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...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2021-11-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/9/22/2930 |
| _version_ | 1827676147860635648 |
|---|---|
| author | Siow Woon Jeng Adem Kiliçman |
| author_facet | Siow Woon Jeng Adem Kiliçman |
| author_sort | Siow Woon Jeng |
| collection | DOAJ |
| 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. |
| first_indexed | 2024-03-10T05:18:35Z |
| format | Article |
| id | doaj.art-3ca982ddfe134bbe98194196282c8b02 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T05:18:35Z |
| publishDate | 2021-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-3ca982ddfe134bbe98194196282c8b022023-11-23T00:15:11ZengMDPI AGMathematics2227-73902021-11-01922293010.3390/math9222930On Multilevel and Control Variate Monte Carlo Methods for Option Pricing under the Rough Heston ModelSiow Woon Jeng0Adem Kiliçman1Institute for Mathematical Research, Faculty of Science, University Putra Malaysia, Serdang 43400, Selangor, MalaysiaInstitute for Mathematical Research, Faculty of Science, University Putra Malaysia, Serdang 43400, Selangor, MalaysiaThe 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.https://www.mdpi.com/2227-7390/9/22/2930rough Heston modelweak convergence error rateMonte Carlo methodcontrol variate methodmultilevel Monte Carlo method |
| spellingShingle | Siow Woon Jeng Adem Kiliçman On Multilevel and Control Variate Monte Carlo Methods for Option Pricing under the Rough Heston Model Mathematics rough Heston model weak convergence error rate Monte Carlo method control variate method multilevel Monte Carlo method |
| title | 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_short | 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 |
| topic | rough Heston model weak convergence error rate Monte Carlo method control variate method multilevel Monte Carlo method |
| url | https://www.mdpi.com/2227-7390/9/22/2930 |
| work_keys_str_mv | AT siowwoonjeng onmultilevelandcontrolvariatemontecarlomethodsforoptionpricingundertheroughhestonmodel AT ademkilicman onmultilevelandcontrolvariatemontecarlomethodsforoptionpricingundertheroughhestonmodel |