Efficient option pricing in the rough Heston model using weak simulation schemes

Rough Heston model possesses some stylized facts that can be used to describe the stock market, i.e., markets are highly endogenous, no statistical arbitrage mechanism, liquidity asymmetry for buy and sell order, and the presence of metaorders. This paper presents an efficient alternative to compute option prices under the rough Heston model. Through the decomposition formula of the option price under the rough Heston model, we manage to obtain an approximation formula for option prices that is simpler to compute and requires less computational effort than the Fourier inversion method. In addition, we establish finite error bounds of approximation formula of option prices under the rough Heston model for 0.1≤H<0.5 under a simple assumption. Then, the second part of the work focuses on the short-time implied volatility behavior where we use a second-order approximation on the implied volatility to match the terms of Taylor expansion of call option prices. One of the key results that we manage to obtain is that the second-order approximation for implied volatility (derived by matching coefficients of the Taylor expansion) possesses explosive behavior for the short-time term structure of at-the-money implied volatility skew, which is also present in the short-time option prices under rough Heston dynamics. Numerical experiments were conducted to verify the effectiveness of the approximation formula of option prices and the formulas for the short-time term structure of at-the-money implied volatility skew.

We characterize the behavior of the Rough Heston model introduced by Jaisson and Rosenbaum (2016, Ann. Appl. Probab., 26, 2860–2882) in the small‐time, large‐time, and (i.e., ) limits. We show that the short‐maturity smile scales in qualitatively the same way as a general rough stochastic volatility model , and the rate function is equal to the Fenchel–Legendre transform of a simple transformation of the solution to the same Volterra integral equation (VIE) that appears in El Euch and Rosenbaum (2019, Math. Financ., 29, 3–38), but with the drift and mean reversion terms removed. The solution to this VIE satisfies a space–time scaling property which means we only need to solve this equation for the moment values of and so the rate function can be efficiently computed using an Adams scheme or a power series, and we compute a power series in the log‐moneyness variable for the asymptotic implied volatility which yields tractable expressions for the implied vol skew and convexity which is useful for calibration purposes. We later derive a formal saddle point approximation for call options in the Forde and Zhang (2017) large deviations regime which goes to higher order than previous works for rough models. Our higher‐order expansion captures the effect of both drift terms, and at leading order is of qualitatively the same form as the higher‐order expansion for a general model which appears in Friz et al. (2018, Math. Financ., 28, 962–988). The limiting asymptotic smile in the large‐maturity regime is obtained via a stability analysis of the fixed points of the VIE, and is the same as for the standard Heston model in Forde and Jacquier (2011, Finance Stoch., 15, 755–780). Finally, using Lévy's convergence theorem, we show that the log stock price tends weakly to a nonsymmetric random variable as (i.e., ) whose moment generating function (MGF) is also the solution to the Rough Heston VIE with , and we show that tends weakly to a nonsymmetric random variable as , which leads to a nonflat nonsymmetric asymptotic smile in the Edgeworth regime, where the log‐moneyness as , and we compute this asymptotic smile numerically. We also show that the third moment of the log stock price tends to a finite constant as (in contrast to the Rough Bergomi model discussed in Forde et al. (2020, Preprint) where the skew flattens or blows up) and the process converges on pathspace to a random tempered distribution which has the same law as the hyper‐rough Heston model, discussed in Jusselin and Rosenbaum (2020, Math. Finance, 30, 1309–1336) and Abi Jaber (2019, Ann. Appl. Probab., 29, 3155–3200).

We combine the rough Heston model and the CIR (Cox–Ingersoll–Ross) interest rate together to form a rough Heston-CIR model, so that both the rough behaviour of the volatility and the stochastic nature of the interest rate can be captured. Despite the convoluted structure and non-Markovian property of this model, it still admits a semi-analytical pricing formula for European options, the implementation of which involves solving a fractional Riccati equation. The rough Heston-CIR model is more general, taking both the rough Heston model and the Heston-CIR model as special cases. The influence of rough volatility and stochastic interest rate is shown to be significant through numerical experiments.

