A. Volatility
Generally, the term volatility refers to a statistical measure of the tendency of a market or security to rise or fall sharply within a short period of time. Volatility is typically calculated by using variance or annualized standard deviation of the price or return. Moreover, it is also characterized as a variable that appears in option pricing formulas. In the option pricing formula, it denotes the extent to which the return of the underlying asset will fluctuate between now and the expiration of the option.
Furthermore, the concept of volatility is also described as a measure of the range of an asset price about its mean level over a fixed amount of time (Abken and Nandi, 1996 21). It follows that volatility is linked to the variance of an asset price. If a stock is labeled as volatile then the price will vary greatly over time. Conversely, a less volatile stock will have a price that will deviate relatively little over time. As we see, the more volatile that a stock is, the harder it is to isolate where it could be on a future date. Since volatility is associated with risk, the more volatile that a stock is, the more risky it is. Consequently, the more risky a stock is, the harder it is to say with any certainty what the future price of the stock will be (Chriss, 1997, 95-96). When people invest, they would like to have no risk. The least amount of risk that is involved, the better the investment is. Since almost every investment has some risk, investors have looked for ways to minimize risk.
B. Stock volatility components
Researchers have sought to analyze the relative importance of economy-wide factors, industry-specific factors, and firm-specific factors on a stock’s volatility. This approach borrows from modern asset pricing theory and its emphasis on so-called factor models, or models that assume a firm’s stock return is governed by factors such as the overall market return, the return on a portfolio of firms sampled from the same industry, or even changes in economic factors such as inflation, changes in oil prices, or growth in industrial production. If returns have a factor structure, then the return volatility will depend on the volatilities of those factors. Campbell, et al. (2001) assumes the factors are the overall market return, an industry return, and an idiosyncratic noise term that captures firm-specific information. They document the important empirical fact that while volatility moves considerably over time, there is not a distinct trend upwards or downwards. More interestingly, however, since 1962, there has been a steady decline in stock market volatility attributed to the overall market factor; that is, the common volatility shared across returns on different stocks has diminished over that period. The variation ascribed to firm-specific sources, by contrast, has risen. The implication for investors, then, is that they need to hold more stocks in their portfolios in order to achieve diversification.
Moreover, stock market volatility tends to be persistent; that is, periods of high volatility as well as low volatility tend to last for months. In particular, periods of high volatility tend to occur when stock prices are falling and during recessions. Stock market volatility also is positively related to volatility in economic variables, such as inflation, industrial production, and debt levels in the corporate sector (Schwert 1989).
The persistence in volatility is not surprising: stock market volatility should depend on the overall health of the economy, and real economic variables themselves tend to display persistence. The persistence of stock market return volatility has two interesting implications. First, volatility is a proxy for investment risk. Persistence in volatility implies that the risk and return tradeoff changes in a predictable way over the business cycle. Second, the persistence in volatility can be used to predict future economic variables. For example, Campbell, et al. (2001) shows that stock market volatility helps to predict GDP growth.
C. What causes the stock volatility
The reason for exploring the causes of stock market volatility is to offer reassurance that the stock market will continue on as one of our most useful investment tools. Of all investment types, the stock market historically has proven to be the most profitable but only for those who had the fortitude to stay the course and earn the “risk premium.” As investors, a portion of what we will enjoy as our long-term rate of return is attributable to our reward for the abuse we have experienced over the past two years. Meanwhile, today’s environment offers some great buying opportunities for those who have long-term goals.
Among the most important determinants of the conditional volatility of the stock market are found to be the conditional volatilities of inflation and interest rates which are directly associated with stock market volatility, and the conditional volatilities of industrial production, the current account deficit and the money supply which are indirectly associated with stock market conditional volatility. Among these variables the strongest effect is found to be from the conditional volatility of the money supply to the conditional volatility of the stock market. By contrast, no evidence is found of a statistically significant relationship between the conditional volatility of the foreign exchange market and the conditional volatility of the stock market.
Moreover, there are also two major factors that create volatility today. The first is the process by which all public companies are valued. They are valued by a process called “marking to market,” which means that the entire company is valued based on the price obtained for the most recent share of stock that was sold. A second component of inefficiency seems to be created by programmed trading. This is a tool used by major financial institutions to buy or sell broad-based blocks of many stocks all at once. A flood of these huge orders amounts to aiming a magnifying glass at the marked-to-market mechanism and accelerates the rising or falling price at which the trade ultimately takes place.
D. Unconditional measures of volatility
The unconditional measure of volatility follows the assumption that stock prices adhere to a random disposition on future markets. The random behavior in the jumps of a stock price movement brings to mind the concept of Brownian motion. Robert Brown who described the abnormal movement put on display by a minute particle absorbed in a liquid or gas identifies the Brownian motion. In 1905, Albert Einstein demonstrated mathematically that Brownian motion may possibly be explained by presupposing the immersed particle was constantly being subjected to a barrage by the smaller molecules that surround it. In a succession of documents written since in 1918, the American applied mathematician Norbert Wiener provided a mathematically concise explanation of Brownian motion. Wiener also derived some of the mathematical properties that were being used in Brownian motion (Ross 1999, 35).
