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# Solving the Options Problem, Part 1

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## Proof that a simple formula can hold true every time.

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One of the interesting features about options is that there's a relationship between calls, puts, and the underlying stock. And because of that relationship, some option positions are equivalent -- that means identical profit/loss profiles -- to others.

Why is that important? You'll discover that some option combinations -- called spreads -- are easier, or less costly to trade than others. Even with today's low commissions, why spend more than you must?

The basic equation that describes an underlying and its options is: Owning one call option and selling one put option (with the same strike price and expiration date) is equivalent to owning 100 shares of stock. Thus,

S = C - P; where S = stock; C = call; P = put

If you want a simple proof that the above equation is true, consider a position that's long one call and short one put. When expiration arrives, if the call option is in the money, you exercise the call and own 100 shares. If the put option is in the money, you're assigned an exercise notice and buy 100 shares of stock. In either case, you own stock.

Note: If the stock is at the money when expiration arrives, you're in a quandary. You don't know if the put owner is going to exercise the put. Therefore, you don't know whether to exercise the call. If you want to maintain the long stock position, the simplest way out is to buy the put -- paying \$0.05 or less -- and exercise the call.

Example of Equivalent Positions

There's one equivalent position that you, the options rookie, should know, because these are strategies you're likely to adopt.

Take a look at a covered-call position (long stock and short one call), or S - C.

From the equation above, S - C = -P. In other words, if you own stock and sell one call option (this is covered-call writing), then your position is equivalent to being short one put option with the same strike and expiration. That position is naked short the put. Amazingly, some brokers don't allow all clients to sell naked puts, but they allow all to write covered calls. The world isn't always efficient (you already knew that).

Thus, writing a covered call is equivalent to selling a naked put. This isn't a big deal to anyone who's an experienced option trader. But to a newcomer to the world of options, this can be an eye-opener.

The more you trade options, the more you'll become aware of other equivalent positions. You may even decide to play with the equation for yourself and discover others while you're at it.
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No positions in stocks mentioned.

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