Parameter Estimation for a Fractional Black–Scholes Model with Jumps from Discrete Time Observations
We consider a stochastic differential equation (SDE) governed by a fractional Brownian motion <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msubsup><mi>B</mi><...
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MDPI AG
2022-11-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/22/4190 |
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| author | John-Fritz Thony Jean Vaillant |
| author_facet | John-Fritz Thony Jean Vaillant |
| author_sort | John-Fritz Thony |
| collection | DOAJ |
| description | We consider a stochastic differential equation (SDE) governed by a fractional Brownian motion <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msubsup><mi>B</mi><mrow><mi>t</mi></mrow><mi>H</mi></msubsup><mo>)</mo></mrow></semantics></math></inline-formula> and a Poisson process <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi>N</mi><mi>t</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> associated with a stochastic process <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi>A</mi><mi>t</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> such that: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><msub><mi>X</mi><mi>t</mi></msub><mo>=</mo><mi>μ</mi><msub><mi>X</mi><mi>t</mi></msub><mi>d</mi><mi>t</mi><mo>+</mo><mi>σ</mi><msub><mi>X</mi><mi>t</mi></msub><mi>d</mi><msubsup><mi>B</mi><mrow><mi>t</mi></mrow><mi>H</mi></msubsup><mo>+</mo><msub><mi>A</mi><mi>t</mi></msub><msub><mi>X</mi><msup><mi>t</mi><mo>−</mo></msup></msub><mi>d</mi><msub><mi>N</mi><mi>t</mi></msub><mo>,</mo><mspace width="1.em"></mspace><msub><mi>X</mi><mn>0</mn></msub><mo>=</mo><msub><mi>x</mi><mn>0</mn></msub><mo>></mo><mn>0</mn><mo>.</mo></mrow></semantics></math></inline-formula> The solution of this SDE is analyzed and properties of its trajectories are presented. Estimators of the model parameters are proposed when the observations are carried out in discrete time. Some convergence properties of these estimators are provided according to conditions concerning the value of the Hurst index and the nonequidistance of the observation dates. |
| first_indexed | 2024-03-09T18:10:56Z |
| format | Article |
| id | doaj.art-837d8c6129104e7c8b8ae203297222e6 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T18:10:56Z |
| publishDate | 2022-11-01 |
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| spelling | doaj.art-837d8c6129104e7c8b8ae203297222e62023-11-24T09:07:27ZengMDPI AGMathematics2227-73902022-11-011022419010.3390/math10224190Parameter Estimation for a Fractional Black–Scholes Model with Jumps from Discrete Time ObservationsJohn-Fritz Thony0Jean Vaillant1Laboratoire de Mathématiques et Informatique et Applications (LAMIA), Université des Antilles, 97157 Pointe-à-Pitre, FranceLaboratoire de Mathématiques et Informatique et Applications (LAMIA), Université des Antilles, 97157 Pointe-à-Pitre, FranceWe consider a stochastic differential equation (SDE) governed by a fractional Brownian motion <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msubsup><mi>B</mi><mrow><mi>t</mi></mrow><mi>H</mi></msubsup><mo>)</mo></mrow></semantics></math></inline-formula> and a Poisson process <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi>N</mi><mi>t</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> associated with a stochastic process <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi>A</mi><mi>t</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> such that: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><msub><mi>X</mi><mi>t</mi></msub><mo>=</mo><mi>μ</mi><msub><mi>X</mi><mi>t</mi></msub><mi>d</mi><mi>t</mi><mo>+</mo><mi>σ</mi><msub><mi>X</mi><mi>t</mi></msub><mi>d</mi><msubsup><mi>B</mi><mrow><mi>t</mi></mrow><mi>H</mi></msubsup><mo>+</mo><msub><mi>A</mi><mi>t</mi></msub><msub><mi>X</mi><msup><mi>t</mi><mo>−</mo></msup></msub><mi>d</mi><msub><mi>N</mi><mi>t</mi></msub><mo>,</mo><mspace width="1.em"></mspace><msub><mi>X</mi><mn>0</mn></msub><mo>=</mo><msub><mi>x</mi><mn>0</mn></msub><mo>></mo><mn>0</mn><mo>.</mo></mrow></semantics></math></inline-formula> The solution of this SDE is analyzed and properties of its trajectories are presented. Estimators of the model parameters are proposed when the observations are carried out in discrete time. Some convergence properties of these estimators are provided according to conditions concerning the value of the Hurst index and the nonequidistance of the observation dates.https://www.mdpi.com/2227-7390/10/22/4190stochastic differential equationfractional Black–Scholesjump processmaximum likelihood estimation |
| spellingShingle | John-Fritz Thony Jean Vaillant Parameter Estimation for a Fractional Black–Scholes Model with Jumps from Discrete Time Observations Mathematics stochastic differential equation fractional Black–Scholes jump process maximum likelihood estimation |
| title | Parameter Estimation for a Fractional Black–Scholes Model with Jumps from Discrete Time Observations |
| title_full | Parameter Estimation for a Fractional Black–Scholes Model with Jumps from Discrete Time Observations |
| title_fullStr | Parameter Estimation for a Fractional Black–Scholes Model with Jumps from Discrete Time Observations |
| title_full_unstemmed | Parameter Estimation for a Fractional Black–Scholes Model with Jumps from Discrete Time Observations |
| title_short | Parameter Estimation for a Fractional Black–Scholes Model with Jumps from Discrete Time Observations |
| title_sort | parameter estimation for a fractional black scholes model with jumps from discrete time observations |
| topic | stochastic differential equation fractional Black–Scholes jump process maximum likelihood estimation |
| url | https://www.mdpi.com/2227-7390/10/22/4190 |
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