On a one time-step Monte Carlo simulation approach of the SABR model: Application to European options

In this work, we propose a one time-step Monte Carlo method for the SABR model. We base our approach on an accurate approximation of the cumulative distribution function of the time-integrated variance (conditional on the SABR volatility), using Fourier techniques and a copula. Resulting is a fast simulation algorithm which can be employed to price European options under the SABR dynamics. Our approach can thus be seen as an alter- native to Hagan’s analytic formula for short maturities that may be employed for model calibration purposes

On an efficient multiple time step Monte Carlo simulation of the SABR model

In this paper, we will present a multiple time step Monte Carlo simulation technique for pricing options under the Stochastic Alpha Beta Rho model. The proposed method is an extension of the one time step Monte Carlo method that we proposed in an accompanying paper Leitao et al. [Appl. Math. Comput. 2017, 293, 461–479], for pricing European options in the context of the model calibration. A highly efficient method results, with many very interesting and nontrivial components, like Fourier inversion for the sum of log-normals, stochastic collocation, Gumbel copula, correlation approximation, that are not yet seen in combination within a Monte Carlo simulation. The present multiple time step Monte Carlo method is especially useful for long-term options and for exotic options

The Heston Stochastic-Local Volatility Model: Efficient Monte Carlo Simulation

In this article we propose an efficient Monte Carlo scheme for simulating the stochastic volatility model of Heston (1993) enhanced by a non-parametric local volatility component. This hybrid model combines the main advantages of the Heston model and the local volatility model introduced by Dupire (1994) and Derman & Kani (1998). In particular, the additional local volatility component acts as a "compensator" that bridges the mismatch between the non-perfectly calibrated Heston model and the market quotes for European-type options. By means of numerical experiments we show that our scheme enables a consistent and fast pricing of products that are sensitive to the forward volatility skew. Detailed error analysis is also provided

Uncertainty quantification and Heston model

In this paper, we study the impact of the parameters involved in Heston model by means of Uncertainty Quantification. The Stochastic Collocation Method already used for example in computational fluid dynamics, has been applied throughout this work in order to compute the propagation of the uncertainty from the parameters of the model to the output. The well-known Heston model is considered and involved parameters in the Feller condition are taken as uncertain due to their important influence on the output. Numerical results where the Feller condition is satisfied or not are shown as well as a numerical example with real market data

The Seven-League scheme: Deep learning for large time step Monte Carlo simulations of stochastic differential equations

We propose an accurate data-driven numerical scheme to solve stochastic differential equations (SDEs), by taking large time steps. The SDE discretization is built up by means of the polynomial chaos expansion method, on the basis of accurately determined stochastic collocation (SC) points. By employing an artificial neural network to learn these SC points, we can perform Monte Carlo simulations with large time steps. Basic error analysis indicates that this data-driven scheme results in accurate SDE solutions in the sense of strong convergence, provided the learning methodology is robust and accurate. With a method variant called the compression–decompression collocation and interpolation technique, we can drastically reduce the number of neural network functions that have to be learned, so that computational speed is enhanced. As a proof of concept, 1D numerical experiments confirm a high-quality strong convergence error when using large time steps, and the novel scheme outperforms some classical numerical SDE discretizations. Some applications, here in financial option valuation, are also presented

A robust spectral method for solving Heston’s model

In this paper, we consider the Heston’s volatility model (Heston in Rev. Financ. Stud. 6: 327–343, 1993]. We simulate this model using a combination of the spectral collocation method and the Laplace transforms method. To approximate the two dimensional PDE, we construct a grid which is the tensor product of the two grids, each of which is based on the Chebyshev points in the two spacial directions. The resulting semi-discrete problem is then solved by applying the Laplace transform method based on Talbot’s idea of deformation of the contour integral (Talbot in IMA J. Appl. Math. 23(1): 97–120, 1979)

Search for lepton-flavor violation at HERA

A search for lepton-flavor-violating interactions $e p \to \mu X$ and $e p\to \tau X$ has been performed with the ZEUS detector using the entire HERA I data sample, corresponding to an integrated luminosity of 130 pb^{-1}. The data were taken at center-of-mass energies, $\sqrt{s}$, of 300 and 318 GeV. No evidence of lepton-flavor violation was found, and constraints were derived on leptoquarks (LQs) that could mediate such interactions. For LQ masses below $\sqrt{s}$, limits were set on $\lambda_{eq_1} \sqrt{\beta_{\ell q}}$, where $\lambda_{eq_1}$ is the coupling of the LQ to an electron and a first-generation quark $q_1$, and $\beta_{\ell q}$ is the branching ratio of the LQ to the final-state lepton $\ell$ ($\mu$ or $\tau$) and a quark $q$. For LQ masses much larger than $\sqrt{s}$, limits were set on the four-fermion interaction term $\lambda_{e q_\alpha} \lambda_{\ell q_\beta} / M_{\mathrm{LQ}}^2$ for LQs that couple to an electron and a quark $q_\alpha$ and to a lepton $\ell$ and a quark $q_\beta$, where $\alpha$ and $\beta$ are quark generation indices. Some of the limits are also applicable to lepton-flavor-violating processes mediated by squarks in $R$-Parity-violating supersymmetric models. In some cases, especially when a higher-generation quark is involved and for the process $e p\to \tau X$, the ZEUS limits are the most stringent to date.Comment: 37 pages, 10 figures, Accepted by EPJC. References and 1 figure (Fig. 6) adde