Options Pricing Principles

You’ve just been introduced to real call and put options and now understand how to interpret their prices when looking at quotes.

But did you notice in Table 1 that some options are more expensive than others? Why is that? And is there a pattern we should understand? The answer is yes, there are many important (and interesting) principles about option pricing that are essential for investors to understand if you are to succeed with options.

The Alpha Trader Certificate Course covers six of the most important option pricing principles you’ll need to understand in order to master options trading. But the goal of this free course is to gain a strong appreciation for what options can do. In order to do that, you’ll need to understand three of these pricing principles, which are labeled Principles 1, 3, and 6 from the Certificate Course.

The motivation for learning these concepts is that they allow you to fully understand and appreciate the ideas behind certain strategies and hedging techniques we’ll be talking about at the end of this course without having to resort to sheer memorization.

Principle #1: Lower Strike Calls (and Higher Strike Puts) are More Expensive

If you look at the prices in Table 1-1, you’ll notice that the lower strike calls are more expensive than the higher strikes. This will always be true assuming, of course, that all other factors are the same. That is, we must be looking at strikes on the same underlying stock and expiration month.

For example, Table 4 below shows the July call prices from Table 1. Why do the prices get cheaper as we move to higher strikes?

Table 4











There are many mathematical reasons why this relationship must hold. However, you already know enough to figure it out intuitively by thinking back to the pizza coupon analogy. Imagine that you walked in to buy a pizza and found the following two coupons lying on the counter:

Notice that both coupons control exactly the same thing (one large three-topping pizza) and have the same expiration date. The only difference is that the coupon on the left allows you to buy the pizza for $10 while the one on the right gives you the right to buy it for $20. Which would you choose and why? Think about it for a moment and then continue reading.

If both pizza coupons allow you to purchase exactly the same thing but one just allows you to get it for a cheaper price, then obviously you would choose to pay the cheaper price. You should pick up the coupon that gives you the right to buy the pizza for $10.

The same thought process occurs in the options markets. For example, both the $32.50 call and $35 call in Table 4 allow the trader to buy 100 shares of eBay so there are absolutely no differences in what those two coupons allow you to buy. However, the $32.50 allows you to buy the 100 shares for less money.

Traders realize the benefit in paying $32.50 rather than $35 so will compete in the market for that coupon. It is a more desirable coupon so its price will be driven higher than the $35 coupon. The same process happens all the way up the line. Each successively lower strike is bid higher than the strike above it. Or conversely, each higher strike is bid lower than the strike below it. In other words, lower strike calls must be priced higher than higher strikes, which is exactly what this first pricing principal states.

When you get into strategies, there will be times when you need to figure out which call option is more valuable. You can always find the answer by asking yourself which is more desirable. The answer to that question is the one that has the lower strike price.

The lower strike call is therefore more desirable to investors and that means they will bid higher to own it. When first attempting to understand option prices, you must remember that “more desirable” means more money with all other factors the same. If you do, you’ll understand many aspects of strategies that many traders must memorize.

Now let’s take a look at why higher strike puts are more expensive. Table 5 is a listing of the July put options from Table 1:

Table 5

With the put options, the reverse appears to be true with the higher strike puts being more expensive. Why is this? Think back to the thought process that made call options more expensive and the same is true for puts. The strikes that are more desirable will be more valuable. Because put options allow you to sell shares of stock for a fixed price, the higher that fixed price (strike price) the more valuable the put option.

If it is more desirable to sell your shares for $40 than for $37.50 then traders should bid the prices of the $40 puts higher than that of the $37.50 puts and the $37.50 puts higher than the $35 puts and so on. And that’s exactly what we see happening. Higher strike puts will always be more expensive than lower strike puts with all other factors the same (same underlying stock and expiration), which is expressed by this first pricing principle.

At the beginning of this course, we compared put options to auto insurance. If you have a $30,000 car and want to insure it for the full value, you will pay a certain premium. However, if you accept a $500 deductible you will pay a lower premium. If you accept a $1,000 deductible, you will pay even less. In other words, in exchange for assuming some of the risk, you will pay a lower premium.

This same relationship is another way of understanding this pricing principal for puts. In Table 5, if a trader owns 100 shares of eBay and buys the July $37.50 put, he is attempting to insure the stock for more than its current value of $37.11. For that coverage he will pay $1.40 premium. However, if he chooses to assume some of the rise, he can pay a lower premium. How can he assume some risk? He can choose lower coverage by selecting a lower strike price.

For instance, if he chooses the July $35 put, he will pay only 50 cents. But in exchange for that lower premium, he is assuming the first $2.11 in damage since the protection on his stock does not start until a stock price of $35. The higher the coverage you elect on an insurance policy, the higher the premium. Therefore, high strike puts cost more than low strike puts.

Arbitrage

We’ve stated that lower strike calls and higher strike puts must always be the more expensive strikes. That’s a pretty bold statement to make. Will these relationships always hold? The answer is yes and the reason is due to a process called arbitrage.

