# Understanding Implied Volatility

**Nick Pritzakis**

In order to price an option you need to know the risk-free interest rate, the cost of carry, time to expiration, the options strike price, the price of the underlying security and the implied volatility.

The Black-Scholes option pricing formula and its various extensions assume that volatility is constant. In addition, it assumes that stock price returns are normally distributed. Simply stated, if you plotted stocks returns on a graph, it would look like a bell-shaped curve.

In reality, volatility is not constant nor do asset returns follow a normal distribution. So how do we calculate implied volatility? Implied volatility is derived from the option prices for a given expiration. Option pricing is derived by the forces of supply and demand for each strike price and expiration by market participants. In order for the theoretical option price to match the market price, one must plug in the implied volatility into their options pricing model.

Remember, models are created to simplify the real-world. Let's see how the model assumptions differ from reality. Below is a distribution chart of Tesla Motors from 10/03/12 to 10/04/13 (daily periodic returns)

Tesla Motors Inc (NASDAQ:TSLA) distribution: Actual vs Normal

The chart compares the actual distribution (blue bars) to the normal distribution (red bell-shaped curve). As you can see, we have more occurrences centered towards the mean. Not only that, but we have more occurrences of extreme price moves (on both sides, up or down) or "fat tails". The assumption that asset returns follow a normal distribution is clearly false.

Implied volatility is derived from supply and demand of each option strike by the participants in the market place. Some people say it's the market's assessment of future volatility, however, this isn't always true and we will explain why later on.

The model assumes a constant volatility amongst each strike price. However, implied volatility is supply and demand driven. The graphing of the implied volatility amongst the various strike prices is called the volatility skew or volatility smile. Below is a volatility skew chart of IBM.

Source: http://www.optionvue.com

As you can see, the implied volatility increases as we go farther out of the money on the put side and decreases as we go farther out of the money on the call side. How can this be explained? If a large institution is long the stock, they would look to buy cheap protection (out of the money puts) to hedge against an adverse price move. If a market maker is accumulating an inventory of these short put options, it would make sense for them to increase the price of the options in order for them to be compensated for their risk and discourage traders from buying more by making the options relatively expensive. The buying pressure causes the prices to rise, hence the implied volatility increases.

Here's another example, let's assume a trader is long the stock and it starts dropping dramatically. They will look to buy puts for protection. They might not have a view of future volatility nor will they care if the options are "cheap" or "expensive", they know that their position is hurting and they need to stop the bleeding. In this case, the demand for put options is not necessarily coming from the trader's view of future volatility.

Another popular strategy amongst institutions is to sell out of the money call options against their long stock position. This allows them to collect some premium against their position. By selling call options, the market maker is collecting an inventory of long call options. To discourage traders from selling more of these options, they will decrease the value of the calls; hence the implied volatility decreases on the call side.

By combining the two strategies, buying puts and selling calls against a long stock position it becomes a "collar". The actions of this strategy helps explain why the implied volatilities will vary amongst out of the money calls and puts, as well as various strike prices.

For the most part, stocks tend to have a "negative skew"; out of the money puts having a greater implied volatility than the equidistant out of the money calls. In addition, implied volatility tends to rise more often when stock prices decline but implied volatility can rise if a stock price rises, especially if it is a high beta or speculative stock.

During periods of complacency, demand for put options decrease and hence volatility tends to drop. During periods of market unrest and panic, demand for put options increases and hence volatility increases. Also, during periods of uncertainty, we'll see implied volatility elevated. You'll often see this prior to a company's earnings announcement. The market anticipates a larger than normal price move in the stock. After the new information (such as earnings releases) is absorbed by the market, the implied volatility will instantly drop. It's no surprise if you look at the average implied amongst the various contract months, you'll see the highest implied volatility in the contract that includes the future earnings announcement.

Supply and demand factors along with the reality that asset price returns don't follow a normal distribution are the main reasons why implied volatility varies amongst calls, puts, strike prices and contract months.

Now, that you understand why there is a volatility smile, let's look at how implied volatility influences our trading.

Whenever you buy an outright call or an outright put option, you're long volatility. Not only are you making a bet on the future direction of the underlying asset, you are also making a bet on whether implied volatility will rise, fall or stay flat.

