because not everyone can borrow at them, or because of collateral and counterparty terms, balance-sheet limits or transaction costs. It is not by itself an arbitrage. A single constant spread summarizes one maturity, while real bases vary with tenor and are modeled with collateralized cross-currency curves, which this lesson does not treat.

In plain words When a quoted forward does not match ideal parity, ask how much the foreign leg’s rate would have to change to close the gap. That number is the basis. It records a measured gap under a named placement and is not a free profit, because the trade that would exploit the gap needs funding that may not be available at the benchmark rates.

Example With the constructed values $S_0 = 1.2$ , $r_d = 0.05$ , $r_f = 0.02$ and $T = 1$ and a spread $b = 0.005$ on the foreign rate,

\begin{aligned} F^{\text{mkt}}_{0,1} &= 1.2\,e^{0.025} \\[0.4em] &\approx 1.2303781, \end{aligned}

about $0.0061673$ below the ideal parity forward $1.2365454$ . The spread raises the effective foreign rate from $0.02$ to $0.025$ and narrows the rate difference in the exponent. With the spread placed on the domestic leg, the same forward needs $-0.005$ , since $1.2\,e^{0.05 - 0.005 - 0.02}$ is the same number.

Check yourself

The spot rate is $1.5$ domestic units per foreign unit, and the constant continuously compounded rates are $r_d = 0.01$ and $r_f = 0.04$ . Under the ideal hypotheses of covered parity, find the six-month forward price, the forward points and the forward in the reciprocal quotation.

Show the answer

The rate difference is $r_d - r_f = -0.03$ , and covered parity gives

\begin{aligned} F_{0,0.5} &= 1.5\,e^{-0.03 \cdot 0.5} \\[0.4em] &= 1.5\,e^{-0.015} \\[0.4em] &\approx 1.4776679 \end{aligned}

domestic units per foreign unit. The forward points are about $-0.0223321$ , negative because the foreign rate is the higher one. As a check, buying $e^{-0.02} \approx 0.9801987$ foreign units costs $1.5 \cdot 0.9801987 \approx 1.4702980$ domestic units, and setting aside the forward payment costs $1.4776679\,e^{-0.005} \approx 1.4702980$ domestic units. The reciprocal forward is $1/1.4776679 \approx 0.6767420$ foreign units per domestic unit, which equals $(1/1.5)\,e^{0.015}$ and lies above the reciprocal spot $0.6666667$ .

A data feed shows the spot rate $0.8$ , quoted as foreign units per domestic unit. The constant continuously compounded rates are $r_d = 0.05$ and $r_f = 0.02$ . A classmate computes the one-year forward as $0.8\,e^{0.03} \approx 0.8243636$ . Find the mistake and give the correct forward in both quotations.

Show the answer

In this quotation the currency being priced is the domestic one and the quote is counted in foreign units, so the exponent is $r_f - r_d$ , not $r_d - r_f$ . The correct forward is $0.8\,e^{-0.03} \approx 0.7763564$ foreign units per domestic unit. In domestic units per foreign unit the spot rate is $1/0.8 = 1.25$ and the forward is

\begin{aligned} F_{0,1} &= 1.25\,e^{0.03} \\[0.4em] &\approx 1.2880682, \end{aligned}

and indeed $1/1.2880682 \approx 0.7763564$ . The classmate’s number corresponds to $1/0.8243636 \approx 1.2130569$ domestic units per foreign unit, a forward below spot although the domestic rate is the higher one, which contradicts covered parity.

Take $S_0 = 1.2$ domestic units per foreign unit, constant continuously compounded rates $r_d = 0.05$ and $r_f = 0.02$ , and a one-year forward quoted at $K = 1.23$ . Under the ideal hypotheses, build a trade that costs nothing today on a notional of $1{,}000{,}000$ foreign units and find its payoff at maturity. Then find the basis $b$ on the foreign leg implied by the quote and explain why the trade need not be an arbitrage in practice.

Show the answer

The parity forward is $1.2\,e^{0.03} \approx 1.2365454$ , above the quote, so the forward is the cheap way to obtain foreign currency. Borrow $1{,}000{,}000\,e^{-0.02} \approx 980{,}198.67$ foreign units, which means owing $1{,}000{,}000$ foreign units at maturity. Exchange them at spot for about $1{,}176{,}238.41$ domestic units and invest these at the domestic rate, which gives about $1{,}236{,}545.44$ domestic units at maturity. Buy $1{,}000{,}000$ foreign units forward at $1.23$ . At maturity pay $1{,}230{,}000$ domestic units under the forward, receive $1{,}000{,}000$ foreign units and repay the foreign loan with them. The trade costs nothing today and pays about $6{,}545.44$ domestic units at maturity in every state, which equals $1{,}000{,}000 \cdot (1.2365454 - 1.23)$ .

The implied basis on the foreign leg is

\begin{aligned} b &= 0.03 - \ln\frac{1.23}{1.2} \\[0.4em] &\approx 0.03 - 0.0246926 \\[0.4em] &\approx 0.0053074 . \end{aligned}

If the trader can borrow foreign currency only at the rate $r_f + b \approx 0.0253074$ , the foreign loan costs more and the payoff of the trade is exactly zero. The gap is an arbitrage only for someone who can really borrow foreign currency at $0.02$ and lend domestic currency at $0.05$ , with no transaction costs, no counterparty or collateral effects and no balance-sheet limits. A quoted basis says that these ideal terms do not hold for the benchmark rates, not that a free profit is available.

Foreign exchange and covered interest parity

Key terms

Exchange-rate quotation

FX forward

Covered interest parity

Domestic pricing measure

Reciprocal quotation

Cross-currency basis

Check yourself

How this lesson connects

Builds on

Used later in