Arbitrage is a type of trading carried out by traders called arbitrageurs. Arbitrageurs are those who look for "free money" lying around in the market. When we say free money, we really mean a guaranteed profit for no cash outlay. Many people incorrectly believe that an arbitrage profit is simply a guaranteed profit, but that is only half correct. If this second condition of "no cash outlay" were not met, the purchase of a government bond would qualify as arbitrage because a profit is guaranteed. However, the government bond requires a cash outlay for a specific period of time before the profit is realized. So while a government bond is guaranteed, the money is far from free.

The classic example of arbitrage is where a stock trader finds shares of IBM asking $100 on the American Exchange and bidding $100.10 on the New York. The arbitrageur could buy them on the American and immediately sell them on the New York for an immediate 10-cent profit.

These actions put buying pressure on the American Exchange and selling pressure on the New York and will continue until IBM is priced the same on both exchanges. While this may not sound like a big profit, bear in mind that large institutions often carry out arbitrage in large block trades and the profits quickly add up.

While arbitrage may sound like the ideal way to make money, it is a lightning fast, high stakes game that is usually reserved for the large institutions. So while it’s not illegal for retail investors to participate in arbitrage, the speed is so fast that you will likely never be able to participate.

The important point to understand about arbitrage is that it is carried out every day and that’s what keeps option prices from getting out of line from where they should be trading in relation to their underlying assets or in relation to other strikes.

Let’s take a look at how arbitrageurs will ensure that lower strike calls and higher strike puts will always be more expensive.

Assume for a moment that the $32.50 call in Table 4 is $4.90 but that the $35 call is, instead, priced at $5. In other words, the $35 call is priced higher than the $32.50 call, which is something we said cannot be possible in the real markets. This is the perfect setup for an arbitrage opportunity since the more valuable call ($32.50) is cheaper than the less valuable one ($35).

In order to exploit a mispricing, arbitrageurs generally buy the underpriced option and simultaneously sell the higher priced option. It is not enough to simply buy the underpriced option (or just sell the overpriced one). In this example, the $32.50 call is a great deal relative to the $35 call; however, just buying the $32.50 call does not guarantee a profit because that option could still lose if the stock’s price falls below $32.50 at expiration.

In order to capitalize on the mispricing, arbitrageurs would buy the $32.50 call and spend $4.90. Then they would immediately sell the $35 call and receive $5 for a net credit of 10 cents to their account:

Buy $32.50 call = - $4.90
Sell $35 call =      $5.00
Net credit =         10 cents

Remember, a net credit of 10 cents may not seem like a lot of money but arbitrageurs do things on a very big scale. They may send hundreds of thousands or even millions of dollars worth of trades to take advantage of such a discrepancy.

The sale of the $35 call more than pays for the $32.50 call so the arbitrageur has zero money invested. In other words, the sale of the $35 call more than financed his purchase of the $32.50 call. To understand why the arbitrage works, think about the arbitrageur’s rights and obligations.

The arbitrageur bought the $32.50 call and sold the $35 call. This means he has the right to buy stock for $32.50 and may have the obligation to sell it for $35, which means he could potentially make an additional $2.50 profit. But even if that doesn’t happen, he still keeps the 10-cent credit. (Remember, when you sell an option, the money you take in from the sale is yours to keep no matter what happens to the stock or option.)

The arbitrageur has therefore capitalized on a trade that resulted in a guaranteed profit for no out-of-pocket expense and that’s the definition of arbitrage. Arbitrageurs will continue to execute the above trades – buy the $32.50 call and sell the $35 call – as long as the opportunity for free money is there. Unfortunately for the arbitrageurs, their actions guarantee that the opportunity will eventually disappear.

As they buy the $32.50 calls they put upward pressure on its price. As they sell the $35 calls they put downward pressure on its price. Eventually the $32.50 calls will be more expensive than the $35 calls and that’s when the opportunity disappears. The free money game is over.

It is the arbitrageurs who guarantee that lower strike calls will always be more expensive than higher strike calls (and that higher strike puts will be more expensive than lower strike puts).

There are many who feel that arbitrage is “unfair” because there’s something that doesn’t seem right about being able to make free money from the market. But arbitrageurs provide an important economic function in that they make sure the relative prices stay fair for the rest of us. You don’t need to understand the process of the arbitrage to trade options. However, you do need to understand that lower strike calls and higher strike puts will always be more expensive due to the process of arbitrage.

Exercise Go to the option quotes page at cboe.com and check out option quotes on several stocks. Are lower strike calls always more expensive than higher strikes? Are higher strike puts always more expensive than lower strikes? What about for different expiration months? Explain in your own words why this occurs.

Principle #3: At Expiration, All Options Must be Worth either Zero or their Intrinsic Value.

At the end of the first section, we said that any intrinsic value must remain with an option at expiration. This means that if an option is in-the-money at expiration that the price must be the difference between the stock price (S) and the exercise price (E), or S – E. For example, if the stock closes at $53 at expiration, the $50 call must be worth exactly $3 since there is $3 worth of intrinsic value and no time value left.

Because an option is either in-the-money or out-of-the-money at expiration, this means that an option can only be worth one of two values at expiration: It is either worth the intrinsic value (intrinsic value zero time value) or it is worth nothing (zero intrinsic value zero time value).
Using our previous example, if the stock is $53, then how can we be sure the $50 call must be worth the $3 intrinsic value at expiration? Once again, the answer is arbitrage.

