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Foreign Exchange Smiles

  1. Robert G. Tompkins

Published Online: 15 MAY 2010

DOI: 10.1002/9780470061602.eqf06008

Encyclopedia of Quantitative Finance

Encyclopedia of Quantitative Finance

How to Cite

Tompkins, R. G. 2010. Foreign Exchange Smiles. Encyclopedia of Quantitative Finance. .

Author Information

  1. Frankfurt School of Finance and Management, Frankfurt, Germany

Publication History

  1. Published Online: 15 MAY 2010

Abstract

It is well known that prices of foreign exchange options systematically diverge from those consistent with the Garman-Kohlhagen and Black option-pricing models. The implied volatilities of options on the same underlying foreign exchange rate differ across strike prices and terms to expiration. When implied volatilities for the same expiration are plotted versus strike prices and a corresponding function relationship is determined, the result is a convex pattern, which is commonly known as a smile. A logical first step in formulating a theory to explain this phenomenon would be to better understand the empirical record of implied volatility smiles. If time invariant regularities are discovered, this will provide a benchmark to compare implied volatility smiles associated with alternative models. In this article, we summarize the empirical record of implied volatility smiles for four alternative currency pairs: British pound, Swiss franc, Deutsche mark, and Japanese yen all versus the US dollar. The time period of analysis is from 1985 to 2000 and it considers options on futures traded at the Chicago Mercantile Exchange.

This article then considers models that have been proposed to explain the existence of implied volatility smiles. Neither stochastic volatility model nor jump-diffusion models are able to generate similar implied volatility smile dynamics. An alternative model that combines both stochastic volatility and jump processes appears to explain the smile surfaces better. While the dynamics are matched, the smiles are not as extreme (in curvature) as the empirical smiles. The differences between the empirical and model generated smile patterns suggest a systematic and substantive negative risk premium. Finally, transaction costs are added to the best fitting theoretical model and the risk premium is substantially reduced. We conclude that both an alternative process for foreign exchange returns and transaction costs explain why implied volatility smiles for options on foreign exchange exist.

Keywords:

  • implied volatility smiles;
  • stochastic volatility;
  • jump diffusion;
  • transaction costs