Estimating the Leverage Parameter of Continuous-Time Stochastic Volatility Models Using High Frequency S&P 500 and VIX

We consider general stochastic volatility models driven by continuous Brownian semimartingales, we show that the volatility of the variance and the leverage component (covariance between the asset price and the variance) can be reconstructed pathwise by exploiting Fourier analysis from the observation of the asset price. Specifying parametrically the asset price model we show that the method allows us to compute the parameters of the model. We provide a Monte Carlo experiment to recover the volatility and correlation parameters of the Heston model.

Download Full-text

ESTIMATION IN CONTINUOUS-TIME STOCHASTIC VOLATILITY MODELS USING NONLINEAR FILTERS

International Journal of Theoretical and Applied Finance ◽

10.1142/s0219024900000139 ◽

2000 ◽

Vol 03 (02) ◽

pp. 279-308 ◽

Cited By ~ 13

Author(s):

JAN NYGAARD NIELSEN ◽

MARTIN VESTERGAARD

Keyword(s):

Differential Equations ◽

Stochastic Volatility ◽

Stock Prices ◽

Continuous Time ◽

Stock Price ◽

Estimation Method ◽

Small Sample ◽

Stochastic Volatility Models ◽

Volatility Models ◽

Filtering Problem

The stylized facts of stock prices, interest and exchange rates have led econometricians to propose stochastic volatility models in both discrete and continuous time. However, the volatility as a measure of economic uncertainty is not directly observable in the financial markets. The objective of the continuous-discrete filtering problem considered here is to obtain estimates of the stock price and, in particular, the volatility using discrete-time observations of the stock price. Furthermore, the nonlinear filter acts as an important part of a proposed method for maximum likelihood for estimating embedded parameters in stochastic differential equations. In general, only approximate solutions to the continuous-discrete filtering problem exist in the form of a set of ordinary differential equations for the mean and covariance of the state variables. In the present paper the small-sample properties of a second order filter is examined for some bivariate stochastic volatility models and the new combined parameter and state estimation method is applied to US stock market data.

Download Full-text

Noncausality in Continuous Time Models

Econometric Theory ◽

10.1017/s0266466600006575 ◽

1996 ◽

Vol 12 (2) ◽

pp. 215-256 ◽

Cited By ~ 22

Author(s):

F. Comte ◽

E. Renault

Keyword(s):

Stochastic Volatility ◽

Continuous Time ◽

Moving Average ◽

Sufficient Conditions ◽

Stochastic Volatility Models ◽

Discrete Sample ◽

Volatility Models ◽

Moving Average Representation ◽

Canonical Representations ◽

Average Representation

In this paper, we study new definitions of noncausality, set in a continuous time framework, illustrated by the intuitive example of stochastic volatility models. Then, we define CIMA processes (i.e., processes admitting a continuous time invertible moving average representation), for which canonical representations and sufficient conditions of invertibility are given. We can provide for those CIMA processes parametric characterizations of noncausality relations as well as properties of interest for structural interpretations. In particular, we examine the example of processes solutions of stochastic differential equations, for which we study the links between continuous and discrete time definitions, find conditions to solve the possible problem of aliasing, and set the question of testing continuous time noncausality on a discrete sample of observations. Finally, we illustrate a possible generalization of definitions and characterizations that can be applied to continuous time fractional ARMA processes.

Download Full-text

The Bickel–Rosenblatt test for continuous time stochastic volatility models

Test ◽

10.1007/s11749-013-0347-1 ◽

2013 ◽

Vol 23 (1) ◽

pp. 195-218

Author(s):

Liang-Ching Lin ◽

Sangyeol Lee ◽

Meihui Guo

Keyword(s):

Stochastic Volatility ◽

Continuous Time ◽

Stochastic Volatility Models ◽

Volatility Models

Download Full-text

Variance-Optimal Hedging in General Affine Stochastic Volatility Models

Advances in Applied Probability ◽

10.1239/aap/1269611145 ◽

2010 ◽

Vol 42 (1) ◽

pp. 83-105 ◽

Cited By ~ 7

Author(s):

Jan Kallsen ◽

Arnd Pauwels

Keyword(s):

Stochastic Volatility ◽

Continuous Time ◽

Stock Price ◽

Forthcoming Paper ◽

Parametric Models ◽

Stochastic Volatility Models ◽

Optimal Hedging ◽

Volatility Models ◽

Optimal Hedge ◽

General Affine

We consider variance-optimal hedging in general continuous-time affine stochastic volatility models. The optimal hedge and the associated hedging error are determined semiexplicitly in the case that the stock price follows a martingale. The integral representation of the solution opens the door to efficient numerical computation. The setup includes models with jumps in the stock price and in the activity process. It also allows for correlation between volatility and stock price movements. Concrete parametric models will be illustrated in a forthcoming paper.

Download Full-text

Functional Relationships Between Price and Volatility Jumps and Their Consequences for Discretely Observed Data

Journal of Applied Probability ◽

10.1017/s0021900200012778 ◽

2012 ◽

Vol 49 (04) ◽

pp. 901-914

Author(s):

Jean Jacod ◽

Claudia Klüppelberg ◽

Gernot Müller

Keyword(s):

Asymptotic Behaviour ◽

Stochastic Volatility ◽

Continuous Time ◽

Discrete Observations ◽

Stochastic Volatility Models ◽

Functional Relationships ◽

Price Process ◽

Functional Relations ◽

Volatility Models ◽

Volatility Jumps

Many prominent continuous-time stochastic volatility models exhibit certain functional relationships between price jumps and volatility jumps. We show that stochastic volatility models like the Ornstein–Uhlenbeck and other continuous-time CARMA models as well as continuous-time GARCH and EGARCH models all exhibit such functional relations. We investigate the asymptotic behaviour of certain functionals of price and volatility processes for discrete observations of the price process on a grid, which are relevant for estimation and testing problems.

Download Full-text