Online Estimation of Time-Varying Volatility Using a Continuous-Discrete LMS Algorithm
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Research Article Online Estimation of Time-Varying Volatility Using a Continuous-Discrete LMS Algorithm Elisabeth Lahalle, Hana Baili, and Jacques Oksman Department of Signal Processing and Electronic Systems Sup´elec, 3 rue Joliot-Curie, Plateau de Moulon, 91192 Gif sur Yvette, France Correspondence should be addressed to Elisabeth Lahalle, [email protected] Received 27 March 2007; Revised 21 December 2007; Accepted 9 July 2008 Recommended by Ioannis Psaromiligkos The following paper addresses a problem of inference in financial engineering, namely, online time-varying volatility estimation. The proposed method is based on an adaptive predictor for the stock price, built from an implicit integration formula. An estimate for the current volatility value which minimizes the mean square prediction error is calculated recursively using an LMS algorithm. The method is then validated on several synthetic examples as well as on real data. Throughout the illustration, the proposed method is compared with both UKF and offline volatility estimation. Copyright © 2008 Elisabeth Lahalle et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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INTRODUCTION
In 1973 Black, Scholes and Merton [1, 2] reasoned that under certain idealized market assumptions the prices of stocks and the derivatives on these stocks are coupled. One of the crucial assumptions is that the traded asset price S follows dSt = μSt dt + σSt dBt ,
(1)
where Bt is a Brownian motion. μ and σ are called, respectively, drift and volatility of the stock; both are deterministic constants. Nevertheless, it turns out that the assumption of constant volatility does not hold in practice. Traders in the market are supposed to assess returns which have different horizon times in order to predict volatility. Researchers in empirical finance have, therefore, developed an increasing interest in the possibility of uncovering the complex volatility dynamics that exist both within and across different financial markets. Even to the most casual observer of markets, it should be clear that volatility is a random variable. Stochastic volatility models provide a framework for such modeling, especially when dealing with high frequency data. Shephard and Andersen trace the origins of the subject in [3] and attributes it to five sets of people. Back in 1995, the ARCH/GARCH models were a hot topic in econometrics research, and their discoverer, Robert Engle, published a collection of papers on the topic.
Now, ten years later, the ARCH/GARCH models are still widely used but their limitations are motivating research into alternative models, specifically, stochastic volatility models (usually abbreviated as SV models). In modern finance, stochastic volatility models represent the latest research which tries to understand financial volatility in continuous time. The resulting process is the nonnegative spot vol
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