In the powder bed fusion process, an accurate prediction of the transient temperature field of a part is essential to calculate the subsequent thermal stress evolution and microstructure propagation in that part. The experimental method is time-consuming and expensive since the temperature field is controlled by many process parameters. Numerical heat transfer models can be used to estimate the temperature field at any time point. However, traditional numerical simulation schemes are not suitable for the layer-wised fabrication process due to the extremely high computational cost. The computational cost mainly relies on the element number and time step size. This research provides a new efficient and part-level simulation scheme based on an open-source finite element library, which is able to adaptively refine and coarsen the mesh and solve finite element equations with multiple processors in a parallel way. Here, a new mesh strategy that aims to reduce the element number while keeping the solution accuracy is developed. The simulation speed is 12× to 18× faster compared with the traditional simulation scheme depending on the scale of the simulated domain and number of processors. Simulation results have been compared with the experimental results of an Inconel 718 component. It is shown that the testing point in the simulation experiences the same thermal cycles of the same point in the experiment. This simulation scheme can also be used to optimize the process parameters such as scanning pattern, scan velocity, and layer thickness and can be easily extended to other additive manufacturing processes.

Geometric Asian options are a type of option where the payoff depends on the geometric mean of the underlying asset over a certain period of time. This paper is concerned with the pricing of such options for the class of Volterra-Heston models, covering the rough Heston model. We are able to derive semi-closed formulas for the prices of geometric Asian options with fixed and floating strikes for this class of stochastic volatility models. These formulas require the explicit calculation of the conditional joint Fourier transform of the logarithm of the stock price and the logarithm of the geometric mean of the stock price over time. Linking our problem to the theory of affine Volterra processes, we find a representation of this Fourier transform as a suitably constructed stochastic exponential, which depends on the solution of a Riccati-Volterra equation. Finally, we provide a numerical study for our results in the rough Heston model.

Affine Forward Variance Models

Statistical model selection and evaluation methods like Akaike information criteria (AIC) and Monte Carlo simulation (MCS) have often established efficient output for reliability analysis with large sample size. Information criterion can provide better model selection and evaluation in small sample sizes setup by considering the well-known measure of bootstrap resampling. Our purpose is to utilize the capabilities of bootstrap resampling in information criterion to check for uncertainty arising from model selection as well as statistics of interest for small sample size using reliability analysis. In this study, therefore, a unique and efficient simulation scheme is proposed which contemplates the best model selection devised from efficient bootstrap simulation or variance reduced bootstrap information criterion to be combined with reliability analysis. It is beneficial to compute the spread of reliability values as against solitary fixed values with desirable statistics of interest for uncertainty analysis. The proposed simulation scheme is verified using a number of sample size focused response functions under repetitions-centred approach with AIC-based reliability analysis for comparison and MCS for accuracy. The results show that the proposed simulation scheme aids the statistics of interest by reducing the spread and hence the uncertainty in sample size-based reliability analysis when compared with conventional methods.

An accurate and efficient finite-difference method for solving the incompressible Navier-Stokes equations on curvilinear grids is developed. This method combines the favorable features of the staggered grid and semistaggered grid approaches. All components of velocity are stored at the corner vertices, and pressure is stored at the grid cell centers. All components of the momentum equations are discretized at cell vertices, facilitating a consistent discretization of the diffusive and convective terms as the boundaries are approached. The Christoffel symbol does not appear in the transformed equations and the cost of computation is comparable to that of the staggered-grid approach. A projection method is used to effectively evolve the discrete system in time, while ensuring a divergence-free velocity field. The discrete divergence and gradient operators of the projection step are constructed on a staggered gird layout leading to exact satisfaction of the discrete continuity. The solution of the Poisson-Neumann equation in the projection step is free of any spurious eigenmodes. The validity of the method is demonstrated on four benchmark problems. The Taylor-Green vortex problem is solved on a curvilinear grid with highly skewed cells and a second-order convergence in l ∞-norm is observed. The mixed convection in a lid-driven cavity is solved on a highly curvilinear grid and excellent agreement with literature is obtained. The results for flow past a cylinder and pusatile flow in a 90° bend are compared with the existing experimental data in the literature. The predictions agree well with the measured data, and validate the approach used.

