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...

Monte Carlo simulation of SDEs using GANs

Generative adversarial networks (GANs) have shown promising results when applied on partial differential equations and financial time series generation. We investigate if GANs can also be used to approximate one-dimensional Ito ^ stochastic differential equations (SDEs). We propose a scheme that approximates the path-wise conditional...

Collocating Volatility: A Competitive Alternative to Stochastic Local Volatility Models

We discuss a competitive alternative to stochastic local volatility models, namely the Collocating Volatility (CV) framework, introduced in [L. A. Grzelak (2019) The CLV framework-A fresh look at efficient pricing with smile, International Journal of Computer Mathematics 96 (11), 2209-2228]. The CV framework consists of two elements, a ...

A neural network-based framework for financial model calibration

A data-driven approach called CaNN (Calibration Neural Network) is proposed to calibrate financial asset price models using an Artificial Neural Network (ANN). Determining optimal values of the model parameters is formulated as training hidden neurons within a machine learning framework, based on available financial option prices. The...

The stochastic collocation Monte Carlo sampler: Highly efficient sampling from ‘expensive’ distributions

In this article, we propose an efficient approach for inverting computationally expensive cumulative distribution functions. A collocation method, called the Stochastic Collocation Monte Carlo sampler (SCMC sampler), within a polynomial chaos expansion framework, allows us the generation of any number of Monte Carlo samples based on only a...

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...

A novel Monte Carlo approach to hybrid local volatility models

We present in a Monte Carlo simulation framework, a novel approach for the evaluation of hybrid local volatility [Risk, 1994, 7, 18–20], [Int. J. Theor. Appl. Finance, 1998, 1, 61–110] models. In particular, we consider the stochastic local volatility model—see e.g. Lipton et al. [Quant. Finance, 2014, 14, 1899–1922], Piterbarg [Risk, 2007,...

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...

Equity and Foreign Exchange Hybrid Models for Pricing Long-Maturity Financial Derivatives

Modelling derivative products in Finance usually starts with the specification of a system of Stochastic Differential Equations (SDEs), that corresponds to state variables like stock, interest rate, Foreign Exchange (FX) rate and volatility. By correlating the SDEs for the different asset classes one can define the hybrid models, and use them...

On cross-currency models with stochastic volatility and correlated interest rates

We construct multi-currency models with stochastic volatility and correlated stochastic interest rates with a full matrix of correlations. We first deal with a foreign exchange (FX) model of Heston-type, in which the domestic and foreign interest rates are generated by the short-rate process of Hull-White [HW96]. We then extend the framework by...

Efficient pricing of commodity options with early-exercise under the Ornstein–Uhlenbeck process

We analyze the efficiency properties of a numerical pricing method based on Fourier-cosine expansions for early-exercise options. We focus on variants of Schwartz’ model [20] based on a mean reverting Ornstein-Uhlenbeck process [23], which is commonly used for modeling commodity prices. This process however does not possess favorable properties...

An equity-interest rate hybrid model with stochastic volatility and the interest rate smile

We define an equity-interest rate hybrid model in which the equity part is driven by the Heston stochastic volatility [Hes93], and the interest rate (IR) is generated by the displaced-diffusion stochastic volatility Libor Market Model [AA02]. We assume a non-zero correlation between the main processes. By an appropriate change of measure the...

Efficient option pricing with multi-factor equity-interest rate hybrid models

On the Heston Model with stochastic interest rates

Incorporating an Interest Rate Smile in an Equity Local Volatility Model

Extension of stochastic volatility models with Hull-White interest rate process