5.5978 x 0.7562 = 4.2330 francs.
In other words, the bank would be ready to sell one Can$ for 4.2330 francs
So the synthetic (SFr/Can$)ask rate
= 4.2330
= 5.5978 x 0.7562
= (SFr/$)ask x ($/Can$)ask
Hence, the synthetic quote is:
SFr/Can$ : 4.2286/4.2330
These rates can be generalized as:
Synthetic (A/C)bid = (A/B)bid x (B/C)bid (Eq. 3.1)
Synthetic (A/C)ask = (A/B)ask x (B/C)ask (Eq. 3.2)
where A, B and C are three currencies.
As in case of implied inverse rate, the synthetic quote and the actual quote between a pair of currencies should overlap (i.e., the bid rate of one should always be lower than the ask rate of the other). There are two reasons for this. First, if the actual rates are too much out of line with the cross rates, then market players in genuine need of a currency would buy and sell through the markets giving them more favorable rates. The second reason is the arbitrage opportunity which would arise in case of a misalignment of actual and cross rates. In both the cases, the resultant demand-supply mismatch would force the synthetic cross rate and the actual rate to come in line with one another. Let us see how the arbitrage process works. As we have seen, the synthetic quote between the franc and the Can$ is:
SFr/Can$ : 4.2286/4.2330
This synthetic quote has been calculated from the following given quotes:
SFr/$ : 5.5971/5.5978
$/Can$ : 0.7555/0.7562
Now suppose that the actual quote between the franc and the Can$ is:
SFr/Can$ : 4.2333/4.2343
As we see, the synthetic ask rate is less than the actual bid rate, giving the arbitrageurs a chance to make a profit by three-point arbitrage (the process of making arbitrage profits involving three markets, where three transactions have to be entered into to achieve the desired results). To make profits, a person should buy low and sell high. The rate at which the arbitrageur, say X, can sell one Can$ against the franc is SFr 4.2333/Can$. The rate at which X can buy one Can$ (through the synthetic market) is SFr 4.2330/Can$. Let us say that X starts with one franc.
With one franc, he can buy: dollars.
Since 0.7562 dollars fetch one Can$, with 1/5.5978 dollars X can buy: Can$
These Can$ can then be sold by X in the SFr/Can$ market for: 4.2333 x francs
= SFr1.000058.
Thus, X makes a profit of SFr 0.000058 for every franc bought and sold.
Now let us see what will happen if the actual rates are
SFr/Can$ : 4.2278/4.2283
The actual ask rate is now lower than the synthetic bid rate. X can now buy Can$ at SFr 4.2283/Can$ and sell them through the synthetic market at SFr 4.2286/Can$. In the SFr/Can$ market, X can sell one franc for
Can$
As each Can$ can be sold for 0.7555 dollars, X can sell the Can$ for:
0.7555 x dollars
Since each dollar fetches 5.5971 francs, X can sell the dollars for:
5.5971 x 0.7555 x francs
= SFr1.000073 francs.
So, X makes a profit of 0.000073 francs for every franc bought and sold.
These arbitrage processes will adjust the rates in both the cases in all the three markets in such a way, that the actual SFr/Can$ rates will come in alignment with the synthetic rates. We can write the conditions for no arbitrage possibility as:
(A/C)bid (actual) (A/C)ask (synthetic) (Eq.3.3)
(A/C)ask (actual) (A/C)bid (synthetic) (Eq.3.4)
Using equations 3.1 and 3.2, we can rewrite the above equations as:
(A/C)bid (A/B)ask x (B/C)ask (Eq. 3.5)
(A/C)ask (A/B)bid x (B/C)bid (Eq. 3.6)
where all the rates are actual rates.
Equations 3.5 and 3.6 are called the no-arbitrage conditions. These signify the limits imposed by the synthetic rates on the actual quote for a pair of currencies (upper and lower limits for the bid and the ask rates respectively). The actual rates only have to be within these limits but they need not necessarily be the same as the synthetic rates. In fact, the synthetic rate having been calculated from two quotes, includes the bid-ask spread of both the quotes. This results in the synthetic rate having a very high bid-ask spread. In reality, a bank giving direct quotes between two such currencies may be able to quote at much lower spread, provided its business volumes in these currencies is high. In the example given above, the actual SFr/Can$ quote may be something like:
SFr/Can$ : 4.2298/4.2313
As in the case of inverse rates, transaction costs allow the actual A/C quote to deviate from the synthetic cross rates to some extent. As mentioned earlier, in the absence of such costs, the bid and the ask rates will be the same. These single rates will force the actual A/C quote to be exactly equal to the synthetic cross rates.
According to Eq. 3.5,
(A/C)bid (A/B)ask x (B/C)ask
It can be rewritten as:
or, (A/B)ask x (B/C)ask x (C/A)ask ³ 1 (Eq. 3.7)
Similarly, Eq. 3.6 says that:
(A/C)ask ³ (A/B)bid x (B/C)bid
It can be rewritten as:
or, (A/B)bid x (B/C)bid x (C/A)bid £ 1 (Eq. 3.8)
Types of Transactions
Foreign exchange transactions can be classified on the basis of the time between entering into a transaction and its settlement. They can basically be classified into spot and forward contracts. Spot transactions are those, which are settled after 2 business days from the date of the contract. A forward contract (also called an outright forward) is one where the parties to the transaction agree to buy or sell a commodity (here, a currency) at a predetermined future date at a particular price. This future date may be any date beyond two business days. The price and the terms of delivery and payment are fixed at the time of entering into the contract. In the forex markets, forward contracts generally mature after 1, 2, 3, 6, 9, or 12 months.
A forward contract is normally entered into to hedge oneself against exchange risk (i.e., the uncertainty regarding the future movements of the exchange rate). By entering into a forward contract, the customer locks-in the exchange rate at which he will buy or sell the currency.