Do Implied Volatility Measures Provide Accurate Forecasts

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02 Nov 2017

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Introduction

Modern option-pricing theory, beginning with Black and Scholes (1973), claims volatility a significant role in determining the intrinsic value for an option (or for any derivative instrument with option features). In the basic Black-Scholes (BS) option-pricing formula, as one of five parameters of the formula, the volatility of the underlying asset is the only one that cannot be directly observed.

The underlying stock price and the other parameters, including the strike price of the option, time to expiration, interest rate, and dividend yield of the underlying asset, are relatively easy to observe. Given that these values are known, the pricing formula relates the option price to the volatility of the underlying asset. Historical stock price data may be used to estimate the volatility parameter, which then can be plugged into the option pricing formula to derive option values. As an alternative, one may observe the market price of the option and then invert the option-pricing formula to determine the volatility implied by the market price. The market's assessment of the underlying asset's volatility, as reflected in the option price, is known as the implied volatility of the option.

Simply, it is a common way to make a forecast with taking the past volatility into consideration. However, as one of several common methods, BS model is obviously not the most accurate forecast. Specially, due to the fact that numerous variations do appear in historical price data which are normally used in predicting volatility. That may affect the accuracy of predicting. Indeed, volatility forecasting plays an important role in evaluating derivatives trading, but it is more like an art rather than a science, particularly among derivatives traders whose behavior are not easy to forecast.

Implied volatility

The concept

An option pricing equation like the basic BS model calculates the fair value using an function including five parameters which are the price of the underlying asset, the option’s strike price and time to expiration, the riskless interest rate, and the volatility. The only one parameter in the BS model cannot be directly observed is the volatility of the stock price which can be estimated from a history of the stock price. In practice, even though an investor cannot observe volatility since the market does reveal option prices. Therefore market’s volatility input can be obtained by solving the model backwards. This is the implied volatility (IV).

Denoting the market price as CMARKET , model value for a given option as CMODEL and implied volatility asσIV , we can write

CMODEL (σIV )= CMARKET

The option’s implied volatility is computed by putting the observable variables into The CMODEL pricing equation and (effectively) inverting it to solve for σIV . Although the inverse function can generally not be written out explicitly, thanks to the monotonic feature in volatility and with the numerical search methods, an iterative search procedure can be used to find the implied σ.

Implied volatilities are used to monitor the markets’ opinion about the volatility of a particular stock. Whereas historical volatilities are backward looking, implied volatilities are forward looking. Traders often choose to quote the IV of an option instead of its price. Unlike the option price, the IV tends to be steadier. In addition, traders often use the IVs of active traded options to estimate IVs for other option.

The relative studies

Because the implied volatility is widely accepted as the market's forecast of future volatility. Movements in implied volatility have been interpreted as reflecting the market's response to new information about the future volatility of the underlying stock. This interpretation has led several authors to conduct event studies examining the impact of new information on the implied volatility of options.

Patell and Wolfson (1981) examined the properties of the implied volatility of equity options at the time of quarterly earnings announcements. Other authors have found historical volatility to be high near earnings announcements. If this volatility is reflected in option prices, then implied volatility should fall after earnings announcements, Patell and Wolfson verified this prediction. A number of authors have examined the behavior of implied volatility in response to stock splits. Historical volatility tends to be high following stock splits. French and Dubofsky (1986) found a small increase in implied volatility in response to stock splits. This finding is contradicted by the results of Klein and Peterson (1988) and Sheikh (1989), who found that implied volatility does not seem to respond to stock splits. Day and Lewis (1988) found that implied volatility is higher around the expiration dates of stock index futures and stock index options. Bailey (1988) examined the response of implied volatility to the release of (Ml) money supply information.

Gemmill (1992) examined the pattern of implied volatility in British markets immediately prior to the election of 1987. Madura and Tucker (1992) considered the effect of U.S. balance-of-trade deficit announcements on the implied volatility of currency options. Levy and Yoder (1993) investigated the behavior of implied volatility around merger and acquisition announcements, and Barone-Adesi, Brown, and Harlow (1994) used the implied volatility of options on target firms to estimate the probability of a successful takeover.

Jayaraman and Shastri (1993) examined the relationship between implied volatility and announcements of dividend increases.

Prior empirical studies provide mixed evidence on the information content of implied volatility relative to historically-based volatility. In general, early papers document that implied volatility is an inefficient predictor of future volatility. Day and Lewis (1992) examine S&P 100 index options with expiries from 1985 through 1989 and find that historical volatility contains predictive power about future volatility beyond that in implied volatility. Lamoureux and Lastrapes (1993) reach a similar conclusion using options on 10 stocks with expiries from 1982 through 1984. On the basis of a sample of S&P 100 index options from March 1983 through March 1987, Canina and Figlewski (1993) find that historical volatility, instead of implied leading them to conclude that implied volatility has no information content.