Robert Brown’s observations were derived from analyzing the erratic motion of a large particle suspended in a fluid. Initially the large particle has no movement, while the movement of the small particles compromising the fluid is random. On the random paths that the small particles are traveling, one of the small particles is bound to crash into the large particle. Because of this collision, the large particle is moved. The direction and size of this movement is independent and random from every other collision. The resulting erratic movement has become known as Brownian motion. To relate stock price movements to Brownian motion, imagine the large particle as the stock price and the smaller particles as the trades that move the stock price. As we see each of the small particles move the large particle, i.e., the trades move the stock price. Each trade moves the price up or down and each trade is independent from the other trades. Thus stock prices do display behavior similar to Brownian motion (Chriss 1997, 96-97).
However, Brownian motion has a few fundamental problems when it is applied to stock prices. The first problem is that if a stock price were to be governed by Brownian motion, and then it can theoretically become negative (Ross 1999, 36). As discussed earlier, stock prices cannot become negative. Another problem is that stock prices often change in proportion to their size. Brownian particles do not have this property (Chriss 1997, 97).
Moreover, Black and Scholes (1973) applied this principle to the strategy of purchasing some stock and simultaneously selling someone else an option to buy that stock at a fixed price. They showed that the risk in owning the stock could be eliminated by continually revising the ratio of options sold to stock owned. The resulting combination should therefore earn the equivalent of the return offered by buying risk-free bonds. From there Black and Scholes backed out the option price from a parabolic partial-differential equation based on the premise that stock prices exhibit Brownian motion. Black held a bachelor’s in physics and a doctorate in applied mathematics, but he was not a specialist in differential equations and only later learned that the equation he solved could be transformed into the heat-diffusion equation of thermodynamics, for which the solution was already known.
E. Conditional Measure of volatility
Stochastic volatility means that the volatility is not constant but undergoes random changes over time. This results in a fat-tailed unconditional distribution of returns, while the conditional distribution (that is, the distribution of returns at time t conditional on all previous returns) remains normal. The most popular class of stochastic volatility models is GARCH. GARCH stands for Generalized Autoregressive Conditional Heteroskedasticity. The model is autoregressive because it involves regressing the variance on its own past values; heteroskedasticity simply means changing variance. The first version of this type of model, known as ARCH, was introduced by Engle (1982) and the generalized, or GARCH, version, was developed by Bollerslev (1986). GARCH relates the variance of asset returns at time t to the variance of asset returns at time t – 1 and the excess return at time t. Where St is the volatility of asset price at time t, Et is the excess return, that is, the difference between the actual return on the asset at time t and the average return, and O, A, and B are constants. This model is referred to as GARCH because volatility at time t depends only on the return and the volatility at time t – 1 and not on the path they have taken in the past. In this model, B is the persistence parameter, which determines how much carryover effect the previous period’s volatility has on today’s volatility. The three parameters can then be estimated from the historical returns data.
GARCH models have a constant mean, but time-varying variance; and the most general models, regime-switching models, allow for both the mean and the variance of a time series to change over time. Moreover, these models have been used extensively in financial applications for series that display time-varying volatility. For instance, researchers have found that while stock market levels are particularly difficult to forecast, stock market volatility displays persistence and is therefore forecastable to some extent. These models have primarily been used with high-frequency data. Similarly, in the GARCH model, a nonnegativity constraint is imposed since variances can never be negative. The EGARCH model uses a log transformation so that a nonnegativity constraint need not be applied. Pagan and Schwert (1990) compared the forecasting performance of a number of different ARCH, GARCH, EGARCH and other more complex semiparametric and nonparametric models. They found that a simple EGARCH model outperformed these other choices.
References:
Abken, Peter A. and Saikat Nandi. (1996) “Options and Volatility.” Economic Review. Dec. p 21-35.
Black, E, and M. Scholes. 1973. The pricing of options and corporate liabilities. The Journal of Political Economy 81:637-659.
Bollerslev, T. 1986. Generalized Autoregressive Conditional Heteroscedasticity. Journal of Econometrics 31: 307-327.
Campbell, J., M. Lettau, B. Malkiel, and Y. Xu. 2001. “Have Individual Stocks Become More Volatile? An Empirical Exploration of Idiosyncratic Risk.” Journal of Finance 56, pp. 1-43.
Chriss, Neil A. Black-Scholes and Beyond, Option Pricing Models. New York: McGraw-Hill, 1997.
Engle, Robert F. 1982. “Autoregressive Conditional Heteroskedasticity With Estimates of the Variance of UK Inflation.” Econometrica, vol. 50, pp. 987-1008.
Ross, Sheldon M. (1999) An Introduction to Mathematical Finance: Options and Other Topics. New York: Cambridge University Press.
Schwert, W. 1989. “Why Does Stock Market Volatility Change Over Time?” Journal of Finance 44, pp. 1,115-1,153.
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