Option prices and risk‐neutral densities for currency cross rates

Abstract

We thank the referee for the helpful advice. This research has benefited from numerous conversations with Söhnke Bartram, and the assistance of Aris Bikos with data collection is gratefully acknowledged. We are indebted to the seminar participants at the OPTION PRICES AND RISK-NEUTRAL DENSITIES FOR CURRENCY CROSS RATES STEPHEN J. TAYLOR YAW-HUEI WANG*The theoretical relationship between the risk-neutral density (RND) of the euro/ pound cross rate and the bivariate RND of the dollar/euro and the dollar/pound rates is derived; the required bivariate RND is defined by the dollar-rate marginal RNDs and a copula function. The cross-rate RND can be used by banks, international businesses, and central bankers to assess market expectations, to measure risks, and to value options, without relying on over-the-counter markets, which may be either non-existent or illiquid. Empirical comparisons are made between cross-rate RNDs estimated from several data sets. Five one-parameter copula functions are evaluated and it is found that the Gaussian copula is the only oneparameter copula function that is ranked highly in all of the comparisons we have made.

Using Copulas to Construct Bivariate Foreign Exchange Distributions with an Application to the Sterling Exchange Rate Index

“…The marginal distributions are given by univariate risk-neutral densities estimated using the Malz (1997) method and the parameter of the copula function is chosen in such a way that the empirical correlation coefficient (computed from the variances of the three margins) equals the implied correlation coefficient (computed from ATM volatilities). A very similar approach has been taken in a recent contribution by Taylor and Wang (2004), who also fit the implied correlation coefficient, but use a more refined setup which ensures that the implied joint density belongs to a common risk-neutral numeraire measure. Both studies (Bikos, and Taylor and Wang) find that one-parameter copulas provide a good fit to the data but essentially use one observation to fit a single parameter.…”

“…There is a very limited literature on the derivation of multivariate risk-neutral distributions and existing approaches are either based on fitting the implied correlation coefficient (Bikos (2000) and Taylor and Wang (2004)), individual option contracts (Bennett and Kennedy (2004)), or historical realised data (Rosenberg (2003)) in order to determine the dependence pattern. In contrast our method is novel in the sense that we exploit all available information in a set of (option-implied) density functions.…”

Bounds and Prices of Currency Cross-Rate Options

“…As suggested by Bakshi, Carr, and Wu (2007), it may be better to specify the generic pricing kernels in each country to derive the prices of derivatives. Nonetheless, Taylor and Wang (2005) analytically show that the prices of derivatives under different measures are equivalent when the law of prices of two related dollar-rate options. Instead of directly exploring the option pricing formula, this study (also using the dollar risk-neutral measure) derives the pricing bounds for cross-rate options by utilizing the exchange option pricing bounds implied in the copula theory.…”

“…Since there is a triangular relationship between the foreign exchange rates among three currencies, Taylor and Wang (2005) show that it is plausible to estimate risk-neutral densities (RNDs) and option prices of a cross-rate under the US dollar measure 2 using the market 1 The motivation for doing this is as follows. It is generally observed that options on dollar-denominated exchange rates are traded under satisfactory liquidity, while cross-rate option markets are much less liquid.…”

“…Thus, the pricing bounds obtained from the liquid market prices of dollar-rate options are useful for pricing, hedging, and arbitraging. 2 Both Taylor and Wang (2005) and this paper commence the analyses under the foreign (dollar) risk neutral measure to price the cross-rate options. As suggested by Bakshi, Carr, and Wu (2007), it may be better to specify the generic pricing kernels in each country to derive the prices of derivatives.…”