But the findings in the papers above are subject to a few problems in their research designs. For example, implied volatility in Day and Lewis (1992) is computed from S&P 100 index options with remaining lives up to 36 trading days, which is related to one-week-ahead future volatility. Lamoureux and Lastrapes (1993) examine the one-day-ahead predictive power of implied volatility based on stock options with maturities up to 129 trading days. Both studies therefore suffer a maturity mismatch problem. In addition, both use overlapping samples, as do Canina and

Figlewski(1993). These papers construct their data on a daily basis, resulting in an extreme degree of overlap in consecutive observations in the time series of historical and future volatility.

Overcoming these problems, more recent papers find evidence that implied volatility embedded in option prices is informationally efficient in forecasting future volatility. Christensen and Prabhala (1998), using monthly non-overlapping data, re-examine the information content of implied volatility of S&P 100 index options. They find that implied volatility outperforms historical volatility in forecasting future volatility. Additionally, they include the historical volatility as the information content in some of their models. They also document that the predictive power of implied volatility improved after the October 1987 stock market crash. Szakmary et al. (2003) find that for a large majority of the 35 futures options markets in the US, implied volatility outperforms historical volatility as a predictor of future volatility in the underlying futures prices over the remaining life of the option. Furthermore, historical volatility is subsumed by implied volatility for most of the 35 markets examined. More recent studies on the implied volatility of S&P 500 index options show that it does have the ability to be informaionally predict the future return on the S&P 500 (Banerjee et al., 2007), and that it has the ability to anticipate the impact of non-continuous price changes (jumps) in the S&P 500 index (Becker et al., 2009).

Evaluating the predictive accuracy of IV

Because all of the parameters that enter the Black-Scholes model and similar valuation equations are observable except for volatility, one can solve for the volatility that would make the market price equal to the model value. Both academics and practitioners regard implied volatility as important information.

Many academics consider IV to be the best forecast of future volatility, because it properly accounts for all publicly available information, including everything that can be gleaned from historical price data. This is because they think of IV as a direct measurement of "the market’s" expectation of future volatility, and the market is informationally efficient. By contrast, traders use an option’s implied volatility as a gauge of how the market is currently pricing it relative to the underlying asset, without worrying too much about whether IV is an accurate forecast of how volatile the underlying will be over the option’s lifetime. The two groups are largely unaware of how differently they think about what information IV contains.

The calculation of IV is seriously affected by a variety of data problems, including the effects of bid-ask spreads in both the option and the underlying, nonsynchronous prices, and transactions costs and other problems that prevent option mispricing induced by imbalances in supply and demand from being arbitraged away. Some data problems can be corrected, or at least partially mitigated by averaging IVs from different options or from multiple transactions in the same option. However, we argue that it is inappropriate to suppress a regular volatility structure, like the "smile," by such averaging. Rather, the existence of a smile pattern implies that the model used to calculate the IVs is not a correct description of how the market is pricing options.

Forecast "rationality" of IV can be tested by regressing realized volatility on implied volatility. For any informationally efficient forecast, the regression constant should be zero and the slope coefficient should be one. The relative information content of two different forecasts, such as IV and historical volatility, can be examined by putting both into an encompassing regression and comparing their coefficients. Such tests have been run for a large number of options markets, using a wide variety of data selection and cleaning techniques. The surprising result from an extensive study of S&P 100 stock index options conducted by Canina and Figlewski [1993] was that not only was IV not a fully rational forecast of future volatility in that market, it appeared to contain no information about it at all. By contrast, historical volatility, while not a rational forecast either, clearly contained more information than IV.

This very negative result led us to reconsider the mechanism by which investors’ volatility expectations are incorporated into option prices, through the trading of arbitrageurs who attempt to exploit option mispricing in the market. If the arbitrage trade is hard to execute, or risky, or entails large transactions costs, this mechanism will be weak. Relatively large pricing errors may be allowed to persist, and implied volatilities computed from market prices can be very different from investors’ true expectations. This suggested a hypothesis, that implied volatilities from different options markets will contain relatively more or less information depending on whether the arbitrage trade in that market is easy or hard.

For the most part, the studies showed that implied volatility contained a statistically significant amount of information about future volatility, and generally more than the historical volatility measures that were examined.

