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Trading options is a complex multidimensional problem.  The option trader must take into account the effects of changes in the underlying asset price, changes in the volatility of the underlying asset price process, and time.

We do not normally think of time as an exposure because time is deterministic.  The problem that arises, however, is the responsiveness of the option to the underlying price; and volatility can interact with time.  

Because an option's price varies with the underlying asset price, volatility, and time to maturity, we can use options to manage or reduce our exposure of a position to these variables.  This is known as 'hedging risk'.

To hedge an existing exposure, you must first identify how the existing position responds to changes in the underlying price or volatility.  For example, does our position become more valuable (less valuable) if the underlying asset price increases (decreases)?

Once the nature of the exposure is identified you can design a hedge by overlaying another instrument (e.g. a long or short option position) onto your position in such a way that it behaves in the opposite manner.  That is, value decreases if underlying asset price increases, and vice versa for the current example.

How do you calculate how many option contracts are required to overlay the original position to hedge the underlying asset price risk?

To answer this question we need to know by how much the option is predicted to change for a 1 unit (e.g. $1) change in the underlying asset price.

Real World Note:  1 option contract for IBM controls 100 shares in IBM.  In the following examples, we use 1 call option in the sense that 1 call option is defined on 1 stock.  This simplifies the interpretation and can easily be extended to 1 real world contract by multiplying by 100.

Suppose the spot price of IBM is $110 and it changes from $110 to $111.  By how much would we expect 1 call option defined on IBM to change?

The answer to this question is given by one of the option hedge parameters referred to as 'delta'.  An option's delta is the predicted change in the option's price given a small change in the underlying asset price.

For example, suppose a call option defined on IBM has a delta equal to 0.70.  The predicted change in the option price, given a $1 change in IBM, equals $0.70.  If we were dealing with an option contract, which is defined on 100 shares of IBM, then the answer would be 100*0.70 = $70.

Clearly, the call option cannot increase by more than $1 per unit because one call option gives its owner the right to buy 1 share.  Hence, the change in the value of this right can never exceed the change in the value of the underlying if option prices are arbitrage free.  

Similarly, the change in the call option's value can never fall below zero because a call option gives its owner the right, but not obligation, to exercise.  So the delta of one call option defined on one stock must always be positive and lie between 0 and 1.

Delta of a Put Option?

A put option has the opposite implications.  An increase in the underlying asset price is undesirable to the holder of a put option.  With this in mind, and then for the same reasons as above, put delta must lie between 0 and -1.


A second important hedge parameter is 'gamma'.  Gamma measures how delta changes in response to a change in the underlying asset price.  

If a trader is hedging a position by making it "delta neutral" (i.e. position delta = 0), then the trader is creating a position value which is not predicted to change in response to a small change in the underlying asset price.  Therefore, it is neutral with respect to the underlying price exposure.

Gamma is important because it provides a trader with information regarding how frequently a delta hedge may have to be adjusted.  That is, if gamma is approximately zero, then delta will remain approximately unchanged in response to a small change in the underlying asset price.  However, if gamma starts exploding (i.e. gets large), then a delta position must be continually changed; and by large amounts.

Other important hedge parameters are: 'vega' (option sensitivity to a small change in the volatility of the underlying process); 'theta'  (option sensitivity to a small change in the time to maturity); and 'rho' (option sensitivity to a small change in interest rates).

Calculating the Hedge Parameters

In this lesson, first repeat the steps for calculating implied volatility in lesson 1.   

Question:  How do I calculate the implied volatility for IBM from an "at-the-money" option using current option prices?

To calculate implied volatility we need the following inputs:

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 (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 option's strike price to approximately equal IBM's current stock price.  This must be in intervals of 5 (i.e. 95, 100, 105, etc.), which is why it will only approximately equal the current stock price.

To get iii;  select the month of your desired maturity.  If in doubt about valid maturities select one month later than the current month.  The calculator will automatically compute the Maturity input for you when you complete step iv.

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.

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 option's time to 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 beside Implied Volatility to indicate that you want this computed.  

Calculating Implied Volatility

You can now click on the button labelled Calculate to see the implied volatility for your IBM option expressed on an annualized basis.  Subsequent lessons will teach you how to apply this information.

Enter Implied Volatility = Volatility

Click on Calculate and the hedge parameters are automatically computed from the implied volatility, given the spot price of the underlying in the marketplace.


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