# Statistical Arbitrage II : Simple FX Arbitrage Models

In the context of the foreign exchange markets, there are several simple no-arbitrage conditions, which, if violated outside of the boundary conditions imposed by transaction costs, should provide the arbitrageur with a theoretical profit when market conditions converge to theoretical normality.

Detection of arbitrage conditions in the FX markets requires access to high-frequency tick data, as arbitrage opportunities are usually short-lived. Various market inefficiency conditions exist in the FX markets. Apart from the basic strategies outlined in the following sections, other transient opportunities may exist, if the trader or trading system can detect and act on them quickly enough.

Round-Trip Arbitrage
Possibly the most well-known no-arbitrage boundary condition in foreign exchange is the covered interest parity condition. The covered interest parity condition is expressed as:

$(1+r_d) = \frac{1}{S_t}(1+r_f)F_t$

which specifies that it should not be possible to earn positive return by borrowing domestic assets at \$$$r_d$$ for lending abroad at $$r_f$$ whilst covering the exchange rate risk through a forward contract $$F_t$$ of equal maturity.

Accounting for transaction costs, we have the following no-arbitrage relationships:

$(1+r_d^a) \geq \frac{1}{S^a}(1+r_f^b)F^b$
$(1+r_f^b) \geq S^b(1+r_d^b)\frac{1}{F^a}$

For example, the first condition states that the cost of borrowing domestic currency at the ask rate ($$1+r_d^a$$) should be at least equal to the cost of converting said currency into foreign currency ($$\frac{1}{s^a}$$) at the prevailing spot rate $$S^a$$ (assuming that the spot quote $$S^a$$ represents the cost of a unit of domestic currency in terms of foreign currency), invested at $$1+r_f^b$$, and finally converted back into domestic currency via a forward trade at the ask rate ($$F^a$$). If this condition is violated, then we can perform round-trip arbitrage by converting, investing, and re-converting at the end of the investment term. Persistent violations of this condition are the basis for the roaring carry trade, in particular between currencies such as the Japanese Yen and higher yielding currencies such as the New Zealand dollar and the Euro.

Triangular Arbitrage
A reduced form of FX market efficiency is that of triangular arbitrage, which is the geometric relationship between three currency pairs. Triangular arbitrage is defined in two forms, forward arbitrage and reverse arbitrage. These relationships are defined below.

$$\left(\frac{C_1}{C_2}\right)_{ask} \left(\frac{C_2}{C_3}\right)_{ask} = \left(\frac{C_1}{C_3}\right)_{bid} \\ \left(\frac{C_1}{C_2}\right)_{bid} \left(\frac{C_2}{C_3}\right)_{bid} = \left(\frac{C_1}{C_3}\right)_{ask}$$

With two-way high-frequency prices, we can simultaneously calculate the presence of forward and reverse arbitrage.

A contrived example follows: if we have the following theoretical two-way tradeable prices: $$\left(\frac{USD}{JPY}\right) = 90/110$$, $$\left(\frac{GBP}{USD}\right) = 1.5/1.8$$, and $$\left(\frac{JPY}{GBP}\right) = 150/200$$. By the principle of triangular arbitrage, the theoretical two-way spot rate for JPY/GBP should be $$135/180$$. Hence, we can see that JPY is overvalued relative to GBP. We can take advantage of this inequality as follows, assuming our theoretical equity is 1 USD:

• Pay 1 USD and receive 90 JPY ;
• Sell 90 JPY ~ and receive $$\left(\frac{90}{135}\right)$$ GBP ;
• Pay $$\left(\frac{90}{135}\right)$$ GBP, and receive $$\left(\frac{90}{135}\right) \times$$ 1.5 USD = 1.32 USD.

We can see that reverse triangular arbitrage can detect a selling opportunity (i.e. the bid currency is overvalued), whilst forward triangular arbitrage can detect a buying opportunity (the ask currency is undervalued).

The list of candidate currencies could be extended, and the arbitrage condition could be elegantly represented by a data structure called a directed graph. This would involve creating an adjacency matrix $$R$$, in which an element $$R_{i,j}$$ contains a measure representing the cost of transferring between currency $$i$$ and currency $$j$$.

Estimating Position Risk
When executing an arbitrage trade, there are some elements of risk. An arbitrage trade will normally involve multiple legs which must be executed simultaneously and at the specified price in order for the trade to be successful. As most arbitrage trades capitalize on small mispricings in asset values, and rely on large trading volumes to achieve significant profit, even a minor movement in the execution price can be disastrous. Hence, a trading algorithm should allow for both trading costs and slippage, normally by adding a margin to the profitability ratio. The main risk in holding an FX position is related to price slippage, and hence, the variance of the currency that we are holding.