Extracting the Joint Volatility Structure of Foreign Exchange and Interest Rates from Option Prices

To companies operating in an ever more globalised marketplace, fluctuating exchange rates and volatile interest rates represent two significant sources of risk. Derivative financial instruments such as options are increasingly used to protect against unwanted exposures, in a way like one would buy insurance to safeguard against adversity. It has long been understood that the dynamics of exchange rates and interest rates are linked by fundamental economic relationships, but to date no attempt has been made to employ a unified framework in which information about the volatilities and correlations of these economic variables can be extracted from the prices of actively traded options. Efficient financial markets subsume all information about an asset in its current price and it is a standing hypothesis in models of these markets that future asset prices are not predictable. The purpose of models for pricing by arbitrage in general and term structure models in particular is to process information available in liquid market prices in order to manage risk. Derivative financial instruments are the vehicle for trading risk; the corresponding hedge portfolio determines both the risk management strategy and the arbitrage–free price for bearing the particular risk embodied in the instrument. However, once derivatives are actively traded in a liquid market, their prices incorporate additional market information to which the model must be calibrated. Standard fixed income derivatives such as caps and swaptions are examples of such a development. Term structure models directly specifying the arbitrage–free dynamics of market observable forward LIBOR or swap rate processes (cf. Miltersen, Sandmann and Sondermann (1997), Brace, Gatarek and Musiela (1997) (BGM), Jamshidian (1997), as well as Musiela and Rutkowski (1997a)) have quickly gained an eminent role in the management of interest rate risk by leading financial institutions. This is in particular due to the fact that they provide a pricing methodology in which the market practice of pricing caps or swaptions by Black/Scholes–like formulas can be applied in a manner consistent with the absence of arbitrage, hence the name Market Models. Musiela and Rutkowski (1997b) give a self–contained and up–to–date treatment of this development, as well as the necessary background, and Schlögl (2002b) extends this methodology to multiple currencies, incorporating exchange rate risk. Closed form solutions which mirror market practice for standard derivatives of course facilitate the calibration to market data, but this resolves the problem only superficially.