Forex Quotes, Spreads, and Triangular Arbitrage

Exchange Rates

An exchange rate is simply the price of one currency in terms of another. For example, a quote of 1.4126 USD/EUR means that each euro costs $1.4126. In this example, the euro is called the base currency and the USD the price currency.

Hence, a quote is the price of one unit of the base currency in terms of the price currency.

A spot exchange rate is the currency exchange rate for immediate delivery, which for most currencies means the exchange of currencies takes place two days after the trade. A forward exchange rate is a currency exchange rate for an exchange to be done in the future. Forward rates are quoted for various future dates (e.g., 30 days, 60 days, 90 days, or one year).

A forward contract is an agreement to exchange a specific amount of one currency for a specific amount of another currency on a future date specified in the forward agreement.

Dealer quotes often include both bid and offer (ask) rates. For example, the euro could be quoted as $1.4124 − 1.4128. The bid price ($1.4124) is the price at which the dealer will buy euros, and the offer price ($1.4128) is the price at which the dealer will sell euros.

By convention, quotes often only include the last two digits for the offer price; in other words, the euro can also be quoted as $1.4124 − 28.

Foreign Exchange Spread

The difference between the offer and bid price is called the spread. Spreads are often stated as “pips.” When the spot quote has four decimal places, one pip is 1/10,000. In the previous example, the spread is $0.0004 (4 pips) reflecting the dealer’s profit.

Dealers manage their foreign currency inventories by transacting in the interbank market (think of this as a wholesale market for currency). Spreads are narrow in the interbank market.

The spread quoted by a dealer depends on:

• The spread in an interbank market for the same currency pair. Dealer spreads vary directly with spreads quoted in the interbank market.
• The size of the transaction. Larger, liquidity demanding transactions generally get quoted a larger spread.
• The relationship between the dealer and client. Sometimes dealers will give favorable rates to preferred clients based on other ongoing business relationships.

The interbank spread on a currency pair depends on:

• Currencies involved. Similar to stocks, high volume currency pairs (e.g., USD/EUR, USD/JPY, and USD/GBP) command lower spreads than do lower volume currency pairs (e.g., AUD/CAD).
• Time of day. The time overlap during the trading day when both the New York and London currency markets are open is considered the most liquid time window; spreads are narrower during this period than at other times of the day.
• Market volatility. Spreads are directly related to the exchange rate volatility of the currencies involved. Higher volatility leads to higher spreads to compensate market makers for the increased risk of holding those currencies.

  Spreads change over time in response to volatility changes.

In addition to these factors, spreads in forward exchange rate quotes increase with maturity. The reasons for this are: longer maturity contracts tend to be less liquid, counterparty credit risk in forward contracts increases with maturity, and interest rate risk in forward contracts increases with maturity.

Cross Rate

The cross rate is the exchange rate between two currencies implied by their exchange rates with a common third currency. It is necessary to use cross rates when there is no active foreign exchange (FX) market in the currency pair being considered. The cross rate must be computed from the exchange rates between each of these two currencies and a major third currency, usually the USD or EUR.

Suppose we have the following quotes:

USD/AUD = 0.60 and MXN/USD = 10.70. What is the cross rate between Australian dollars and pesos (MXN/AUD)?

So our MXN/AUD cross rate is 6.42 pesos per Australian dollar. The key to calculating cross rates is to make sure the common currency cancels out.

Cross Rates with Bid and Ask Spreads

Bid and ask spreads complicate the calculation of cross rates considerably. Suppose we are given three currencies A, B, and C; we can have three pairs of currencies (i.e., A/B, A/C, and B/C).

Rules:

To compute the cross rate for A/C, given A/B and B/C, we can follow these rules to obtain the bid and offer prices. If we are instead given A/B and C/B rates, we will have to make adjustments to obtain the B/C bid and offer rates from the C/B bid and offer rates, because A/B × C/B ≠ A/C. The process is as follows:

Triangular Arbitrage

Real world currency dealers will maintain bid/ask quotes that ensure a profit to the dealer, regardless of which currencies customers choose to trade. If this was not the case, customers could earn profits through the process of triangular arbitrage. In triangular arbitrage, we begin with three pairs of currencies, each with bid and ask quotes, and construct a triangle where each node in the triangle represents one currency.

To check for arbitrage opportunities, we go around the triangle clockwise (and later, counterclockwise) until we reach our starting point. As before, we follow the up the bid and multiply, down the ask and divide rule.

