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Option Algebra: A Primer


Learn the basic equations that allow you to create synthetic options positions.

Oh no, more math? Yes, but it's worth it, I promise. And after all, options attract math geeks, so admit it – you're a little curious, aren't you?

The kind of algebra I'm talking about lets us create any option position or underlying position we want out of something else. Believe it or not, there are some times when this is useful.

Here's the basic equation:

Stock + Put = Call

Using proper math geek notation, we could write this as:

S + P = C

Here's what it means. Say that trader Amy owns some stock worth $100. She makes money if its price goes up, and she loses if it goes down. Now suppose that Amy is worried about an upcoming event that might hurt the stock. For insurance, she buys a protective put with a strike price of $98. She pays $ 4.00 for the put.

Now, Amy no longer can lose up to $100 on the stock. Her loss is limited. The put option gives her the right to sell the stock for $98 no matter what. So her maximum loss on the stock is $2.00, from the current $100 price down to the $98 strike. Adding to that $2.00 loss the $4.00 that she paid for the put, Amy's worst possible outcome is a $6.00 reduction in her net worth. Her possible profit is still unlimited in case the event doesn't hurt the stock. Her upside profit is reduced, however, by the $4.00 cost of the insurance. The stock now has to go up in price not just by one cent, but by four dollars and one cent, for her to make a profit on it. So her new break-even price is $104.
Hmm. Unlimited upside, limited downside, and a breakeven price that's higher than the current price. That sounds familiar. Oh yeah, that sounds just like a long call option.

And it is. Imagine another trader named Bert. Instead of a s tock-plus-put combo, Bert buys a call at a strike price of 98, paying $6 for it. What is his P/L situation? Well first, the stock has to go up for him to make money. Since he paid $6 for the call, if he keeps it until expiration, the stock would have to rise above the $98 call strike by that amount. That is, to $104. This is his break-even price, which is the same as Amy's.

If at option expiration the stock is above $104, Bert's call makes money, and his upside is unlimited. Just like Amy's.

Bert's worst case loss would occur if his option were worthless at expiration. That would occur if the stock ended up at the $98 call strike or anywhere below that. His loss would be his cost for the call, which is $6 – exactly the same price and maximum loss as Amy.

In fact, up until option expiration, at any price Bert and Amy's profit or loss would be identical. The profit for the stock plus the put, equals the profit of the long call alone.

In other words, stock plus put equals call: [S + P = C]. By buying a put [+P] , Amy turned her long stock position [S] into a synthetic call [C]. Synthetic in option lingo means "made out of something else."

Things don't just work out this way because I chose the numbers for this made-up example. These are relationships that must hold in order for the option market to work. This is a very simplified description of a principle called put-call parity, about which you can read more here or here.

Calls are not the only thing that can be created like this, or synthesized. We can make any position out of the other ingredients. All we have to do is rearrange the algebra.

If [S + P = C], then all of the following also must be true, and they are:

[-S - P = –C] Short Stock plus Short Put = Synthetic Short Call

[C - P = S] Long Call + Short Put = Synthetic Long Stock

[P - C = -S] Long Put + Short Call = Synthetic Short Stock

[C – S = P] Long Call + Short Stock = Synthetic Long Put

[S – C = -P] Long Stock + Short Call = Synthetic Short Put

How can this information be used? Well, first, although each actual position has the same P&L as its synthetic equivalent, the amount of capital required will be different. In the example of Bert and Amy above, Bert's cost was $6, while Amy's was $104. Same P&L, 17 times the cost.

Secondly, sometimes we can combine a synthetic with something else we already have to create a composite position that we want. One of the most common examples is the collar. There are a couple of different ways to use a collar. One of them is as another means of protecting an existing position.

Assume Amy wants another alternative to protect her $100 stock. She thinks the danger to the stock is short term. After the event, either the stock will be crushed or it will be fine. She would like just to freeze her position until after the event passes. She could create a collar by adding a synthetic short stock to her actual long stock. Then what she makes or loses on the actual long stock will be exactly equal and opposite to what happens to the synthetic. The net effect will be zero.

To accomplish this, Amy checks the table above and sees that [P - C = -S]. Long Put + Short Call = Synthetic Short Stock. She needs a short call and a long put, both at the $100 strike price. The short call is no problem – since she owns the stock, the call is covered. She doesn't need to put up any additional capital to secure the call. In fact, the call will probably bring in nearly the same amount of cash that she needs to buy the put. The collar is a no-cost insurance policy. (There is a tradeoff though – although Amy now can not lose, she can not gain anything either. The meter only starts running again once she removes the collar).

Editor's note: This story by Russ Allen originally appeared on Online Trading Academy.

To read more from Online Trading Academy, see:

The Nerve to Win

Risk Management Basics

Program Trading and the Markets
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