This paper uses a disaggregated approach to study the volatility of common stocks at the market, industry, and firm levels. Over the period 1962–1997 there has been a noticeable increase in firm-level volatility relative to market volatility. Accordingly, correlations among individual stocks and the explanatory power of the market model for a typical stock have declined, whereas the number of stocks needed to achieve a given level of diversification has increased. All the volatility measures move together countercyclically and help to predict GDP growth. Market volatility tends to lead the other volatility series. Factors that may be responsible for these findings are suggested.

We undertake a comprehensive investigation of price and volume co-movement using daily New York Stock Exchange data from 1928 to 1987. We adjust the data to take into account well-known calendar effects and long-run trends. To describt tbe process, we use a seminonparametric estimate of the joint density of current price change and volume conditional on past price changes and volume. Four empirical regularities are found: 1) positive correlation between conditional volatility and volume, 2) large price movements are followed by high volume, 3) conditioning on lagged volume substantially attenuates the "leverage " effect, and 4) after conditioning on lagged volume, there is a positive risk/return relation.

It is widely known that conditional covariances of asset returns change over time.

We study optimal investment strategies given investor access not only to bond and stock markets but also to the derivatives market. The problem is solved in closed form. Derivatives extend the risk and return tradeoffs associated with stochastic volatility and price jumps. As a means of exposure to volatility risk, derivatives enable non-myopic investors to exploit the time-varying opportunity set; and as a means of exposure to jump risk, they enable investors to disentangle the simultaneous exposure to diffusive and jump risks in the stock market. Calibrating to the S&P 500 index and options markets, we find sizable portfolio improvement from derivatives investing.

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The paper studies inference for volatility type objects and its implications for the hedging of options. It considers the nonparametric estimation of volatilities and instantaneous covariations between diffusion type processes. This is then linked to options trading, where we show that our estimates can be used to trade options without reference to the specific model. The new options “delta ” becomes an additive modification of the (implied volatility) Black-Scholes delta. The modification, in our example, is both substantial and statistically significant. In the inference problem, explicit expressions are found for asymptotic error distributions, and it is explained why one does not in this case encounter a bias-variance tradeoff, but rather a variance-variance tradeoff. Observation times can be irregular. Some key words and phrases: volatility estimation, statistical uncertainty, small interval asymptotics, mixing convergence, option hedging

Stochastic volatility (SV) is the main concept used in the fields of financial economics and mathematical finance to deal with the endemic time-varying volatility and codependence found in financial markets. Such dependence has been known for a long time, early comments include Mandelbrot (1963) and Officer (1973). It was also clear to the founding fathers of modern continuous time finance that homogeneity was an unrealistic if convenient simplification, e.g. Black and Scholes (1972, p. 416) wrote “... there is evidence of non-stationarity in the variance. More work must be done to predict variances using the information available. ” Heterogeneity has deep implications for the theory and practice of financial economics and econometrics. In particular, asset pricing theory is dominated by the idea that higher rewards may be expected when we face higher risks, but these risks change through time in complicated ways. Some of the changes in the level of risk can be modelled stochastically, where the level of volatility and degree of codependence between assets is allowed to change over time. Such models allow us to explain, for example, empirically observed departures from Black-Scholes-Merton prices for options and understand why we should expect to see occasional dramatic moves in financial markets. The outline of this article is as follows. In section 2 I will trace the origins of SV and provide links with the basic models used today in the literature. In section 3 I will briefly discuss some of the innovations in the second generation of SV models. In section 4 I will briefly discuss the literature on conducting inference for SV models. In section 5 I will talk about the use of SV to price options. In section 6 I will consider the connection of SV with realised volatility. A extensive reviews of this literature is given in Shephard (2005). 2 The origin of SV models The origins of SV are messy, I will give five accounts, which attribute the subject to different sets of people.

This paper relates variation in stock market volatility to regime shifts in stock market returns. We apply a Markov switching model to market returns and examine the variation in volatility in different return regimes. We find that stock returns are best characterized by a model containing six regimes with significantly different volatility across the regimes. Volatility is higher when returns are either above or below the normal regime-- the further returns deviate from the normal regime, the higher the volatility. Furthermore, volatility is higher in negative return regimes than in positive return regimes. These observations lead us to conclude that return and volatility are related nonlinearly and that the relationship is asymmetric. Markowitz (1952, p.89) notes that, "The concepts `yield' and `risk' appear frequently in financial 1 writings. Usually if the term `yield' were replaced by ... `expected return,' and `risk' by `variance of return,' little change of apparent meaning wo...

: When volatility feedback is taken into account, there is strong evidence of a positive tradeoff between stock market volatility and expected returns on a market portfolio. In this paper, we ask whether this intertemporal tradeoff between risk and return is responsible for the reported evidence of mean reversion in stock prices. There are two relevant findings. First, price movements not related to the effects of Markov-switching market volatility are largely unpredictable over long horizons. Second, time-varying parameter estimates of the long-horizon predictability of stock returns reject any inherent mean reversion in favour of behaviour implicit in the historical tradeoff between risk and return. JEL classification: G12; G14 Keywords: Volatility Feedback; Mean Reversion; Markov Switching; TimeVarying Parameter 1 1. Introduction More than a decade has passed since Fama and French (1988) and Poterba and Summers (1988) reported that price movements for market portfolios...

We consider the strategic timing of information releases. We develop a dynamic disclosure model in which a firm may privately receive information. Because investors don’t know whether the firm is informed, the firm will not necessarily disclose immediately. We show that bad market news can trigger the immediate release of information by firms. Conversely, good market news can slow the release of information by firms. Thus, our model generates clustering of negative announcements. Surprisingly, this result holds only when firms can preempt the arrival of external information. These results have implications for conditional variance and skewness of stock returns. (JEL D8, G3, M4) One of the most important ingredients to the process of price discovery in financial markets is the flow of new information. The importance of information flow is perhaps most apparent during times of market “crisis, ” when it often seems that bad news is being reported simultaneously from multiple sources. This clustering of news could occur because firms learn more during bad times, or because firms strategically time the release of information. Indeed, it has long been recognized in the literature that corporate news disclosures are controlled by selfinterested