A hybrid tree/finite-difference approach for Heston-Hull-White type models

We study a hybrid tree-finite difference method which permits to obtain efficient and accurate European and American option prices in the Heston Hull-White and Heston Hull-White2d models. Moreover, as a by-product, we provide a new simulation scheme to be used for Monte Carlo evaluations. Numerical results show the reliability and the efficiency of the proposed method

A hybrid approach for the implementation of the Heston model

We propose a hybrid tree-finite difference method in order to approximate the Heston model. We prove the convergence by embedding the procedure in a bivariate Markov chain and we study the convergence of European and American option prices. We finally provide numerical experiments that give accurate option prices in the Heston model, showing the reliability and the efficiency of the algorithm

Multilevel Monte Carlo methods for applications in finance

Since Giles introduced the multilevel Monte Carlo path simulation method [18], there has been rapid development of the technique for a variety of applications in computational finance. This paper surveys the progress so far, highlights the key features in achieving a high rate of multilevel variance convergence, and suggests directions for future research.Comment: arXiv admin note: text overlap with arXiv:1202.6283; and with arXiv:1106.4730 by other author

A new approach for the black-scholes model with linear and nonlinear volatilities

Since financial engineering problems are of great importance in the academic community, effective methods are still needed to analyze these models. Therefore, this article focuses mainly on capturing the discrete behavior of linear and nonlinear Black-Scholes European option pricing models. To achieve this, this article presents a combined method; a sixth order finite difference (FD6) scheme in space and a third-order strong stability preserving Runge-Kutta (SSPRK3) over time. The computed results are compared with available literature and the exact solution. The computed results revealed that the current method seems to be quite strong both quantitatively and qualitatively with minimal computational effort. Therefore, this method appears to be a very reliable alternative and flexible to implement in solving the problem while preserving the physical properties of such realistic processes. © 2019 by the authors

Applications of Gaussian Process Regression to the Pricing and Hedging of Exotic Derivatives

Traditional option pricing methods like Monte Carlo simulation can be time consuming when pricing and hedging exotic options under stochastic volatility models like the Heston model. The purpose of this research is to apply the Gaussian Process Regression (GPR) method to the pricing and hedging of exotic options under the Black-Scholes and Heston model. GPR is a supervised machine learning technique which makes use of a training set to train an algorithm so that it makes predictions. The training set is composed of the input vector X which is a n × p matrix and Y an n×1 vector of targets, where n is the number of training input vectors and p is the number of inputs. Using a GPR with a squared-exponential kernel tuned by maximising the log-likelihood, we established that this GPR works reasonably for pricing Barrier options and Asian options under the Heston model. As compared to the traditional method of Monte Carlo simulation, GPR technique is 2 000 times faster when pricing barrier option portfolios of 100 assets and 1 000 times faster computing a portfolio of Asian options. However, the squared-exponential GPR does not compute reliable hedging ratios under Heston model, the delta is reasonably accurate, but the vega is off