However, in almost no case did IV appear to be a fully rational forecast.

conclusion

This essay has offered a critical evaluation on the widely held belief that implied volatility computed from market options prices is an informationally efficient forecast of the volatility that will actually be experienced by the underlying asset from the present through expiration date. We began by noting that this was actually two separate hypotheses. The first holds that the market price for an option will be its model value based on the market’s expectation of future volatility, so that the implied volatility reveals the market’s true volatility forecast. The second is that investors as a group are rational in evaluating the information available to them, so that the market’s volatility forecast is the correct conditional expected value of the future volatility, given the available information.

The second hypothesis relates to the rationality of expectations formation. We are loath to question investor rationality, both because it is one of the fundamental principles of all of financial theory, and also because of the large body of empirical evidence that supports market efficiency in other markets. But the first hypothesis relates to the performance of the trading mechanism in the options markets. In theory, the reason the implied volatility computed from an option’s market price could reveal the market’s forecast of future volatility is that arbitrageurs stand ready to take positions aggressively based on their (rational) volatility beliefs. Their trading will stabilize the market so that imbalances in supply and demand do not push option prices away from their model values. Otherwise, if the market price for the option is not its model value, the implied volatility will not be the market’s volatility forecast.

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The widely used regression-based tests for forecast rationality and for relative information content in competing forecasts were then discussed. A close look at empirical results from running these regressions on implied volatilities from a large sample of OEX call options, published in Canina and Figlewski [1993], revealed that at least for that market and time period, implied volatility from the options prices appeared to contain no information at all about the future volatility of the underlying stock index. We argued that these highly negative results should be interpreted not as evidence that OEX option traders are irrational, but rather, that the aggressive arbitrage trading needed to hold option market prices close to their theoretical values is especially hard, costly, and risky to do in this market.

At the same time, we described how the volatility-based options arbitrage, that plays such an important role in the theoretical market environment in which the valuation model holds, may be less appealing than other trading strategies in the real world. Actual options market makers may well find such arbitrage trading based on their volatility expectations to be much less profitable than simply setting bids and offers around the current market prices, even when those prices are very different from the theoretical model values.

This interpretation of the CF results led to the hypothesis that the performance of implied volatility in the rationality test regression should vary across markets according to how difficult the arbitrage trade is to execute. Stock index options represent a polar case, where the arbitrage trade is complicated to execute at the outset, and the resulting position is both costly and risky to hedge over time. At the other extreme would be futures options, where the option and the underlying are traded side by side on the same trading floor, with low transactions costs (along with other practical advantages not present in other options markets, such as more favorable margin treatment for hedged positions). A selection of results from the finance literature covering a variety of markets provided some support for this hypothesis. However, only for a single maturity in a single market—nearby at-the-money options on crude oil futures—did the statistical analysis not reject the hypothesis that implied volatility was a fully rational forecast.

Lastly, we addressed the issue of how to obtain the necessary volatility input for an option valuation model when the available choices all seem to be irrational. In particular, we argued strongly against the common practice of running the rationality test regression on implied volatilities and alternative forecasts from models based on historical price data, finding that IV has the highest R2 and receives the largest weight in an encompassing regression, and concluding that IV is therefore the best volatility input for the valuation formula. If the bias in the volatility forecast is not corrected, information content, as indicated by a high R2 in these regression tests, does not translate directly into forecast accuracy and correct model option values. The results from Lamoureux and Lastrapes [1993] illustrated that point clearly. Unfortunately, attempting to correct for bias based on the results of the rationality regression need not be successful, since the bias can vary over time. This is an area where more research is needed.

The rationality test regression shows that implied volatility is biased, as are the volatility forecasts from other prediction methods. This is somewhat disturbing to our belief in efficient markets. Even though we can explain this result without abandoning the assumption that investors evaluate information rationally, we also cannot rule out irrationality without actually observing and testing their expectations.

However, if the evidence had shown the bias in implied volatility to be persistent and easily correctable, it would have been much stronger evidence against investor rationality than what we actually observe. Bias that is time-varying and hard to adjust for is consistent with a financial market in which investors act rationally, but can only learn about new market informational factors gradually over time. Once evidence of a systematic expectations error accumulates, they will try to correct for it in making predictions, so that last period’s mistakes are not perpetuated. But investors can not adjust immediately to each new factor as it arises. This will lead to a continual series of short run biases, but things that persist long enough will become understood and incorporated into future expectations.

Unfortunately, the general failure of implied volatility to pass the rationality test, and the difficulty in correcting IV to turn it into a rational forecast, means the question of how to obtain the best prediction of future volatility from observed implied volatilities remains open.



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