The following example will illustrate triangular arbitrage.

Worked example: Triangular arbitrage

The following quotes are available from the interbank market:

Quotes:

Currency pair	Bid	Offer
USD/AUD	0.6000	0.6015
USD/MXN	0.0933	0.0935

1. Compute the implied MXN/AUD cross rate.

2. If your dealer quotes MXN/AUD = 6.3000 − 6.3025, is an arbitrage profit possible? If so, compute the arbitrage profit in USD if you start with USD 1 million.

Solution

1. To compute implied cross rates, we need:

$${\left(\frac{\mathrm{MXN}}{\mathrm{AUD}}\right)}_{\mathrm{bid}}={\left(\frac{\mathrm{USD}}{\mathrm{AUD}}\right)}_{\mathrm{bid}}\times {\left(\frac{\mathrm{MXN}}{\mathrm{USD}}\right)}_{\mathrm{bid}}$$

Since we are given USD/MXN quotes instead of MXN/USD quotes, we first invert these quotes:

$${\left(\frac{\mathrm{MXN}}{\mathrm{USD}}\right)}_{\mathrm{bid}}=\frac{1}{(\mathrm{USD}/\mathrm{MXN}{)}_{\mathrm{offer}}}=\frac{1}{0.0935}=10.70MXN/USD$$

$${\left(\frac{\mathrm{MXN}}{\mathrm{USD}}\right)}_{\mathrm{offer}}=\frac{1}{(\mathrm{USD}/\mathrm{MXN}{)}_{\mathrm{bid}}}=\frac{1}{0.0933}=10.72MXN/USD$$

Now, the implied cross rates:

$$(MXN/AUD{)}_{\mathrm{bid}}=(USD/AUD{)}_{\mathrm{bid}}(MXN/USD{)}_{\mathrm{bid}}=0.60\times 10.70=6.42$$

$$(MXN/AUD{)}_{\mathrm{offer}}=(USD/AUD{)}_{\mathrm{offer}}(MXN/USD{)}_{\mathrm{offer}}=0.6015\times 10.72=6.4481$$

2. Since the dealer quote of MXN/AUD = 6.3000 − 6.3025 falls outside of these cross rates, arbitrage profit may be possible (we have to check this). Remember to use the dealer quotes in the triangle and not the cross rates we computed.

To label the arrows in this triangle, we follow the “up the bid, down the offer” rule. To convert from USD to MXN, (“down” with respect to the USD/MXN quote), we use the offer rate of 0.0935.

Going clockwise and starting with USD 1 million:

1. Convert USD 1 million into MXN @ 0.0935 USD/MXN. Note that the quote is USD/MXN and hence we are going down, and thus need to use the ask. Also remember: down, divide.

We get 1 million/0.0935 = 10,695,187 MXN.

2. Next, we convert 10,695,187 MXN into AUD @ 6.3025 MXN/AUD to get 1,696,975 AUD.

3. Finally, we convert AUD 1,696,975 into USD @ 0.6000 USD/AUD. Here the quote is USD/AUD and we are converting from AUD to USD, so we are going “up the quote” and need to multiply by the bid. (Remember: up, multiply.) We get 1,696,975 × 0.60 = 1,018,185 USD − a profit of 18,185 USD.

We can also check for arbitrage in the counter clockwise direction (even though we can never earn an arbitrage profit in both directions):

1. Convert USD 1 million into AUD using 0.6015. Again, the quote is USD/AUD and we are going down, so use the ask price and divide. We get 1 million/0.6015 = 1,662,510 AUD.

2. Next, we convert 1,662,510 AUD into MXN using 6.3000 to get 10,473,814 MXN.

3. Finally, we convert MXN 10,473,814 into USD at 0.0933 to get 977,207 USD − a loss of 22,793 USD.

A currency is quoted at a forward premium relative to a second currency if the forward price (in units of the second currency) is greater than the spot price. A currency is quoted at a forward discount relative to a second currency if the forward price (in units of the second currency) is less than the spot price. The premium or discount is for the base currency (i.e., the currency at the bottom of the quote).

For example, if the spot price is 1.20$/€ and the forward price is 1.25$/€, we say that the euro is trading at a forward premium.

Given a quote of A/B, if this equation results in a positive value, we say that currency B (i.e., the base currency) is trading at a premium in the forward market.

In the FX markets, forward quotes are often presented as a premium or discount over spot rates. The following example illustrates this convention.