CSI 771 - Computational Statistics, Fall 2005

Course Project Webpage by Wei Sun

11/28/2005: Final Project Milestone

• Final report
• Presentation

11/6/2005: Project preliminary report .

09/26/2005: Feasibility study -- software, etc

• Basically, I will use Matlab as the tool to implement the methods, but also I may use other software such as SAS or R for analysis if needed.
• The most difficult part of the project is to go over those theories about the stochastic volatility model and make myself familar with the related financial backgruoud. However, I think I can achieve it at some level.

09/19/2005: Design a plan to replicate and extend one of the studies

• I am going to replicate part of works done by Toshiaki Watanabe in the paper "A Non-linear Filtering Approach to Stochastic Volatility Models with an Application to Daily Stock Returns" published in the Journal of Applied Econometrics in 1999.
• A number of methods are discussed and compared in the paper such as GMM, QMLE, SMM, also they propose the new NFML.
• Theory about the stochastic volatility model need to be reviewed and I need make myself familiar with the related background in finance
• Possible extension
  • try to use a new nonlinear filtering method named Unscented Kalman Filter to estimate the SV model
  • apply the methods in US stock index such as S&P 500 or Nasdaq

09/12/2005: Two articles in statistics literature that reports monte carlo studies

• 1. A Non-linear Filtering Approach to Stochastic Volatility Models with an Application to Daily Stock Returns
  By Toshiaki Watanabe, Journal of Applied Econometrics,
  Vol.14, No.2 (Mar. - Apr., 1999), p101-121

  Stochastic volatility model (SV) has recently attracted the attention of financial economists and researchers. Since volatility is a latent variable in SV models, it is difficult to evaluate the exact likelihood. The maximum likelihood method is not easy to implement in SV models, but several alternatives are now available in the community, such as the method of moment including simple moment matching, generalized method of moments(GMM) and various simulated methods of moment (SMM). Another simple approach is quasi-maximum likelihood (QML) estimation by Nelson (1988) and Ruiz(1994) using the standard kalman filter to the linear state space representation of a SV model to obtain the quasi-likelihood. In this paper, author developed a new likelihood-based method for the analysis of SV model using a non-linear filter which yields the exact likelihood of SV models. The non-linear filter is described by a series of integrals. Solving these integrals by piecewise linear approximation with randomly chosen nodes provides the likelihood of SV models. Critically, the accurary of the density approximation depends on the number of nodes used for the piecewise linear approximation, and the number of nodes is limited by the computational demands. They choose nodes by generating random numbers based on approximations of the volatility densities computed by executing the standard kalman filter to the linear state space representation of the SV model
  conditional on the QML estimates of the parameters. Solving the non-linear filter using this technique yields the exact likelihood of SV models. In addtion, a non-linear smoothing algorithm for volatility estimation is constructed in the paper too.

  The Monte Carlo simulations analyzing the finite sample performance of the proposed method show that the new method performs well even when the number of nodes is reasonably small. In parameter estimation, the new method outperformed QML and GMM estimator.

• 2. Statistical Inference for Random-Variance Option Pricing
  by Sergio Pastorrello, Eric Renault, Nizar Touzi,
  Journal of Business & Economic Statistics, Vol. 18, No.3, July 2000, p358-367

  The classical Black and Scholes (BS) model for option pricing relies on the assumption that asset prices are generated by a constant volatility diffusion process. Empirical work on valuation and hedging, however, has shown that a nonnegligible property of stock prices is that their variance changes through time. In a continuous time framework, a random diffusion coefficient can be modeled through a second stochastic differential equation, with functional form known up to a vector of parameters, describing the dynamics of a one-to-one transform of the volatility of stock prices. Such a generalization of the basic BS model, however, raises several interesting issues related to the pricing and hedging of derivateive asset based on a stock with stochastic volatility and to the estimation and testing of this continious-time model with an unobservable state variable.

  In this paper, authors address the problems associated with the estimation and testing of this continuous-time stochastic volatility models of option pricing. At first sight, it may seem natural to estimate the parameters of the model from the observation of a time series of stock prices. But they argue that option price are much more informative about the parameters than are the asset prices by a Monte Carlo experiment comparing two very simple strategies based on the different information sets. They also show that, given their option-pricing model, estimation based on option prices is much more precise in samples of typical size, without increasing the compuational burden.