Smiles all around: FX joint calibration in a multi-Heston model

Abstract

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 ...

An Affine Multicurrency Model with Stochastic Volatility and Stochastic Interest Rates

“…, N, for S d indicating the set of d × d symmetric matrices, and finally T r denotes the trace operator. The diffusion term exhibits a structure that is completely analogous to that introduced in Heath and Platen [43] and De Col, Gnoatto, and Grasselli [26]: in the present case we have that the dynamics of the exchange rate is driven by a linear projection of the variance factor Σ(t) along a direction parametrized by the symmetric matrix A i . As a consequence the total instantaneous variance is T r…”

“…We observe that, as in the multi-Heston case of De Col, Gnoatto, and Grasselli [26], the functional form of the model is invariant under the measure change between Q 0 and the ith risk-neutral measure. The inverse FX rate under the Q i risk-neutral measure follows from Ito calculus, recalling that S j,i = S i,j −1 :…”

“…The component of dB(t) that is orthogonal to the spot driver dZ(t) is not affected by the measure change; this is a typical feature of FX stochastic volatility models, implying that the model is consistent with the foreign-domestic symmetry; see De Col, Gnoatto, and Grasselli [26] and Del Baño Rollin [28]. We are now able to derive the riskneutral dynamics of the factor process Σ(t) governing the volatility of the exchange rates under Q i , which is given by…”

“…Using Theorem 4.1 in Mayerhofer [59], we immediately conclude that the stochastic exponential above is a true martingale. In the following, we proceed along the lines of De Col, Gnoatto, and Grasselli [26]. The possibility of buying the foreign currency and investing it at the foreign short rate of interest is equivalent to the possibility of investing in a domestic asset with price processB i j = B j S i,j , where i is the domestic economy and j is the foreign one.…”

“…Or, equivalently, the dynamics of (S i,l S l,j ) computed by applying Ito's rule to the product (S 0,l /S 0,i ) × (S 0,j /S 0,l ) must give the dynamics of a process that shares the same functional form of both S i,l and S l,j . This symmetry property is fundamental in being able to joint calibrate and consistently price multicurrency options (see, e.g., De Col, Gnoatto, and Grasselli [26] in a multifactor stochastic volatility framework and Eberlein, Papapantoleon, and Shiryaev [32] in a general semimartingale setting).…”