The Stochastic Intrinsic Currency Volatility Model: A Consistent Framework for Multiple FX Rates and Their Volatilities

The SABR and the Heston stochastic volatility models are widely used for foreign exchange (FX) option pricing. Although they are able to reproduce the market's volatility smiles and skews for single FX rates, they cannot be extended to model multiple FX rates in a consistent way. This is because when two FX rates with a common currency are described by either SABR or Heston processes, the stochastic process for the third FX rate associated with the three currencies will not be a SABR or a Heston process. A consistent description of the FX market should be symmetric so that all FX rates are described by the same type of stochastic process. This article presents a way of doing that using the concept of intrinsic currency values. To model FX volatility curves, the intrinsic currency framework is extended by allowing the volatility of each intrinsic currency value to be stochastic. This makes the framework more realistic, while preserving all FX market symmetries. The result is a new SABR-style option pricing formula and a model that can simultaneously be calibrated to the volatility smiles of all possible currency pairs under consideration. Consequently, it is more powerful than modelling the volatility curves of each currency pair individually and offers a methodology for comparing the volatility curves of different currency pairs against each other. One potential application of this is in the pricing of FX options, such as vanilla options on less liquid currency pairs. Given the volatilities of less liquid currencies against, for example, USD and EUR, the model could then be used to calculate the volatility smiles of those less liquid currencies against all other currencies.