Demystifying Volatility, Part 2 -- Implied Volatility
Jun 13, 2005
A deeper dig into implied volatility and its applications to stock-based compensation valuation, including an explanation of the "volatility smile."
Background
In Part 1 of Demystifying Volatility, we explored the meaning of volatility in the context of option pricing theory and its importance in the valuation of stock-based compensation. We discussed the necessity for developing a reasonable estimate of volatility for valuation under FAS 123R and showed graphically the impact it has on stock returns. Finally, we surveyed the historical volatility for a recently public company, its peers, and their index.
This article will examine implied volatility and its applications to stock-based compensation valuation. FAS 123R requires the volatility assumption to be the volatility expected over the life of the option. Historical volatility can be a starting point for this assumption, but additional information may be needed to produce a reasonable assumption, in particular for companies with limited historical data or companies that expect future data to differ significantly from historical data. This is where implied volatility comes to the rescue. This article will cover the following topics:
- Definition of implied volatility
- Calculation of implied volatility using Black-Scholes
- Volatility smiles
- FAS 123R discussion of implied volatility
Implied Volatility Defined
In financial mathematics, the implied volatility of a stock option is defined as the theoretical value for volatility embedded in the market price of an exchange-traded option. It is a representation of the market's best estimate of future volatility.
The value is determined from a theoretical option pricing model. For exchange-traded options on shares of stock, the Black-Scholes formula is most commonly used. The Black-Scholes formula returns a theoretical value for European put and call stock options based on the following inputs:
- Current stock price
- Option expiration date
- Option exercise price
- Risk-free rate of return
- Dividend yield for the stock
- Volatility
Implied volatility is determined by using the current option price, often readily available from public sources, and solving for the volatility input. Let's examine the math behind the Black-Scholes formula, and then we'll take a look at a couple of examples using actual market data.
Black-Scholes and Implied Volatility
The data used to calculate implied volatility is the current information available for an exchange-traded option. The risk-free rate is the continuous Treasury Yield for the time to expiration. For a European call option, the option price using the Black-Scholes Formula is:
The Black-Scholes Formula
N(x) is the cumulative probability distribution function for a standardized normal distribution, values of which are commonly found in published tables.
Black-Scholes Inputs
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Continuous annual volatility of the change in stock price |
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TCM rate over the expected life of the option |
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Expected life of the option |
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Stock price at grant date |
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Price option holder will pay to exercise. |
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Continuous annual dividend yield as a % of stock price |
While the formula may look daunting, it can easily be programmed into spreadsheet tools. A solver tool can be used to find the volatility that makes the theoretical option value equal to the market price of the option. Let's look at some examples where we have used market traded options to solve for the implied volatility.
Implied Volatility Examples Using Black-Scholes
The table below illustrates the inputs needed to calculate the implied volatility for a stock using the Black-Scholes formula based on three different options. This stock is assumed to pay a dividend of $0.16 on the sixteenth day of every third month beginning on August 16, 2005.
It is important to consider the expected life of the employee stock option when using implied volatility to develop a volatility assumption. In the table above, you can see that the longer the time until maturity, the more someone is willing to pay for an option. The expected life also has an impact on the implied volatility. Another consideration in relying on implied volatility is factor in the smiling face of volatility.
Volatility Smiles
One needs to exercise some caution when choosing which traded option to use to calculate an implied volatility. The Black-Scholes model assumes that the underlying stock price follows a lognormal distribution. However, in the real world, lower stock prices are more likely than a lognormal distribution would suggest and higher stock prices are less likely. One potential explanation of this deals with leverage. When the value of a company's equity decreases, the company's leverage increases. This causes the volatility of the stock to increase further, making lower stock prices more probable. Similarly, when the value of a company's equity increases, the company's leverage decreases. In this scenario, the volatility of the stock decreases, making further increases in the stock price less likely. Let's call this real world stock distribution the actual distribution.
A volatility smile refers to the relationship between the implied volatility of an option and its strike price, given the current stock price. It is a function of the difference between the actual distribution of the stock price and the lognormal distribution. For stock options, higher strike prices relative to the current stock price correspond to lower implied volatilities. In other words, the further an option is out-of-the-money, the lower the implied volatility. A hypothetical graph of the volatility smile for stocks is below.
Consider an employee stock option that is deep out-of-the-money. It will have a high strike price relative to the current stock price. Let's assume that the stock price distribution follows the actual distribution discussed above. When this distribution is used, the value of the option decreases because there is less of a chance that the stock price will increase enough to be in the money. This relatively low option price leads to an implied volatility that is relatively low, which is exactly what the volatility smile suggests.
Due to the varying implied volatilities across strike prices, options that are at-the-money should be used to calculate implied volatilities that are to be used in stock option valuation. The Securities and Exchange Commission (SEC) recently issued Staff Accounting Bulletin No. 107 in which they also recommended using at-the-money options.
FAS 123R and Implied Volatility
In estimating expected volatility, FAS 123R recommends several factors to consider:
- Historical stock prices
- Historical and current implied volatilities
- Changes in corporate and capital structure
- Limitations on availability of historical data
- Similar entities
- Intervals of price observation (daily, weekly, or monthly)
- Weighting of items for consideration
As mentioned earlier in the article, implied volatility often can be quite useful in developing a volatility assumption. However, there is often uncertainty regarding the level of confidence that can be placed in an implied volatility calculation. The SEC Staff Bulletin added some clarification to help with this issue. It suggested four things to consider when determining the level of confidence to place in an implied volatility calculation:
- If the volume of market activity in the underlying stock and traded options is high, prices for these securities are more likely to reflect a marketplace participant's expected volatility expectations.
- Implied volatility should be derived using synchronized variables. This means that whenever possible, market prices of the traded options and stock should be measured at the same point in time. This point in time should also correspond to the grant date of the employee stock options. If this is not reasonably practicable, the implied volatility should be derived as of a point in time as close to the grant date as possible.
- When valuing an at-the-money employee stock option, at- or near-the-money options should be used to derive implied volatility. If this is not possible, multiple traded options with an average exercise price close to the exercise price of the employee stock option should be used. The varying implied volatilities across strike prices exhibited by the volatility smile lend support to this recommendation.
- The remaining term of the traded option should be as close as possible to the expected or contractual term, as applicable, of the employee stock option. However, the bulletin acknowledges that this is somewhat unreasonable and that other relevant information should be considered when estimating expected volatility in relation to options with a term of less than one year. Closed-form models, such as the Black-Scholes model, use the expected term of the employee stock option rather than the contractual term.
When applicable, the examples shown earlier in this article illustrated these four recommendations.
Conclusion
To summarize, we have:
- Discussed the meaning of implied volatility
- Explored the calculation of implied volatility using the Black-Scholes model and looked at a few examples
- Examined the concept of volatility smiles and its relation to stock option valuation
- Surveyed the role of implied volatility suggested by FAS 123R and considered the relevant advice given by the Staff Accounting Board
In developing an estimate of volatility for FAS 123R, we have seen that both historical and implied volatility play important roles in supporting the volatility assumption.
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JPMorgan Compensation and Benefit Strategies will continue to provide information about stock option valuation and related issues as they emerge. For more information, visit our stock option practice website, www.jpmorgan/am/cbs/stock_options.com, which features substantive resources and up-to-date FAS 123-related news.
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