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Estimating Volatility
Volatility is the standard deviation of stock returns, which is a measure of the spread of stock returns relative to a central trend or drift. Expected volatility is based on an estimate or forecast of future volatility. For the purpose of forecasting volatility from the option grant date forward over the term of the option, your company has to make critical choices because of the large impact that volatility has on option value.
Volatility forecasts typically rely on historical volatility and/or implied volatility, which refers to the volatility forecast that is implied by the prices of traded options observed in the market. There is extensive research on volatility modeling with the more sophisticated models falling into three categories: Deterministic local volatility models that assume volatility is a function of the stock price level and time, but with risk factors that are not correlated with the stock. (GARCH models are an example.) Second, continuous mean reverting stochastic volatility models, which assume that volatility is driven by a risk factor, which is likely correlated with the stock. Third, stochastic volatility models where the volatility can jump causing a discontinuity in the stock price.
Because there is no definitive proof that adding more technical sophistication and realism enhances the ability to predict volatility over the long term, FAS123 Solutions balances technical sophistication with common sense evidence. This includes evidence from the options market, fundamental analysis of a company's prospective operations and financials, and industry and market trends in volatility. Our approach is consistent with the statement by FASB in its March 2004 exposure draft stated:
"An entity that uses historical share price volatility as its estimate of expected volatility without considering the extent to which future experience is reasonably expected to differ from historical experience (and the other factors cited in this paragraph) would not comply with the requirements of this Statement."
Using Historical Volatility to Forecast the Future
In simple terms, historical volatility is informative to the extent that the spread of observed stock returns up to the present is predictive of the future spread of stock returns. It is important to consider qualifiers before using historical volatility to forecast future volatility, such as the following:
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If a company has not changed much over time, then a sample with more data is better.
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If a company has changed with respect to capital structure, lines of business, industry effects, etc., then observations from the distant past are less important. Additionally, implied volatility and comparable company data might be informative.
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If a company's recent history resulted in volatility that is unusually high (or low), then implied volatility and longer historical sampling might be necessary to make better forecasts.
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If a company has only recently become public, then its history is inadequate for estimating volatility. Similar companies that are growing from newly public to more mature companies with more stock price history can be informative.
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For nonpublic companies FAS 123(R) recommends looking to estimates of volatility based on historical volatility of an appropriate set of comparable public companies. FASB refers to associated option valuation as the "calculated value method."
Volatility estimation using historical data should carefully consider how far into the past to look, how frequently to sample stock returns, and whether to put equal weight on all past returns. A common — and overused — rule of thumb is to match the historical sample period to the expected term of the option. The qualifiers above suggest that this is not always reliable. FAS 123(R) explicitly requires that companies using numerical models, such as lattices and Monte Carlo simulation, take into account volatility commensurate with the contractual term of the option. This does not suggest that one use a ten-year historical or implied volatility when modeling a ten-year option with a lattice or simulation model. Rather, it means that the model should incorporate the term structure of volatility over the full ten years.
Constant vs. Time-varying Volatility
Constant volatility is usually estimated as an average of the spread of the returns that are observed over time. However, the spread changes, often in unpredictable ways. The price of a stock might go through periods of low volatility and then suddenly experience volatility bursts that dissipate quickly, or that persist for a period of time. Changes in volatility correspond to company specific factors, or uncertainty in financial markets and the economy, generally. The most consistent pattern, for which there is econometric evidence, is the tendency for changes in volatility to dissipate in the long run. This means that volatility tends to revert to a long run average, though it does not usually do so immediately after a burst of volatility. The pattern is called "mean reversion" of volatility over the long run. If volatility is estimated over a year or longer, large changes are found to be less common.
Dividends
In estimating historical volatility, it is also necessary to adjust for discrete dividend payments during the sampling period. Price data is often dividend adjusted, but if it isn't, then dividend payments have to manually be added back to the stock price in the period immediately following the ex-dividend date. This adjustment is pre-tax, but it gives a reasonable approximation for the effect of dividend payments. Another method is to discard data for intervals that include ex-dividend dates.
Using Implied Volatility to Forecast the Future
The volatility that is "implied" by observed prices of exchange-traded call and put options on a company's stock is an important alternative to solely using historical data. If certain conditions are met, then implied volatility ought to impound the relevant information from historical volatility, as well as the market's assessment of current and future circumstances. In other words, implied volatility is forward-looking and ought to include at least as much information as historical volatility. The most relevant observed prices are those of short-lived options and long-term equity anticipation securities ("LEAP", a registered trademark of the Chicago Board Options Exchange). In theory, the prices of long-lived instruments such as warrants and convertible bonds are useful, but might not be practical due to lack of liquidity and other market imperfections.
Implied volatility is inferred from observed exchange-traded option prices by answering the question: What level of volatility, when input into an option valuation model — typically Black-Scholes — would generate the observed market prices? This process of reverse engineering volatility from prices is possible for two reasons: (i) the other parameters that are input into the model (stock and exercise price, option term, risk-free rate and dividend yield) are observable, and (ii) given the other inputs, option market prices are strictly increasing in the volatility parameter. There are three necessary conditions for implied volatility to provide a reliable forecasting tool:
There are three necessary conditions for implied volatility to provide a reliable basis for forecasting volatility in the context of employee stock options:
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There must be a liquid market for traded options, insuring that observed near-the-money prices are current and frequent. Stale prices and large bid-ask spreads can result in significant pricing errors.
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The model that is used to infer volatility from observed prices must be well specified and consistent with the market in which prices are observed. On a topical level, this means, for example, that prices on American put options in a market with significant frictions should not be used to derive implied volatility with the Black-Scholes model.
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The term and strike of the exchange-traded options should approximately correspond to the forecast. In the case of a constant volatility forecast the longest-dated actively traded options provide the most useful information. In the case of forecasting a forward local volatility, as needed for implied numerical methods strike and term are critical.
These conditions are non-trivial. Therefore it is important to use implied volatility estimates properly. Intraday options market activity is not documented as well as stock market activity. The majority of exchange-traded options are short-term relative to employee stock options and thus, implied volatilities typically reflect short-term expectations. Furthermore, for a given term, implied volatility varies with strike price yielding a volatility "smile" or "skew." In the case of a constant volatility forecast, where the volatility smile and volatility term structure are assumed away, there are several averaging considerations. Using implied volatility properly requires expert guidance that FAS123 Solutions stands ready to provide.
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