Parameter Estimation of the Heston Volatility Model with Jumps in the Asset Prices
Abstract
:1. Introduction
2. Heston Model without and with Jumps
2.1. Model Characterisation
2.2. Euler–Maruyama Discretisation
3. Estimation Framework
3.1. Regular Heston Model
3.1.1. Estimation of $\mu $
3.1.2. Estimation of $\kappa $, $\theta $, and $\sigma $
3.1.3. Estimation of $\rho $
3.1.4. Estimation of $v\left(t\right)$—Particle Filtering
 The particle with the smallest value is the first in the new sequence, i.e.,$${\tilde{V}}_{1}^{sort}(k\Delta t)=\underset{j\in \{1,2,\dots ,N\}}{min}\left\{{\tilde{V}}_{j}(k\Delta t)\right\},$$
 The particle with the largest value is the last in the new sequence, i.e.,$${\tilde{V}}_{N}^{sort}(k\Delta t)=\underset{j\in \{1,2,\dots ,N\}}{max}\left\{{\tilde{V}}_{j}(k\Delta t)\right\},$$
 For any $j\in \{2,3,\dots N1\}$, we have$${\tilde{V}}_{j1}^{sort}(k\Delta t)<{\tilde{V}}_{j}^{sort}(k\Delta t)<{\tilde{V}}_{j+1}^{sort}(k\Delta t).$$
3.2. Heston Model with Jumps
3.3. Estimation Procedure
Algorithm 1:Estimating the Heston model 
Require:

Algorithm 2:Estimating the Heston model with jumps 
Require:

4. Analysis of the Estimation Results
4.1. Exemplary Estimation
4.2. Important Findings
4.3. Towards RealLife Applications
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A. Table of Symbols
Quantity  Explanation 

T  max time (i.e., $t\in [0,T]$) 
$S\left(t\right)$  asset price (with $S\left(0\right)\equiv {S}_{0}$) 
$v\left(t\right)$  volatility (with $v\left(0\right)\equiv {v}_{0}$) 
${B}^{S}\left(t\right)$  Brownian motion for the price process 
${B}^{v}\left(t\right)$  Brownian motion for the volatility process 
$\mu $  drift 
$\kappa $  rate of return to the longtime average 
$\theta $  longtime average 
$\sigma $  volatility of the volatility 
$\rho $  correlation between prices and volatility 
$Z\left(t\right)$  size of the jump 
${\mu}^{J}$  mean of the jump size 
${\sigma}^{J}$  standard deviation of the jump size 
$q\left(t\right)$  Poisson process counting jumps 
$\lambda $  intensity of jumps 
$\Delta t$  time step (also known as the discretisation constant) 
n  number of time steps (also known as the length of data) 
${\epsilon}^{S}\left(t\right)$  price process random component 
${\epsilon}^{v}\left(t\right)$  volatility process random component 
${\epsilon}^{add}\left(t\right)$  additional random component—see Equation (10)) 
$\eta $  regression parameter for drift estimation—see Equation (11) 
$R\left(t\right)$  ratio between neighbouring prices—see Equations (12) and (84) 
${y}^{S}\left(t\right)$  series of dependent variables for the drift estimation—see Equation (14) 
${x}^{S}\left(t\right)$  series of independent variables for the drift estimation—see Equation (15) 
${\mathbf{y}}^{S}$  vector of the dependent variable for the drift estimation—see Equation (17) 
${\mathbf{x}}^{S}$  vector of the independent variable for the drift estimation—see Equation (18) 
${\mu}_{0}^{\eta}$  mean of the prior distribution of $\eta $ 
${\sigma}_{0}^{\eta}$  standard deviation of the prior distribution of $\eta $ 
Quantity  Explanation 

