Parameter Estimation for a Fractional Black–Scholes Model with Jumps from Discrete Time Observations

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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Bibliographic Details
Main Authors: John-Fritz Thony, Jean Vaillant
Format: Article
Language:English
Published: MDPI AG 2022-11-01
Series:Mathematics
Subjects:
stochastic differential equation
fractional Black–Scholes
jump process
maximum likelihood estimation
Online Access:https://www.mdpi.com/2227-7390/10/22/4190
Description
Summary: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.
ISSN:2227-7390

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