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Cross Currency Valuation and Hedging in the Multiple Curve Framework

2021; Society for Industrial and Applied Mathematics; Volume: 12; Issue: 3 Linguagem: Inglês

10.1137/20m1324375

1945-497X

Alessandro Gnoatto, Nicole Seiffert,

Financial Markets and Investment Strategies

We generalize the results of [T. Bielecki and M. Rutkowski, SIAM J. Financial Math., 6 (2015), pp. 594--655] on funding and collateralization to a multicurrency framework and link their results with those of [V. Piterbarg, Risk, Aug. 2012, pp. 46--51], [N. Moreni and A. Pallavicini, Quant. Finance, 14 (2014), pp. 199--210], and [M. Fujii, Y. Shimada, and A. Takahashi, On the Term Structure of Interest Rates with Basis Spreads, Collateral and Multiple Currencies, SSRN preprint 1556487, 2010]. In doing this, we provide a complete study of absence of arbitrage in a multicurrency market where, in each single monetary area, multiple interest rates coexist. We first characterize absence of arbitrage in the case without collateral. After that we study collateralization schemes in a very general situation: the cash flows of the contingent claim and those associated to the collateral agreement can be specified in any currency. We study both segregation and rehypothecation and allow for cash and risky collateral in arbitrary currency specifications. Absence of arbitrage and pricing in the presence of collateral are discussed under all possible combinations of conventions. Our work provides not only a reference for the analysis of wealth dynamics but also valuation formulas that are a useful foundation for cross currency curve construction techniques. Our framework provides also a solid foundation for the construction of multicurrency simulation models for the generation of exposure profiles in the context of xVA calculations.