Sensitivity studies of cloud microphysical schemes of WRF-ARW model in the simulation of trailing stratiform squall lines over the Gangetic West Bengal region

Despite the many advances in the use of weakly compressible smoothed particle hydrodynamics (SPH) for the simulation of incompressible fluid flow, it is still challenging to obtain second-order convergence even for simple periodic domains. In this paper, we perform a systematic numerical study of convergence and accuracy of kernel-based approximation, discretization operators, and weakly compressible SPH (WCSPH) schemes. We explore the origins of the errors and issues preventing second-order convergence despite having a periodic domain. Based on the study, we propose several new variations of the basic WCSPH scheme that are all second-order accurate. Additionally, we investigate the linear and angular momentum conservation property of the WCSPH schemes. Our results show that one may construct accurate WCSPH schemes that demonstrate second-order convergence through a judicious choice of kernel, smoothing length, and discretization operators in the discretization of the governing equations.

Today, better numerical approximations are required for multi-dimensional SDEs to improve on the poor performance of the standard Monte Carlo pricing method. With this aim in mind, this paper presents a method (MSL-MC) to price exotic options using multi-dimensional SDEs (e.g. stochastic volatility models). Usually, it is the weak convergence property of numerical discretizations that is most important, because, in financial applications, one is mostly concerned with the accurate estimation of expected payoffs. However, in the recently developed Multilevel Monte Carlo path simulation method (ML-MC), the strong convergence property plays a crucial role. We present a modification to the ML-MC algorithm that can be used to achieve better savings. To illustrate these, various examples of exotic options are given using a wide variety of payoffs, stochastic volatility models and the new Multischeme Multilevel Monte Carlo method (MSL-MC). For standard payoffs, both European and Digital options are presented. Examples are also given for complex payoffs, such as combinations of European options (Butterfly Spread, Strip and Strap options). Finally, for path-dependent payoffs, both Asian and Variance Swap options are demonstrated. This research shows how the use of stochastic volatility models and the θ scheme can improve the convergence of the MSL-MC so that the computational cost to achieve an accuracy of O(ε) is reduced from O(ε−3) to O(ε−2) for a payoff under global and non-global Lipschitz conditions.

An accurate and efficient Chebyshev expansion method for large-scale transient heat conduction problems

Many engineering applications require efficient computation of the solution of the Poisson equation defined over one, two or three dimensions, representing a wide variety of physical systems. The most popular method used for this purpose is the finite element method (FEM). An important class of applications in modeling and simulation require repeated computation of the solution over a given geometry, for different values of the relevant material property in a region within the domain. They can arise in the areas of nondestructive testing and evaluation, field-control in electromagnetic applications, as well as computer-aided design of devices and systems. In this work, a linear algebraic study of the dependence of the coefficient (stiffness) matrix on the varying material property is used to develop a general expression of the finite element solution. Based on this, a novel modeling and simulation scheme is suggested which significantly reduces the computational cost of the repetitive solution stage, as well as the size of the numerical model to be stored. Numerical simulations are presented for a simple illustrative case and a few possible applications are suggested in different engineering fields.

In this article, we provide a general methodology to compute the resolvent kernel as well as the density when available for a one-dimensional second-order differential operators with discontinuous coefficients. In a sequel, the computed resolvent kernel will be used to setup an efficient and accurate simulation scheme.

To simulate synthetic aperture radar (SAR) raw data of extended scenes in the presence of arbitrary trajectory deviations, an efficient simulation scheme based on parallel computation in hybrid domain is proposed. We firstly present a novel target-to-antenna distance model which takes the three-dimensional components of the trajectory deviations into consideration. Then a general and compact hybrid domain formulation of the SAR raw data in the presence of arbitrary trajectory deviations is obtained. Based on the formulation, a simulation scheme which relies on 1-D range Fourier domain processing followed by azimuth time domain integration is proposed. Then a parallelized implementation is introduced to solve the problem that the hybrid domain method becomes less efficient with the increase of the sampling number of the extended scene in the azimuth direction. Some simulation results are finally presented to verify the validity and effectiveness of the proposed simulation scheme.