Pricing Multi-Dimensional FX Derivatives via Stochastic Local Correlations

Abstract

We present a hybrid Heston local correlation model for pricing multi-dimensional FX derivatives. The model is symmetric under inversion and triangulation and yields prices that are consistent with the one-dimensional vanilla markets of both main and cross FX rates. Irrespective of the choice of numeraire, the model retains its functional form. Important for practitioners, the model is easy to calibrate and can be scaled to any desired dimensionality (i.e., it can include an arbitrary amount of FX rates). The strength of the model -pricing both typical terminal and path-dependent options such as worst-ofs, quanto double no-touches, and third currency barrier options -is shown in test results comparing the Heston local correlation model with other multi-dimensional FX models. Keywordscorrelation, FX, stochastic-local volatility, triangular relation In troductionPricing financial derivatives with multiple underlying components, such as basket options [1], requires a sound modeling of the correlations among the individual underliers. Unfortunately, for most asset classes these correlations are not observable in the market. This leads to material uncertainties and complications in the valuation and risk management of even rather simple multi-component derivatives. Nevertheless, there exists a notable example in which the market actually allows us to infer significant information about the correlation structure: derivatives depending on multiple FX rates. In the FX framework, the market yields potentially more information on where correlations trade than any standard model (e.g., a Gaussian copula model) can handle. This valuable information can be extracted from the FX vanilla markets by making use of the fact that appropriate multiplications and divisions of FX rates are still FX rates and, hence, their implied volatilities are often traded and observable in the market. In order to capture the market implied correlation structure, we need a model that is capable of consistently pricing vanilla options across different FX pairs. At the same time, given the potential high dimensionality of the problem, we seek to retain analytical tractability. Here, we discuss several approaches to model correlations between FX rates and present a multi-dimensional extension of the hybrid Heston plus local volatility model presented in [2,3].Neglecting volatility smiles, the standard Black-Scholes (BS) model can easily be extended to model multi-dimensional FX rate pairs. As an example, we may consider an FX triangle, consisting of three underlying FX rates like EURUSD, GBPUSD, and EURGBP (we shall use the standard FX terminology where the notation ccy1ccy2 stands for the amount of ccy2 paid per unit of ccy1). We then obtain the following relationship between the three FX rates S EURUSD (t), S GBPUSD (t), and S EURGBP (t): S EURGBP (t) = S EURUSD (t) / S GBPUSD (t). From the perspective of a USD investor, we denote EURUSD and GBPUSD as the main FX rates and EURGBP as the cross FX rate. As shown in detail in [3, p. 226], we may...

A new class of multidimensional Wishart-based hybrid models

“…Moreover, the pricing of derivatives with a multi-asset risk exposure requires a satisfactory modelling of the correlation structure among the involved assets, consistently with an accurate reproduction of individual plain vanilla market evidences. This leads to the introduction of multi-asset SLV models as done in De Col and Kuppinger (2014), where a simplified multi-asset Heston framework is extended to accommodate for hybrid asset covariations. In order to incorporate a more flexible dependence structure among state variables, we also introduce the Wishart Stochastic Local Covariance model (WSLC) as hybrid generalization of the pure SV model in Da Fonseca et al (2007).…”