Dec

21

Options 101 # 56 Chapter Five Put-Call Parity & Synthetic Options

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Up to this point, calls and puts appear to be polar opposites. Calls represent the right to buy while puts represent the right to sell. And if you look at option quotes, there doesn’t appear to be any connection between the price of the call and the same-strike put. However, call and put prices are highly dependent and are not arbitrarily chosen. Although it may not appear possible, there is a strong, mathematical relationship that describes the connection between calls and puts known as put-call parity. This is one of the most important concepts for beginning and advanced option traders alike, so we will devote an entire chapter to it. Once you understand put-call parity, you’ll find that calls and puts are really one in the same; it just depends on how they’re paired with the underlying stock. So, strange as it may sound, calls are really puts and puts are really calls if properly paired with the underlying stock. In fact, when options were first introduced in 1973, only call options were traded on 16 stocks. Put options did not appear until 1977. The reason is that puts were created by “put-call brokers” who created puts by properly pairing calls with the underlying stock and put-call parity show them how to do just that.

The word “parity” means “equivalence.” There are many types of parity relationships in the financial world such as purchasing power parity and interest rate parity, for example. Parity relationships simply show the connections between two or more variables. As stated before, put-call parity just shows that calls and puts are “equivalent” when properly paired with the underlying stock. Once you understand put-call parity, you will have increased insights into the mechanics of the options market and how options are priced. As you gain experience, you will also find more efficient ways of trading options by using this simple relationship between calls and puts.

There are many ways to present the put-call parity relationship. But perhaps the easiest way to understand it is by looking at how option orders are filled on the floor of an exchange. Once we get the basic put-call parity relationship, we can alter it to get some intriguing looks at how call and put prices are connected.

Filling an Option Order
Let’s say you want to buy 10 calls to open of the ABC $50 strike at market. ABC stock is also trading at $50. In order to make the calculations easy, we’ll assume there is one year to expiration.

When this buy order is received on the floor, the market maker must become the seller so that the transaction can be completed. This means the market maker must be willing to be short a call. While you may be totally comfortable in speculating by buying 10 calls, the market maker may not be so eager to be on the short side of the transaction. We found in Chapter Three that the call owner’s gains are the call seller’s losses. If you buy a call, you have the potential for unlimited gains, and that means the call seller is exposed to unlimited losses.

If the market maker is to be on the short side of 10 one-year calls, his risk is that the stock price rises. In order to protect himself from this risk, he can purchase 1,000 shares of ABC stock. Now, no matter how high the stock moves, he will always be able to deliver 1,000 shares of stock (represented by the 10 calls) at expiration.

However, because the market maker now owns the stock, there is now a new risk; the stock price may fall. To protect himself from this risk, he can buy 10 $50 puts with one year to expiration.

Our market maker is now long 1,000 shares of stock, long 10 $50 put options and short 10 $50 call options. By using these options, the market maker is now fully hedged (protected) against any stock price movement at expiration. This means he is guaranteed to receive $50 per share ($50,000 total) for his stock no matter what happens one year from now. How is this possible?

Think back to the first chapter when we talked about rights and obligations. The market maker paid $50 for the stock and has the right to sell it for $50 by exercising the put. The market maker will exercise the put if the stock price is below $50 at expiration and receive $50 per share. By selling the call, he has the obligation to sell the stock for $50 at expiration. Of course, the only time someone would exercise the call is if the stock’s price is above $50 at that time. If the stock is above $50 at expiration, the market maker will be assigned (“called away”) on the short call and receive $50 per share. What happens if the stock is exactly $50 at expiration? In this case, both options expire worthless and the market maker can sell the stock in the open market for $50. So no matter what happens to the price of the stock at expiration, the market maker is guaranteed to receive $50 per share in one year. It is kind of ironic that, using these speculative derivatives of puts and calls, we can actually create a risk-free portfolio. This is perfect proof of our statement in the first chapter that options are not necessarily risky; it all depends on how they’re used.

The market maker can now sell you the 10 calls with confidence knowing that all stock price risk has been removed. But what price should he charge you for the 10 calls?

We stated the market maker is guaranteed to receive $50,000 in one year regardless of the stock price. Because this is guaranteed, we can easily figure out what the value of this package is worth today. Let’s assume that the risk-free interest rate is 2%. Chapter Two showed us that the present value of $50,000 in one year is $50,000 / (1.02) = $49,020 today. The market maker should therefore pay $49,020 today for the package containing long stock, long put and short call positions. If he pays $49,020 and receives $50,000 in one year, his return on investment will be 2%, which is exactly the interest rate he should receive for a risk-free investment.

The market maker will spend $50,000 for the 1,000 shares of stock trading at $50. Let’s also assume he pays $3 for the put. Now he will spend an additional $3,000 for the put for a total cash outlay of $53,000. We already figured that the fair price for this package of three assets should be worth $49,020 yet he’s paying $53,000 for it.

The market maker has overpaid by $53,000 – $49,020 = $3,980, so he will need to bring in a credit for this amount. How can the market maker receive a credit of $3,980? Easy, he will fill your order on the 10 $50 calls for roughly $3.98. Doing so, he will receive the necessary credit to make his $53,000 cash outlay equal to $49,020. Of course, the market maker will try to make a little profit and quote a price of $4.00 or $4.10 on the $50 calls (remember, options above $3 must quote in ten-cent increments so $3.98 or $4.05 would not be allowed but $3.98 would be the theoretical price). Why won’t he scalp you and charge a much higher price, say $10? The reason is competition. If the market maker quotes too high of a price, somebody else will step in and offer a lower price. Market makers get very competitive with risk-free packages. The reason is they can, in this example, borrow at 2% and loan it risk-free at a higher percent. The reason their rate of return is higher than 2% is because of the additional few cents they tack on for profit, which is reflected in the bid-ask spread. In effect, they end up buying the put for less than the theoretically fair price and sell the call for a little more than the theoretically fair price, thus paying less for the package than the present value would suggest it is worth. However, if they are able to borrow at 2% and are earning a much larger risk-free return, market makers will want to create more of these packages and put buying pressure on the puts and selling pressure on the calls until the rate of return is much closer to the risk-free rate.

To summarize, the market maker’s initial position looks like this:

Buy 1000 shares at $50 = -$50,000
Buy 10 $50 puts at $3 = -$ 3,000
Sells 10 $50 calls at $3.98 = +$3,980
Equals -$49,020 cash outlay by market maker.

This is guaranteed to grow to a value of $50,000 in one year since $49,020 * 1.02 = $50,000 because of the full hedge provided by the long put and short call.

This three-sided position (long stock + long put + short call) established by the market maker is called a conversion. If he had done the reverse (short stock + short puts + long calls) then it would be called a reverse conversion or reversal. Reversals are one of many “locked” trades. A locked trade simply means that it cannot lose. In this case, the market maker buys the three assets at a discount from the exercise price and receives the full exercise price at expiration.

To be continued…..

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