We introduce a novel multi-factor Heston-based stochastic volatility model, which is able to reproduce consistently typical multi-dimensional FX vanilla markets, while retaining the (semi)-analytical tractability typical of affine models and relying on a reasonable number of parameters. A successful joint calibration to real market data is presented together with various in-and out-of-sample calibration exercises to highlight the robustness of the parameters estimation. The proposed model preserves the natural inversion and triangulation symmetries of FX spot rates and its functional form, irrespective of choice of the risk-free currency. That is, all currencies are treated in the same way. arXiv:1201.1782v3 [q-fin.PR] 13 Jun 2013 2 A MULTIFACTOR HESTON-BASED EXCHANGE MODEL 3 set of risk neutral measures and the relation among model parameters under different probability measures. Rather remarkably, as a consequence of the specific Heston-type dynamics, the model remains functionally invariant, after parameter rescaling, when the risk-neutral measure is changed. This is a key feature that allows obtaining a calibration with reasonable computational effort. We will then test the model on real market data and show how a joint calibration of the volatility smiles of EUR/USD/JPY and AUD/USD/JPY triangles is possible. In-and out-of-sample calibration tests will be reported to comment on the robustness of the parameter estimation.Previous analyses of the multi-dimensional FX volatility smile problem have used different approaches to recover the risk neutral probability distribution of the cross exchange rate, either by means of joint densities or copulas, see Austing (2011), Bennett and Kennedy (2004), Schneider (2006), andHurd et al. (2005). Such contributions may be seen as a generalization to the multi-dimensional setting of the classical idea of Breeden and Litzenberger (1978), see also Bliss and Panigirtzoglou (2002) and the Gram-Charlier based approach in Schlögl (2012). The shortcoming of these techniques is that they provide only a distribution for the cross rate and not an explicit specification of the dynamics. In the context of stochastic volatility models, Carr and Verma (2005) propose a model with a single joint stochastic factor, which however limits the flexibility to achieve satisfactory joint calibrations. Another approach, in the presence of a SABR stochastic volatility specification, is studied in Shiraya and Takahashi (2012) where asymptotic formulae are presented.The approach in this paper is fundamentally different. Instead of putting the currency pairs at the basis of our model, we start from the observation that any exchange rate may be seen as a ratio between two quantities, the value of the currencies with respect to some universal numéraire, and include this feature in the specification of the model. Flesaker and Hughston (2000) introduced the idea of a "natural numeraire", the value of which can be expressed in different currencies, thus leading to consistent expressions for the FX rates as ...

We introduce a novel multi-factor Heston-based stochastic volatility model, which is able to reproduce consistently typical multi-dimensional FX vanilla markets, while retaining the (semi)-analytical tractability typical of affine models and relying on a reasonable number of parameters. A successful joint calibration to real market data is presented together with various in-and out-of-sample calibration exercises to highlight the robustness of the parameters estimation. The proposed model preserves the natural inversion and triangulation symmetries of FX spot rates and its functional form, irrespective of choice of the risk-free currency. That is, all currencies are treated in the same way.

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...

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