On Multilevel and Control Variate Monte Carlo Methods for Option Pricing under the Rough Heston Model

The rough Heston model is a form of a stochastic Volterra equation, which was proposed to model stock price volatility. It captures some important qualities that can be observed in the financial market—highly endogenous, statistical arbitrages prevention, liquidity asymmetry, and metaorders. Unlike...

Descripción completa

Guardado en:

Detalles Bibliográficos
Autores principales:	Siow Woon Jeng, Adem Kiliçman
Formato:	article
Lenguaje:	EN
Publicado:	MDPI AG 2021
Materias:	rough Heston model weak convergence error rate Monte Carlo method control variate method multilevel Monte Carlo method Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/3ca982ddfe134bbe98194196282c8b02
Etiquetas:	Agregar Etiqueta Sin Etiquetas, Sea el primero en etiquetar este registro!

id	oai:doaj.org-article:3ca982ddfe134bbe98194196282c8b02
record_format	dspace
spelling	oai:doaj.org-article:3ca982ddfe134bbe98194196282c8b022021-11-25T18:17:15ZOn Multilevel and Control Variate Monte Carlo Methods for Option Pricing under the Rough Heston Model10.3390/math92229302227-7390https://doaj.org/article/3ca982ddfe134bbe98194196282c8b022021-11-01T00:00:00Zhttps://www.mdpi.com/2227-7390/9/22/2930https://doaj.org/toc/2227-7390The rough Heston model is a form of a stochastic Volterra equation, which was proposed to model stock price volatility. It captures some important qualities that can be observed in the financial market—highly endogenous, statistical arbitrages prevention, liquidity asymmetry, and metaorders. Unlike stochastic differential equation, the stochastic Volterra equation is extremely computationally expensive to simulate. In other words, it is difficult to compute option prices under the rough Heston model by conventional Monte Carlo simulation. In this paper, we prove that Euler’s discretization method for the stochastic Volterra equation with non-Lipschitz diffusion coefficient <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">E</mi><mo>[</mo><mo>\|</mo><msub><mi>V</mi><mi>t</mi></msub><mo>−</mo><msubsup><mi>V</mi><mi>t</mi><mi>n</mi></msubsup><msup><mo>\|</mo><mi>p</mi></msup><mo>]</mo></mrow></semantics></math></inline-formula> is finitely bounded by an exponential function of <i>t</i>. Furthermore, the weak error <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>\|</mo><mi mathvariant="double-struck">E</mi><mo>[</mo><msub><mi>V</mi><mi>t</mi></msub><mo>−</mo><msubsup><mi>V</mi><mi>t</mi><mi>n</mi></msubsup><mo>]</mo><mo>\|</mo></mrow></semantics></math></inline-formula> and convergence for the stochastic Volterra equation are proven at the rate of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo>(</mo><msup><mi>n</mi><mrow><mo>−</mo><mi>H</mi></mrow></msup><mo>)</mo></mrow></semantics></math></inline-formula>. In addition, we propose a mixed Monte Carlo method, using the control variate and multilevel methods. The numerical experiments indicate that the proposed method is capable of achieving a substantial cost-adjusted variance reduction up to 17 times, and it is better than its predecessor individual methods in terms of cost-adjusted performance. Due to the cost-adjusted basis for our numerical experiment, the result also indicates a high possibility of potential use in practice.Siow Woon JengAdem KiliçmanMDPI AGarticlerough Heston modelweak convergence error rateMonte Carlo methodcontrol variate methodmultilevel Monte Carlo methodMathematicsQA1-939ENMathematics, Vol 9, Iss 2930, p 2930 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	rough Heston model weak convergence error rate Monte Carlo method control variate method multilevel Monte Carlo method Mathematics QA1-939
spellingShingle	rough Heston model weak convergence error rate Monte Carlo method control variate method multilevel Monte Carlo method Mathematics QA1-939 Siow Woon Jeng Adem Kiliçman On Multilevel and Control Variate Monte Carlo Methods for Option Pricing under the Rough Heston Model
description	The rough Heston model is a form of a stochastic Volterra equation, which was proposed to model stock price volatility. It captures some important qualities that can be observed in the financial market—highly endogenous, statistical arbitrages prevention, liquidity asymmetry, and metaorders. Unlike stochastic differential equation, the stochastic Volterra equation is extremely computationally expensive to simulate. In other words, it is difficult to compute option prices under the rough Heston model by conventional Monte Carlo simulation. In this paper, we prove that Euler’s discretization method for the stochastic Volterra equation with non-Lipschitz diffusion coefficient <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">E</mi><mo>[</mo><mo>\|</mo><msub><mi>V</mi><mi>t</mi></msub><mo>−</mo><msubsup><mi>V</mi><mi>t</mi><mi>n</mi></msubsup><msup><mo>\|</mo><mi>p</mi></msup><mo>]</mo></mrow></semantics></math></inline-formula> is finitely bounded by an exponential function of <i>t</i>. Furthermore, the weak error <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>\|</mo><mi mathvariant="double-struck">E</mi><mo>[</mo><msub><mi>V</mi><mi>t</mi></msub><mo>−</mo><msubsup><mi>V</mi><mi>t</mi><mi>n</mi></msubsup><mo>]</mo><mo>\|</mo></mrow></semantics></math></inline-formula> and convergence for the stochastic Volterra equation are proven at the rate of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo>(</mo><msup><mi>n</mi><mrow><mo>−</mo><mi>H</mi></mrow></msup><mo>)</mo></mrow></semantics></math></inline-formula>. In addition, we propose a mixed Monte Carlo method, using the control variate and multilevel methods. The numerical experiments indicate that the proposed method is capable of achieving a substantial cost-adjusted variance reduction up to 17 times, and it is better than its predecessor individual methods in terms of cost-adjusted performance. Due to the cost-adjusted basis for our numerical experiment, the result also indicates a high possibility of potential use in practice.
format	article
author	Siow Woon Jeng Adem Kiliçman
author_facet	Siow Woon Jeng Adem Kiliçman
author_sort	Siow Woon Jeng
title	On Multilevel and Control Variate Monte Carlo Methods for Option Pricing under the Rough Heston Model
title_short	On Multilevel and Control Variate Monte Carlo Methods for Option Pricing under the Rough Heston Model
title_full	On Multilevel and Control Variate Monte Carlo Methods for Option Pricing under the Rough Heston Model
title_fullStr	On Multilevel and Control Variate Monte Carlo Methods for Option Pricing under the Rough Heston Model
title_full_unstemmed	On Multilevel and Control Variate Monte Carlo Methods for Option Pricing under the Rough Heston Model
title_sort	on multilevel and control variate monte carlo methods for option pricing under the rough heston model
publisher	MDPI AG
publishDate	2021
url	https://doaj.org/article/3ca982ddfe134bbe98194196282c8b02
work_keys_str_mv	AT siowwoonjeng onmultilevelandcontrolvariatemontecarlomethodsforoptionpricingundertheroughhestonmodel AT ademkilicman onmultilevelandcontrolvariatemontecarlomethodsforoptionpricingundertheroughhestonmodel
_version_	1718411373422051328

On Multilevel and Control Variate Monte Carlo Methods for Option Pricing under the Rough Heston Model

Ejemplares similares