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__THE
IMPLIED VOLATILITY SMILE__

The implied volatility smile refers to the graph of implied volatility against strike prices for options holding maturity, exercise style and type (put/call) constant. In advanced lessons on options you will learn how to interpret the implied volatility smile.

The objective of this lesson is to learn how to calculate the implied volatility smile and then use it to estimate the price of nearby options.

For example, consider five options that vary only by strike price (i.e., we hold maturity, style, type (put/call), and style (American/European) constant. Denote the different strike prices as: K - 10, K - 5, K, K + 5, K+10 where K is approximately at-the-money. Suppose you have prices for all of these except K - 5, which is what you need to make market on. How can you go about estimating the price of K - 5?

The answer to this requires this and the next lesson. In this lesson, we will calculate and plot implied volatilities against strike prices. This allows us to interpolate the implied volatility for the K - 5. We can then estimate the market price of this option in the next lesson.

More specifically, we will calculate the implied volatility for an IBM call option for four different strike prices. Let this be a call option and suppose the current stock price for IBM is 112. The approximate (closest) at-the-money strike price (K) is 110.

Call options that are nearby to K are out of the money if the strike prices are K+5 and K+10. Similarly, in-the-money call options would include K-5 and K-10.

Our objective is to now calculate te implied volatility for each strike price except K - 5. To do this for the first option you must first repeat the calculations performed in lesson one which has been reproduced below:

**Step 1:** The relevant inputs are:

i. The underlying asset's current price (i.e., IBM's stock price)

ii. The option's strike price (At-the-money implies strike price (approximately) equals IBM's stock price)

iii. The option's maturity date

iv. The option's current price

v. The risk free rate of interest

vi. The dividend yield for the underlying asset

By completing the following steps using the Option Calculator you can get this data as follows:

To get i. leave the underlying ticker symbol as IBM or overwrite with IBM if
this has been changed. Next click on the button ** Get Stock Quote**.
You will observe that the source web site (i.e., CBOE) comes up in the bottom
right hand segment of your screen and the data is extracted automatically for
you.

To get ii. enter the first option's **Strike Price**.

To get iii. look in the box with the months listed, then select the month
in which you would like your option to mature. *To get this
information automatically computed and entered into the calculator for you it
is important that you complete step iv. below. *

To get iv. first check that you have selected either a Put or a Call whichever
is desired (i.e., call gives you the right to buy IBM at the strike price
during the option's life and put gives you the right to sell). Finally,
enter IBM beside the Option Symbol box and then click on the button ** Get Option
Quote.**

The result, is that now either the latest put or call price is automatically
retrieved for you and entered into **Market Option Price.**

To get v. select from the drop down menu (in the bottom input box) the first
URL, Bloomberg's treasury.html and then double click on it. You will
observe the yield curve for US treasuries. Enter the yield for the
closest maturity on this yield curve to the box indicating **Interest Rate **(e.g.,
if the option matures in 1 month, copy the yield for 3 months because it is the
closest duration of time and therefore the closest maturity).

Finally, to get vi. select from the drop down menu Yahoo's finance page.
Then enter IBM in the box titled Get Quotes and change basic to Detailed from
the drop down menu beside this box on Yahoo's site. You will now see
under Yield the **Dividend Yield** for IBM.

This has now completed all input requirements for the IBM option.

**Final Issues**

Check that you have selected American for IBM stock options. Virtually all stock options in the US are of the American style (i.e., can be exercised on any day during the life of the option).

Click the little box beside ** Implied Volatility** in the calculator to indicate that you want this
computed.

**Calculating Implied Volatility**

You can now click on the button calculate to see the implied volatility for your IBM option expressed on an annualized basis. You probably want to copy and paste results of each calculation to a spreadsheet or other application. First, click inside the results section (e.g., value) and then right click the mouse to see the copy menu. Now left click the mouse to copy the results to the window's clip board. You can then paste this to your spreadsheet.

In a subsequent module the calculation of a smile is automated for you but for now you need to learn where these numbers come from.

**Repeating for Remaining Options**

You will not need to enter all inputs again. The two key inputs to change are the strike price and then click on Option price to bring a new Option quote. Finally leave all other settings as is and click on Calculate. Copy and paste these results (and inputs) to your spreadsheet and then continue.

Repeat this for the remaining strike/option prices to reveal what the relationship looks like for your four options.

**Interpolating
Implied Volatility**

Plot your four implied volatilities against strike price using Excel or some other application. Now you can interpolate what you think the implied volatility would be for strike K - 5. In the next lesson you will learn how to apply this implied volatility to estimate the value of the strike K - 5 call option.

OS Financial Trading System, PO Box 11356, Pittsburgh, PA 15238 USA, Phone 1-800-967-9897, Fax 1-412-967-5958, Email fts@ftsweb.com,

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