On the consistency of jump-diffusion dynamics for FX rates under inversion

In this note we investigate the consistency under inversion of jump diffusion processes in the Foreign Exchange (FX) market. In other terms, if the EUR/USD FX rate follows a given type of dynamics, under which conditions will USD/EUR follow the same type of dynamics? In order to give a numerical description of this property, we first calibrate a Heston model and a SABR model to market data, plotting their smiles together with the smiles of the reciprocal processes. Secondly, we determine a suitable local volatility structure ensuring consistency. We subsequently introduce jumps and analyze both constant jump size (Poisson process) and random jump size (compound Poisson process). In the first scenario, we find that consistency is automatically satisfied, for the jump size of the inverted process is a constant as well. The second case is more delicate, since we need to make sure that the distribution of jumps in the domestic measure is the same as the distribution of jumps in the foreign measure. We determine a fairly general class of admissible densities for the jump size in the domestic measure satisfying the condition.Comment: 14 pages, 3 figure

High-dimensional limits of eigenvalue distributions for general Wishart process

In this article, we obtain an equation for the high-dimensional limit measure of eigenvalues of generalized Wishart processes, and the results is extended to random particle systems that generalize SDEs of eigenvalues. We also introduce a new set of conditions on the coefficient matrices for the existence and uniqueness of a strong solution for the SDEs of eigenvalues. The equation of the limit measure is further discussed assuming self-similarity on the eigenvalues.Comment: 28 page

Recent advances on eigenvalues of matrix-valued stochastic processes

peer reviewedSince the introduction of Dyson's Brownian motion in early 1960s, there have been a lot of developments in the investigation of stochastic processes on the space of Hermitian matrices. Their properties, especially, the properties of their eigenvalues have been studied in great detail. In particular, the limiting behaviours of the eigenvalues are found when the dimension of the matrix space tends to infinity, which connects with random matrix theory. This survey reviews a selection of results on the eigenvalues of stochastic processes from the literature of the past three decades. For most recent variations of such processes, such as matrix-valued processes driven by fractional Brownian motion or Brownian sheet, the eigenvalues of them are also discussed in this survey. In the end, some open problems in the area are also proposed

An Affine Multicurrency Model with Stochastic Volatility and Stochastic Interest Rates

We introduce a tractable multicurrency model with stochastic volatility and correlated stochastic interest rates that takes into account the smile in the foreign exchange (FX) market and the evolution of yield curves. The pricing of vanilla options on FX rates can be efficiently performed through the FFT methodology thanks to the affine property of the model. Our framework is also able to describe many nontrivial links between FX rates and interest rates: a calibration exercise highlights the ability of the model to simultaneously fit FX implied volatilities while being coherent with interest rate product