Original articles
A new calibration of the Heston Stochastic Local Volatility Model and its parallel implementation on GPUs

https://doi.org/10.1016/j.matcom.2020.04.001Get rights and content

Abstract

In this article we propose a new more general calibration of the Heston Stochastic-Local Volatility (HSLV) model. More precisely, the main contribution is to perform the direct calibration of the whole set of parameters at the same time instead of the usual two steps procedure. Moreover, the proposed approach allows to use exotic options to calibrate the HSLV model, thus making it more flexible and general. However, as there are no analytical formulas available to price exotic options to calibrate the model, the cost function (the HSLV pricer) involved in the calibration process must be computed using Monte Carlo methods, thus leading to a highly demanding computational problem. Therefore, we also propose efficient parallel GPU implementations of Monte Carlo techniques for the pricers. Furthermore, for solving the resulting global optimization problem, we develop customized parallel multi-CPU implementations of two of the most common stochastic metaheuristic global optimization algorithms: Differential Evolution and Simulated Annealing. A comparison between both algorithms has been made. This second level of parallelization has been carried out by the implementation of the cost function as a single GPU kernel and keeping the OpenMP parallelization for the optimization algorithm, thus leading to a hybrid multi-GPU implementation of the calibrator. All these implementations have been tested with real market data for European and barrier options in the context of foreign exchange markets.

Introduction

It is well-known that local volatility (LV) models (starting from the seminal paper [4]) can match exactly the market implied volatility surface for European vanilla options, that is, they can reproduce the volatility smile, a phenomenon that the classical Black–Scholes (BS) model could not explain. While this is an improvement with respect to the classical BS model, LV models still exhibit some limitations. For example, they produce a flattening in implied forward volatilities [25], which is an undesirable property. This does not occur in Stochastic Volatility (SV) models, where the variance of the asset satisfies its own stochastic differential equation (SDE). One example is the classical Heston Stochastic Volatility (HSV) model, that arises from the seminal work [14]. Nevertheless, since SV models have a parametric form, they cannot be accurately calibrated to a set of European vanilla prices. As an attempt to obtain a model which can both calibrate exactly European vanilla options and reproduce the stochastic behavior of the volatility, Stochastic Local Volatility (SLV) models have been proposed in the literature. SLV models were originated during the late 90 s and the early 2000s (see [15] and [18], for example), and nowadays they constitute a standard for pricing foreign exchange (FX) options in the market (see [3] and [19], among others). Although these SLV models present several desirable properties, its calibration can turn out to be complicated due to the conditional expectation that appears when relating the local component of the SLV model with the local volatility. Many attempts have been made to efficiently compute this expectation: Ren et al. [26] proposed a lognormal model for the spot and its volatility with zero correlation, and in [5] the authors apply an extension of this technique to the Heston Stochastic Local Volatility (HSLV) model using a finite volume scheme. Another common approach comes from the discretization of the forward Kolmogorov equation by using appropriate finite differences schemes to obtain the joint probability distribution of the asset and its volatility (see [29] and [3], for example). However, this type of methods are not well suited for the calibration and pricing of multi-factor models due to the curse of dimensionality. In order to avoid this curse of dimensionality, in the works by Henry-Labordère [13] and van der Stoep et al. [30] Monte Carlo based approaches are proposed.

The previous models depend on a set of parameters that need to be calibrated to match observed past market data in order to be able to use it to predict future market prices. Actually, note that a precise model jointly with an accurate numerical method can yield totally wrong results with poorly or non calibrated parameters. In this sense, calibration is the tool for embedding reality in a numerical simulation, allowing the model to learn and profit from real measured data. In a general setting, a calibration problem, for a given model, can be posed as an unconstrained global optimization problem in a bounded domain. Stochastic global metaheuristic algorithms are useful to solve this kind of problems. They have the advantage of requiring little information about the function, and also allow to escape from local optima, their main disadvantage being the slow rate of convergence due to their stochastic nature. Classical well known examples of these methods are Simulated Annealing (SA, see [1], [17]), Differential Evolution (DE, see [28]) or Particle Swarm (PS, see [16]).

In most of the works in the literature the HSLV model calibration follows two steps. Particularly, in [30] the parameters in the HSV model are calibrated in a first step, and then an original non-parametric technique is proposed to obtain the leverage function. As for the calibration of HSV parameters only European options are considered, due to the existence of semianalytical formulas. Moreover, the authors illustrate that even when using non calibrated parameters, the non-parametric computation of the leverage function can fit market prices. This means that the parameters and the leverage are not uniquely determined.

In this article we propose an efficient machinery for the direct and joint calibration of all the HSV parameters and the leverage involved in the HSLV model at the same time. For this purpose, we use multi-CPU parallel and efficient global optimization methods, together with GPU and multi-GPU implementations of Monte Carlo numerical methods for the HSLV pricer. This technique allows to calibrate all the parameters at the same time and does not rely on the use of analytical formulas, thus allowing to directly calibrate the model even against exotic options. Finally, we compare the obtained results with HSV and HSLV models for both European vanilla and exotic barrier options.

The outline of this paper is as follows. In Section 2 we make a brief introduction to the HSLV model. In Section 3 we present the Monte Carlo method for the HSLV models. In Section 4 we pose the calibration problem and introduce the optimization algorithms we used for the calibration and how we implemented their multi-CPU parallelization using OpenMP. In Section 5 we present the numerical results while in Section 6 we summarize the conclusions. In the Appendix we detail the GPU parallel implementation of the Monte Carlo technique for the HSLV pricing models.

Section snippets

HSLV model

The dynamics of the HSLV model are governed by the following system of SDEs (see [29]): dSt=rStdt+L(St,t)vtStdXtS,dvt=κ(θvt)dt+μσvtdXtv, where t denotes time, St the price of the underlying asset and vt its associated variance. Concerning the involved parameters, r is the constant risk-free interest rate, κ is the mean-reversion speed of the variance, θ the long-term mean variance, σ is the volatility of the variance (also known as vol of vol) and μ is the mixing fraction. Moreover, the two

Monte Carlo simulation

In this section we describe the main steps of the Monte Carlo (MC) numerical discretization of the HSLV model for option pricing. The key component is the computation of the conditional expectation involved in the leverage expression (3).

Model calibration

Once we have introduced the numerical scheme for the pricing with the HSLV model, we describe the model calibration procedure. In order to define the cost function f, we use the square of the root mean square error (RMSE) between the market prices Cmarket and the model prices Cmodel: f(p)=j=0JCmarketjCmodelj(p)2,where J is the number of European options per maturity and p=(κ,θ,σ,ρ,μ) is the vector of model parameters.

In terms of the cost function, the calibration problem can be formulated as

Numerical results

In this section we present the numerical experiments to validate the proposed methodology. More precisely, in Section 5.1 we present an experiment to show the computational speedup of the GPU pricer (the cost function) and the multi-CPU calibration with the optimization algorithms.

In Section 5.2 we apply this technique to calibrate the HLSV model to FX market data. Finally, in Section 5.3 we use the obtained calibrated model to price European and exotic options, more precisely to one-touch and

Conclusions

In this paper we have proposed a technique for the calibration of the whole set of parameters of the HSLV model in the framework of FX rate options. To the best of our knowledge, this is the first time that the direct calibration at once of the whole HSLV has been proposed, also including the market information about barrier options. This technique involves the MC discretization of the HSLV options pricing model and the use of efficient global stochastic optimization algorithms.

Thus, the

Acknowledgments

This work has been funded by Spanish MINECO with the grant MTM2016-76497-R and by Galician Government, Spain with the grant ED431C2018/033, both including FEDER financial support.

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