For someone new to options, this could be a dangerous thing. For example, before a company earnings announcement is released, implied volatility increases because of the uncertainty of the outcome. For example, let's assume STO stock is trading at $100, 30 days till expiration and the trader buys the at the money $100 call at a 50% implied volatility for $5.75 before their earnings announcement. STO announces a positive quarter and the next day the stock moves from $100 to $102. However, the implied volatility drops from 50% to 40% and hence the $100 call option is priced at $5.60. Being right on the directional move and being wrong on the volatility assumption can turn a good idea into a losing trade. Now, if you are going to be trading outright calls or outright puts it's critical that you understand what volatility levels are trading.

Of course, there are strategies that can be implemented to reduce the role of volatility. For example, if the trader bought the $100 call and sold the $105 call they would have actually been profitable. By selling the $105 call, they reduced the role of volatility by being short volatility in the $105 call strike; it offset some of the value lost from being long volatility in the $100 strike.

Favourable strategies when implied volatility is relatively expensive are butterflies, condors, iron butterflies and iron condors, call spreads and put spreads.

Unfavourable strategies when volatility is relatively expensive are buying straddles, buying outright calls and buying outright puts.

Favourable strategies when volatility is relatively cheap are buying straddles, buying outright calls, buying outright puts and ratio-back spreads.

Let's go back to our example of STO stock, assuming the stock price is at $100, there are 30 days till expiration and the trader is long the $100 call strike.

At a 60% implied volatility the call option is priced around $6.85

At a 30% implied volatility the call option is priced around $3.45

At a 15% implied volatility the call option is priced around $1.75

As you can see, the volatility levels have a huge influence on the price of an option.

Implied volatility and time decay tend to have an inverse relationship. The higher the volatility, the more time decay in the options and vice versa. With that said, implied volatility plays a larger influence on medium to long term options. This is because they have more sensitivity to implied volatility.

So what does implied volatility really imply? Firstly, implied volatility is expressed in annualized terms. However, most traders are aware of how a stock behaves in shorter time frames. For the most part, there are 252 trading days in the year. To get annualized volatility into a shorter time frame you must square root the number of trading days.

For example, the square root of 252 is 15.87. If STO stock is trading at a 39% volatility and the stock is trading at $100. We would take .39 and divided it by 15.87. This gives us a number of .0245, to get it in percentage terms we simply multiply by 100 and get 2.45%

That +/- 2.45% is representative of one standard deviation or 68.2%

Source: http://en.wikipedia.org/wiki/Standard_deviation

Also, if you were long the at the money $100 call and at the money $100 put (straddle) we would approximately need the stock to move +/- $2.45 to break even.

Now, to get our annualized volatility into weekly terms we would simply do the same calculation. However, we would use the square root of 52 (the number of weeks in a year)

.39/7.2= .0542 or 5.42%

If we had the $100 straddle, with 7 days till expiration we need the stock to approximately move +/- 5.42% for us to break even.

Lastly, to get our annualized volatility into monthly terms we would simply do the same calculation. However, we would use the square root of 12 (the number of months in a year)

.39/3.5= .1114 or 11.14%

If we had the $100 straddle, with 30 days till expiration we would need the stock to move +/- 11.14% for us to break even. Using these calculations is a useful way to figure out how the options market is pricing volatility.

In conclusion, implied volatility is derived from supply and demand forces. The shape of the volatility smile gives us an indication of how the large players are positioned in the stock. Whenever you buy an option, you are long volatility and time decay is working against you. Your strategy selection should include a view on volatility. If you don't have a view on volatility, implement a strategy that offsets the role of volatility. By observing the price of the at the money straddle you are able to get an idea of how the market is pricing volatility.

Another way to gauge if volatility is expensive or cheap is to compare the market's realized volatility or historical volatility to its implied volatility. Stay tuned for future articles as we dig deeper into this subject matter.

**Nick Pritzakis is a proprietary trader based out of New York,**

specializing in option strategies. You can follow him on twitter @questoptions

specializing in option strategies. You can follow him on twitter @questoptions

**Further Reading:**

http://www.utdallas.edu/~tday/6310FAQ3.html

http://www.macroption.com/implied-volatility-excel/