>> For this week’s Learning Curve in PDF format, click here. The number of multi-currency exotic options is large and growing. They naturally appeal to large international corporations who need to hedge their exposures in different currencies. Multi-currency options also attract sophisticated speculators having particular views on correlated movements of different foreign exchange rates, or FXRs. The most popular contracts include basket and barrier type options. It is common practice to estimate “true” option value via the expectation of the option payoff with respect to the so-called risk-neutral densities of the underlying assets.
The value of a multi-currency option appears to depend on correlations between underlying FXRs. In practice, it can be difficult to calculate correlations from historical returns because one has to decide on the length and frequency of time series and how to weight past returns. The estimates may not be stable: if the data are split into groups then the correlations for each group may differ significantly. Also, the option value should depend on correlations during the life of the option, which are predicted future correlations. Another possibility is to find implied correlations from market prices of contracts that estimate the market’s perception of correlation.
In this Learning Curve, we assume there are liquid vanillas for all currencies involved in the multi-currency option and estimate correlations using implied volatilities of these vanillas. To price options written on several FXRs with the same denominating currency, financial practitioners and traders often use implied correlations calculated from implied volatilities of FXRs that form “currency triangles”. Some options, however, may have underlying FXRs with different denominating currencies. In this Learning Curve, we present the formulas for the implied correlations between FXRs when denominating currencies are the same and different. The latter is not widely known. Using implied volatilities at different maturities, it is possible to approximate time dependent instantaneous correlations to price barrier options more accurately. Also, given arguments that objective and risk-neutral dependence structures are the same under certain general conditions (see e.g. J. V. Rosenberg, Non-Parametric Pricing of Multivariate Contingent Claims, The Journal of Derivatives 10 (3), pp.9-26, 2003) the implied correlations can be used in forecast analysis instead of correlations estimated from historical returns. The implied correlation can better estimate the predicted future correlation since it reacts quickly to market changes (see e.g. Campa, J.M. and Chang, P.H.K. (1998), The Forecasting Ability of Correlations Implied in Foreign Exchange Options, The Journal of International Money and Finance 17, pp.855-880).
Model And Notation
It is common practice--for FX option valuation purposes--to assume FXRs follow a multivariate geometric Brownian risk-neutral process
These correlations are commonly used to price multi-currency options where the payoff depends on several FXRs with the same denominating currency, e.g. basket options.
Part 2 of this Learning Curve will look at the implied correlation of foreign exchange rates with different denominating currencies.
This week’s Learning Curve was written by Dr Pavel Shevchenko, at the Commonwealth Scientific and Industrial Research Organisation--Mathematical and Information Sciences, in Sydney, Australia.