${\tau}_{0}^{\eta}$  precision of the prior distribution of $\eta $ 
$\widehat{\eta}$  OLS estimator of $\eta $—see (21) 
${\mu}^{\eta}$  mean of the posterior distribution of $\eta $—see (20) 
${\tau}^{\eta}$  precision of the posterior distribution of $\eta $—see (19) 
${\eta}_{i}$  ith sample of $\eta $—see (22) 
${\mu}_{i}$  ith estimate of the drift—see (23) 
${\beta}_{1}$  regression parameter for volatility parameters estimation—see Equation (25) 
${\beta}_{2}$  regression parameter for volatility parameters estimation—see Equation (26) 
$\mathit{\beta}$  vector of regression parameters for volatility parameters estimation—see Equation (29) 
${\mathbf{y}}^{v}$  vector of the dependent variable for the volatility parameter estimation—see Equation (30) 
${\mathbf{x}}_{1}^{v}$  vector of the independent variable for the volatility parameter estimation—see Equation (31) 
${\mathbf{x}}_{2}^{v}$  vector of the independent variable for the volatility parameter estimation—see Equation (32) 
${\mathbf{X}}^{v}$  matrix of the independent variable for the volatility parameter estimation—see Equation (34) 
${\mathbf{\epsilon}}^{v}$  noise vector of the volatility parameter estimation—see Equation (35) 
${\mathit{\mu}}_{0}^{\beta}$  mean vector of the prior distribution of $\mathit{\beta}$ 
${\mathbf{\Lambda}}_{0}^{\beta}$  precision matrix of the prior distribution of $\mathit{\beta}$ 
${\mathit{\mu}}^{\beta}$  mean vector of the posterior distribution of $\mathit{\beta}$—see Equation (37) 
${\mathbf{\Lambda}}^{\beta}$  precision matrix of the posterior distribution of $\mathit{\beta}$—see (36) 
$\widehat{\mathit{\beta}}$  OLS estimator of $\mathit{\beta}$—see (38) 
${\mathit{\beta}}_{i}$  ith sample of $\mathit{\beta}$—see (39) 
${\kappa}_{i}$  ith estimate of $\kappa $—see (40) 
${\theta}_{i}$  ith estimate of $\theta $—see (41) 
${a}_{0}^{\sigma}$  shape parameter of the prior distribution of ${\sigma}^{2}$ 
${b}_{0}^{\sigma}$  scale parameter of the prior distribution of ${\sigma}^{2}$ 
${a}^{\sigma}$  shape parameter of the posterior distribution of ${\sigma}^{2}$ 
${b}^{\sigma}$  scale parameter of the posterior distribution of ${\sigma}^{2}$ 
${\sigma}_{i}$  ith estimate of the $\sigma $—see (42) 
${e}_{1}^{\rho}\left(t\right)$  series of residuals of the price equation—see (45) 
${e}_{2}^{\rho}\left(t\right)$  series of residuals of the volatility equation—see (46) 
$\psi $  regression parameter for $\rho $ estimation, $\psi =\sigma \rho $—see (47) 
$\omega $  regression parameter for $\rho $ estimation, $\omega ={\sigma}^{2}\left(1{\rho}^{2}\right)$—see (47) 
${e}_{1}^{\rho}\left(t\right)$  series of independent variables for the estimation of $\rho $—see (45) 
${e}_{2}^{\rho}\left(t\right)$  series of dependent variables for the estimation of $\rho $—see (46) 
${\mathbf{e}}_{1}^{\rho}$  vector of the independent variables for the estimation of $\rho $—see (50) 
${\mathbf{e}}_{2}^{\rho}$  vector of the dependent variables for the estimation of $\rho $—see (51) 
${\mathbf{e}}^{\rho}$  matrix of residuals—see (52) 
${\mathbf{A}}^{\rho}$  auxiliary matrix for solving $\rho $ regression—see (53) 
${\mu}_{0}^{\psi}$  mean of the prior distribution of $\psi $ 
${\tau}_{0}^{\psi}$  precision of the prior distribution of $\psi $ 
${\mu}^{\psi}$  mean of the posterior distribution of $\psi $—see (54) 
${\tau}^{\psi}$  precision of the posterior distribution of $\psi $—see (55) 
${a}_{0}^{\omega}$  shape parameter of the prior distribution of $\omega $ 
${b}_{0}^{\omega}$  scale parameter of the prior distribution of $\omega $ 
${a}^{\omega}$  shape parameter of the posterior distribution of $\omega $—see (56) 
${b}^{\omega}$  scale parameter of the posterior distribution of $\omega $—see (57) 
${\psi}_{i}$  ith sample of $\psi $—see (59) 
${\omega}_{i}$  ith sample of $\omega $—see (58) 
${\epsilon}_{j}\left(t\right)$  jth sample of particle filtering independent errors—see (61) 
${z}_{j}\left(t\right)$  jth sample of particle filtering residuals—see (62) 
${w}_{j}\left(t\right)$  jth sample of particle filtering correlated—see (62) 
${\tilde{V}}_{j}\left(t\right)$  jth raw volatility particle—see (64) 
${\tilde{W}}_{j}\left(t\right)$  jth particle likelihood measure—see (65) and (77) 
${\stackrel{\u02d8}{W}}_{j}\left(t\right)$  jth particle probability—see (66) 
${\mathbf{U}}_{j}\left(t\right)$  jth particleprobability vector—see (67) 
${\tilde{V}}_{j}^{sort}\left(t\right)$  jth raw volatility particle in a sorted sequence—see (68), (69), and (70) 
${\stackrel{\u02d8}{W}}_{j}^{sort}\left(t\right)$  probability of the jth particle in the sorted sequence—see (71) 
${F}_{{\tilde{V}}^{sort}}\left(v\right)$  continuous CDF of resampled particles—see (72) 
Quantity  Explanation 

