# Volatility window

For modeling purposes, volatility is probably the single most important parameter because the value of an option is more sensitive to volatility than any other parameter except the current stock price.  Please pay close attention to the information on this page.

## Reminder

An option's fair value (its mathematically optimal price) is determined by six factors:

• S, the current stock price
• K, the option's strike price
• T, the time remaining until the option expires
• R, the risk-free interest rate that can be earned on perfectly secure bonds or cash
• Y, the stock's dividend yield
• V, the stock's "price volatility"

So there's a mathematical function we can compute to determine the value of an option:

fair value = f  (S, K, T, R, Y, V)

Fair value is what the option is really worth, subject to some assumptions we needn't discuss here except than to observe that they are reasonable, based on published research, widely accepted, and are universally applied in finance. The detailed formula for f won a Nobel prize.

Volatility V is a statistical measurement of how much the stock's price varies from day to day. It is used to calculate how much the stock's price might cumulatively "wander" as the days pass. Bigger values of V represent more uncertainty, or an expectation of greater variability over time. A surprising fact about option theory is that although volatility is a sort of statistical average (standard deviation), it can be used to calculate an exact value for an option.

You can see that of these six parameters, only V is uncertain; the others are determinate. But given a good estimate of future stock price volatility, you can get an excellent estimate of what an option is worth today or at any future date.

We can also invert this logic. If the market price of an option is assumed to be its fair value, then we can compute what future volatility this price implies:

V = (S, K, T, R, Y, market price)

This formula means: if we assume the market price is right, then g tells us how volatile investors who are trading this option expect the stock price to be, until the option expires. We can directly infer investors' expectations from option prices. The inferred value of V is called implied volatility.

## Compare options by comparing implied volatilities

In particular, by comparing implied volatilities within an option chain rather than the prices themselves, we can determine whether two options are relatively cheap or expensive compared to each other, and compared to fair value. Comparing implied volatilities has the effect of normalizing everything with respect to the other parameters, so we can directly compare options with different K, T, R and Y.  The stock price S is of course the same, since these are options on the same underlying stock.

Consider two options A and B on the same stock, having different strikes, expiration dates, and market prices.

• When A and  B are both priced at fair value, they have exactly the same implied volatilities even though their other parameters are different.
• If the implied volatility of A is less than that of B, then A is relatively cheap compared to B.
• If the implied volatility of A or B is less than fair value, that option is relatively cheap and might represent a purchase opportunity.
• If implied volatility is greater than fair value, that option is expensive and might be a candidate for selling.
• If A is relatively cheap and B is relatively expensive, that differential might represent a trading opportunity.

## The scatter plot

When you consider an entire option chain, the individual options will have different implied volatilities. That is, the prices at which the options have been trading imply somewhat different expectations about future volatility.

For example, an option expiring in one month might have a lower implied volatility than an option with the same strike price, expiring in nine months. This could be perfectly rational because investors might expect a different situation in nine months than they expect next month. Such differences always exist, and they convey information about what market participants expect the future to hold. Certain option strategies are intended to exploit these differences.

So which number is right? There is no "right" here, only differing expectations revealed by price differences. However, you can examine the overall volatility structure of the chain to get the "big picture".

In this plot, option strike prices are spread along the horizontal axis, and the vertical axis measures implied volatility. Each option in the chain is plotted, unless it was not possible to determine an implied volatility (for instance if the quoted price is ridiculous). The tiny points represent contracts that have not traded today - their volume is zero. The larger red dots represent puts, blue dots are calls.

You can organize the plot with expirations, rather than strikes, on the horizontal axis by selecting the by expirations radio button. You can choose which strikes and expirations you want to see by selecting checkboxes. Here we plotted All options for which an implied volatility could be computed. By clicking the Nearby options radio button in the Environment panel, this plot can immediately be restricted to options striking near the current stock price, that will expire in 3 months or less.

The horizontal solid line marks the current setting of Options Laboratory's implied volatility scroll bar in the Environment panel. It moves up and down as you slide that scroll bar. By default, the program sets the scroll bar to a number representing the average implied volatility of all the "nearby" options. Nearby options are those with a strike price near the current stock price, and expiring within three months. The somewhat U-shaped "smile" shown in the illustration if pretty typical.

You can see that the average volatility for nearby strikes may be biased upward by a few outlying options that traded at high implied volatilities today (relatively expensive options). You might therefore choose to reduce the volatility setting slightly.

You can also see a very visible band structure in the higher strike prices, where the options simply haven't been trading. Market makers generate these putative bid/ask prices using mathematical models like the ones in this program. The band structure is, to some extent, related to differences between your interest rate assumption and that of the market makers. If your risk-free interest rate setting is too high compared to theirs, the band structure will be more widely separated and rise more steeply. This structure also reflects that the puts are far out of the money in this particular example, and that the bid/ask spread is rather wide for these thinly traded options.

Each strike has a "stack" of options with successively higher implied volatilities. This can happen when options that have longer to run, such as LEAPS, are priced higher because there is relatively more uncertainty about how the stock price will behave in the long run. Options that are "cheap" will be found below the horizontal green line representing the average, market implied volatility; that is, their implied volatilities are low. It usually happens that nearby options trade less expensively than far-out ones. However, nearby options also tend to react more sharply to short-term stock price shifts more than option theory would predict; so, after a sudden stock price change in either direction, the nearby options may have relatively higher implied volatilities for a while.