## Thursday, July 17, 2008

### Simplifying Compound Options

Preet over at Where Does All My Money Go? wrote an interesting post about compound options. It shows that the financial world has become very complex, but I think I can simplify it a little.

A stock option is a side bet on stock performance. We can also think of it as a form of insurance. To get a call option you pay a small amount of money (say \$2) called a premium to buy the right, but not the obligation, to purchase stock at a certain strike price (say \$50) at some time in the future.

If the stock goes above \$50, then you will exercise your option to buy the stock at \$50 and then sell it for the higher price. If the stock stays below \$50, then your option will expire worthless.

Now that we have call options figured out, what happens if we have options to buy other options? These are called compound options. This may seem far-fetched, but our financial world has all sorts of such side bets.

So, we can buy a compound option for, say \$1, which gives us the right to buy a call option for \$2, which in turn gives us the right to buy the stock for \$50. Whew! Are you still with me? Maybe there can even be options on compound options.

However, for the case where the compound option has the same expiry date as the call option, we can show that the following two investments are exactly the same (except possibly for small differences in commissions):

A: A compound option with \$1 premium for the right to pay \$2 for a call option at \$50.

B: A regular call option with \$1 premium struck at \$52.

To show that these investments are the same, let’s look at a few different cases to see what happens. Each case will have a different stock price when the options expire.

Stock Price \$49:

A: The compound option expires worthless. The \$1 premium is lost.

B: The \$52 call option expires worthless. The \$1 premium is lost.

Stock Price \$51:

A: We’re above \$50, but it will cost \$2 to buy the call option. This isn’t worthwhile because we’ll only make \$1 on the stock. The \$1 premium on the compound option is lost.

B: The \$52 call option expires worthless. The \$1 premium is lost.

Stock Price \$52.50:

A: We exercise the compound option and pay \$2 to get the call option at \$50. We then exercise the call option by paying \$50 to get the stock, and sell the stock for \$52.50. We paid \$53 in total and have lost fifty cents.

B: We exercise the call option by paying \$52 to get the stock, and sell the stock for \$52.50. Taking into account the \$1 premium, we have lost fifty cents.

Stock Price \$55:

A: We exercise the compound option and pay \$2 to get the call option at \$50. We then exercise the call option by paying \$50 to get the stock, and sell the stock for \$55. We paid \$53 in total and have made \$2.

B: We exercise the call option by paying \$52 to get the stock, and sell the stock for \$55. Taking into account the \$1 premium, we made \$2.

In every case, the investments gave the same return. So, instead of buying compound options, you can just buy an option at a different strike price. Having said all this, though, I think it’s a bad idea for the average investor to use stock options. They only make sense in specialized circumstances as a form of insurance.