Currency trios - using geometric concepts to visualise and interpret relationships between currencies

Master Thesis


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University of Cape Town

A currency trio is a set of three currencies and their respective exchange rates, which have a relationship fixed by a triangular arbitrage condition. This condition forms the basis for the derivation of a geometric interpretation of the relationships between the exchange rates. In the geometric framework, the three currencies in a currency trio are represented by a triangle, where each of the vertices represents a currency. The volatilities of the exchange rates are represented by the lengths of the sides joining the respective currencies and the cosine of each angle represents the correlation between the two exchange rates depicted by the angle's adjacent sides. The geometric approach is particularly useful when dealing with implied data as it allows the calculation of implied correlation using implied volatility. This is valuable as implied volatility is frequently quoted in the foreign exchange market; whereas, implied correlation is not directly quoted and is more difficult to extract from market data. This dissertation aims to thoroughly investigate the geometric framework and use it to visualise and interpret the relationships between currencies in a currency trio. The analysis will initially look at currency trios with realised spot data before moving on to implied data. In the implied data context, the framework will be used to extract and evaluate implied correlation estimates using implied volatility data extracted from the foreign exchange market. The framework will be extended to investigate whether an illiquid option can be proxy hedged using options on the two other currencies in a currency trio. Finally, the findings will be discussed and the feasibility of the applications of the framework will be considered.