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 ...

This is the accepted version of the paper.This version of the publication may differ from the final published version. This article presents lower and upper bounds on the prices of basket options for a general class of continuous-time financial models. The techniques we propose are applicable whenever the joint characteristic function of the vector of log-returns is known. Moreover, the basket value is not required to be positive. We test our new price approximations on different multivariate models, allowing for jumps and stochastic volatility. Numerical examples are discussed and benchmarked against Monte Carlo simulations. All bounds are general and do not require any additional assumption on the characteristic function, so our methods may be employed also to non-affine models. All bounds involve the computation of one-dimensional Fourier transforms, hence they do not suffer from the curse of dimensionality and can be applied also to high dimensional problems where most existing methods fail. In particular we study two kinds of price approximations: an accurate lower bound based on an approximating set and a fast bounded approximation based on the arithmetic-geometric mean inequality. We also show how to improve Monte Carlo accuracy by using one of our bounds as a control variate. Permanent repository link

Abstract. We propose a general framework for modeling multiple yield curves which have emerged after the last financial crisis. In a general semimartingale setting, we provide an HJM approach to model the term structure of multiplicative spreads between FRA rates and simply compounded OIS risk-free forward rates. We derive an HJM drift and consistency condition ensuring absence of arbitrage and, in addition, we show how to construct models such that multiplicative spreads are greater than one and ordered with respect to the tenor's length. When the driving semimartingale is an affine process, we obtain a flexible and tractable Markovian structure. Finally, we show that the proposed framework allows to unify and extend several recent approaches to multiple yield curve modeling.

We provide a general and tractable framework under which all multiple yield curve modeling approaches based on affine processes, be it short rate, Libor market, or Heath-Jarrow-Morton modeling, can be consolidated. We model a numéraire process and multiplicative spreads between Libor rates and simply compounded overnight indexed swap rates as functions of an underlying affine process.Besides allowing for ordered spreads and an exact fit to the initially observed term structures, this general framework leads to tractable valuation formulas for caplets and swaptions and embeds all existing multicurve affine models. The proposed approach also gives rise to new developments, such as a short rate type model driven by a Wishart process, for which we derive a closed-form pricing formula for caplets. The empirical performance of two specifications of our framework is illustrated by calibration to market data. K E Y W O R D Saffine processes, forward rate agreement, Libor rate, multiple yield curves, multiplicative spread J E L C L A S S I F I C A T I O N :E43, G12 568

Abstract. We propose a general framework for modeling multiple yield curves which have emerged after the last financial crisis. In a general semimartingale setting, we provide an HJM approach to model the term structure of multiplicative spreads between FRA rates and simply compounded OIS risk-free forward rates. We derive an HJM drift and consistency condition ensuring absence of arbitrage and, in addition, we show how to construct models such that multiplicative spreads are greater than one and ordered with respect to the tenor's length. When the driving semimartingale is an affine process, we obtain a flexible and tractable Markovian structure. Finally, we show that the proposed framework allows to unify and extend several recent approaches to multiple yield curve modeling.