98 CHAPTER 3. SPOT MARKETS FOR FOREIGN CURRENCY
convertgbpinto Aussies, onegbpshould buy 1.5× 1.6 = 2.4aud. With this
aud/gbprate and assuming a zero spread,
- nobody can make a free-lunch profit by any sequence of transactions, and
- everyone is indifferent between direct conversions between two currencies and
indirect, triangular transactions.
Below, we see what the implications of arbitrage and shopping around are when
there are bid-ask spreads. In order to simplify matters, we shall first show how
to compute the implied rates from an indirect route. We shall call these implied
rates thesyntheticrates. Having identified these synthetic rates, we can then invoke
the same mechanisms that enforce the Law of One Price as when we studied the
relationship between the quotes made by various market makers.
Computing Synthetic Cross-Rates
In general, a synthetic version of a contract is a combination of two or more other
transactions that achieves the same objective as the original contract. That is, the
combination of the two or more contractsreplicates the outcome of the original
contract. We shall use the notion of replication repeatedly in this textbook. For
now, consider a simple spot transaction: a Japanese investor wants to convertjpy
intogbp.
- The investor can use the direct market and buygbpagainstjpy. We will call
this the original contract. - Alternatively, the investor can first buyusdwithjpy, and then immediately
exchange theusdforgbp. This is a combination of two contracts. It replicates
the original contract since, by combining the two transactions, the investor
initially paysjpy, and ultimately ends up withgbp. Thus, this is a synthetic
way of achieving the original transaction.
Note that the synthetic contract may be the more efficient way to deal, since
theusdmarket has a lot of volume (or depth) in every country, and therefore has
smaller spreads. (This is why theusdis involved in 90 percent of the trades). Let
us see how the syntheticjpy/gbprates can be computed.
Example 3.13
What are the syntheticjpy/gbprates, bid and ask, if the quotes arejpy/usd101.07
- 101.20 andusd/gbp1.3840 -1.3850?
Step 1: multiply or divide? The dimension of the rate we are looking for is
jpy/gbp. Because the dimensions of the two quotes given to us areusd/gbpand