${V}_{j}\left(t\right)$  jth final volatility particle—see (73) 
${\lambda}^{th}$  proportion of particles encoding a jump 
${\tilde{J}}_{j}\left(t\right)$  jth raw momentofajump particle—see (75) 
${\mu}_{0}^{J}$  mean of the raw sizeofajump particle 
${\sigma}_{0}^{J}$  standard deviation of the raw sizeofajump particle 
${\tilde{Z}}_{j}\left(t\right)$  jth raw sizeofajump particle—see (76) 
${Z}_{j}\left(t\right)$  jth resampled sizeofajump particle—see (78) 
$\lambda \left(t\right)$  probability of a jump—see (79) 
${\lambda}_{i}$  ith estimate of $\lambda $—see (80) 
$Z\left(t\right)$  estimate of an average size of a jump—see (81) 
${\mu}_{i}^{J}$  ith estimate of ${\mu}^{J}$—see (82) 
${\sigma}_{i}^{J}$  ith estimate of ${\sigma}^{J}$—see (83) 
Notes
1  Equation (65) is the reason we cannot run this procedure for $k=n$, as we would not be able to obtain $R\left((n+1)\Delta t\right)$, since the last available value is $R(n\Delta t)$. 
2  For applications in finance, this task is sometimes easier than for some other fields of science, as numerous works have been published already, presenting the results of the estimates of wellknown stocks or market indices within various models. See, e.g., Eraker et al. (2003) 
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Prior Parameter  Value 

${\mu}_{0}^{\eta}$  1.00125 
${\sigma}_{0}^{\eta}$  0.001 
${\mathbf{\Lambda}}_{0}$  $\left[\begin{array}{cc}10& 0\\ 0& 5\end{array}\right]$ 
${\mathit{\mu}}_{0}$  $\left[\begin{array}{c}35\times {10}^{6}\\ 0.988\end{array}\right]$ 
${a}_{0}^{\sigma}$  149 
${b}_{0}^{\sigma}$  0.025 
${\mu}_{0}^{\psi}$  $0.45$ 
${\sigma}_{0}^{\psi}$  0.3 
${a}_{0}^{\omega}$  1.03 
${b}_{0}^{\omega}$  0.05 
${\lambda}^{th}$  0.15 
${\mu}_{0}^{J}$  $0.96$ 
${\sigma}_{0}^{J}$  0.3 
Parameter  True Value  Estimated Value  Relative Error [%] 

$\mu $  0.1  0.09829  1.77 
$\kappa $  1  1.2190  21.90 
$\theta $  0.05  0.0493  1.92 
$\sigma $  0.01  0.0108  8.55 
$\rho $  −0.5  0.4379  12.40 
$\lambda $  1  1.3349  33.49 
${\mu}_{J}$  −0.8  −0.9651  20.64 
${\sigma}_{J}$  0.2  0.2298  14.88 
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Gruszka , J.; Szwabiński, J. Parameter Estimation of the Heston Volatility Model with Jumps in the Asset Prices. Econometrics 2023, 11, 15. https://doi.org/10.3390/econometrics11020015
Gruszka J, Szwabiński J. Parameter Estimation of the Heston Volatility Model with Jumps in the Asset Prices. Econometrics. 2023; 11(2):15. https://doi.org/10.3390/econometrics11020015
Chicago/Turabian StyleGruszka , Jarosław, and Janusz Szwabiński. 2023. "Parameter Estimation of the Heston Volatility Model with Jumps in the Asset Prices" Econometrics 11, no. 2: 15. https://doi.org/10.3390/econometrics11020015