# SCIENCE CHINA Information Sciences, Volume 61, Issue 4: 042202(2018) https://doi.org/10.1007/s11432-017-9215-8

## Parameter estimates of Heston stochastic volatility model with MLE and consistent EKF algorithm • AcceptedAug 8, 2017
• PublishedMar 5, 2018
Share
Rating

### Abstract

Heston model is the most famous stochastic volatility model in finance. This paper considers the parameter estimation problem of Heston model with both known and unknown volatilities. First, parameters in equity process and volatility process of Heston model are estimated separately since there is no explicit solution for the likelihood function with all parameters. Second, the normal maximum likelihood estimation (NMLE) algorithm is proposed based on the Itôtransformation of Heston model. The algorithm can reduce the estimate error compared with existing pseudo maximum likelihood estimation. Third, the NMLE algorithm and consistent extended Kalman filter (CEKF) algorithm are combined in the case of unknown volatilities. As an advantage, CEKF algorithm can apply an upper bound of the error covariance matrix to ensure the volatilities estimation errors to be well evaluated. Numerical simulations illustrate that the proposed NMLE algorithm works more efficiently than the existing pseudo MLE algorithm with known and unknown volatilities. Therefore, the upper bound of the error covariance is illustrated. Additionally, the proposed estimation method is applied to American stock market index S&P 500, and the result shows the utility and effectiveness of the NMLE-CEKF algorithm.

### Acknowledgment

This work was in part supported by National Key Research and Development Program of China (Grant No. 2016YFB0901902), National Natural Science Foundation of China (Grant No. 61622309), and National Basic Research Program of China (973 Program) (Grant No. 2014CB845301).

• Figure 1

(Color online) Values of $V_{k}$ and $(\sqrt{V_{k}})^2$ in the condition that $\delta=0.01$, $n=1000$, where ${\rm~max}(|V_k-(\sqrt{V_k})^2|)=2.94\times10^{-2}$, $k~=~1,2,\ldots,n.$

• Figure 2

(Color online) Values of $V_{1+ks}$ and $(\sqrt{V_{1+ks}})^2$ in the condition that $\tilde{n}=1000\times~100$, $\tilde{\delta}=~\frac{\delta}{100}$, where ${\rm~max}(|V_k-(\sqrt{V_k})^2|)=1.26\times10^{-3}$, $k=0,1,2,\ldots,n-1$.

• Figure 3

(Color online) Estimation of $\kappa,~\theta,~\sigma$ with NMLE and PMLE.

• Figure 4

(Color online) Estimation of $\rho$ with NMLE.

• Figure 5

(Color online) (a) Volatility tracking with CEKF and EKF; (b) square errors of CEKF and EKF.

• Figure 6

(Color online) The relationship of MSE and $P_k$ for CEKF and EKF.

• Figure 7

(Color online) (a) Volatility tracking results with NMLE-CEKF and PMLE-CEKF; (b) volatility errors with NMLE-CEKF and PMLE-CEKF.

• Figure 8

(Color online) Estimation of $\kappa,~\theta,~\sigma$ with NMLE-CEKF and PMLE-CEKF.

• Figure 9

(Color online) The logarithm price of S&P 500 (SPX) index value and VIX.

• Figure 10

(Color online) Return of S&P 500 (SPX) index value.

• Figure 11

(Color online) The volatility trackings of S&P 500 of Heston model compared with VIX index and history volatilities.

Copyright 2020 Science China Press Co., Ltd. 《中国科学》杂志社有限责任公司 版权所有