In order to understand the idea of the arbitrage, think back to the pizza coupons. Imagine that pizza coupons do have value and are traded in the streets (the marketplace). Now assume that pizzas are $15, which means our $10 coupon has $5 of immediate value, which is the same thing as saying it has $5 worth of intrinsic value. However, let’s assume they are trading for only $4. Can anything be done to capitalize on the missing $1 intrinsic value?

The answer is yes. The way the market corrects for this missing value is that enterprising individuals (arbitrageurs) would buy the pizza coupon for $4 and then take it to the store and buy the pizza for $10. They have spent a total of $14 to get the pizza ($4 for the coupon $10 for the pizza). Then they'd walk out in the street and sell the pizza for $15, thus making a $1 guaranteed profit. This $1 profit is exactly the amount of the missing intrinsic value.

As individuals figure this out, they will compete in the market for these coupons thus raising their price. At what point will the competition for coupons stop? When the price of the coupon reaches at $5 (or more), which means that the full intrinsic value is now reflected in the price of the coupon.

At expiration, all in-the-money options must trade for their intrinsic value otherwise a similar set of transactions would take place in the market by arbitrageurs. For instance, assume that the stock is $53 and the $50 call is trading for $2 in the final minutes of trading, which means there is $1 missing from the intrinsic value. Arbitrageurs would short the stock and buy the call for a net credit of $51 to their account:

Short stock =   $53
Buy $50 call =  -$2
Net credit =      $51

Notice that the sale of stock generates a $53 credit which more than covers the cost of the $50 call. In other words, there is no cash outlay required on the part of the arbitrageur.

However, because they’ve shorted the stock, they have an obligation to buy it back and can do so by exercising the call and paying $50 out of the $51 net credit they received. This leaves them with a guaranteedminimum profit of $1 for no out-of-pocket expense, which is exactly the amount of missing intrinsic value.

Of course, if the stock price falls below $50, the arbitrageur would just let the call expire worthless and buy the stock in the open market to close out the short stock position. This would result in a profit greater than one dollar. So whether the stock price rises or falls, the arbitrageur is guaranteed a minimum profit of one dollar.

As with all arbitrages, the arbitrageurs’ actions restore the proper pricing relationship. In this example, the above transactions (shorting the stock, buying the call) put selling pressure on the stock and buying pressure on the call and will continue until the intrinsic value is restored. When the full intrinsic value is returned, the arbitrage is over.

Expiration Values for Put Options

Just like call options, put options get one of two values at expiration. At expiration, put options must be worth either zero or their intrinsic value, which is found by taking the exercise price (E) minus the stock price (S), or E - S.

For example, assume the stock is $53. The $60 put must be trading for $60 - $53 = $7 at expiration since that is the amount of intrinsic value. However, if the stock is above $60 at expiration, the put will expire worthless since there is no reason to exercise a put and collect $60 when you can just sell the stock in the open market for more money.

If a put option is in-the-money at expiration (stock price is below the strike price) and not trading for the intrinsic value then arbitrage is possible. Assume the stock is $53 but that the $60 put is trading for only $5 thus missing $2 worth of intrinsic value. Arbitrageurs would buy the stock and buy the put for a net cash outlay of $58:

Buy stock =           -$53
Buy $60 put =         -$5
Net debit =            -$58

The arbitrageur would then immediately exercise the put and receive the $60 strike price thus making an immediate, guaranteed minimum profit of $2 for no cash outlay, which is exactly the amount of missing intrinsic value. The missing intrinsic value can only be restored if the stock price increases to $55 or if the put price increases to $7 or some combination of the two. Notice that the above transactions (buying stock, buying puts) will place buying pressure on the stock and the $60 put and thus increase their prices. These are the forces necessary to restore intrinsic value.

The important point to understand is that, at expiration, all calls and puts are either worth their intrinsic value or they are worthless. There are no in-between values. It is not a matter of courtesy or tradition by the market makers; it is forced through the process of arbitrage.

Principle #6: For any two call options (or any two puts) on the same stock with the same expiration, the difference in their prices cannot exceed the difference in their strikes.

This relationship says that for any two call options, the difference in their prices cannot be greater than the difference in their strikes. This is assuming that both options have the same underlying stock and expiration date. Assume that one day we see the following two option quotes on the same underlying stock:

$50 Call = $10
$55 Call = $4

We know from the first pricing relationship that the $50 call should be worth more than the $55, which it is. However, Principle #6 says there cannot be this much of a difference, because the difference in strikes is $5 yet the difference in price is $6 as shown by the following diagram:



The difference in prices has exceeded the difference in strikes, which is a violation of this principle. Before we show you how the markets would correct for this, let’s understand intuitively why this cannot happen.

Think back to the pizza coupon examples. Assume two coupons are identical except that one allows you to pay $10 while another let’s you pay $15. Now assume the market places a $1 value on the $15 coupon. What’s the maximum value of the $10 coupon?

We know the $10 coupon must be worth more than the $15 coupon so it must be worth more than one dollar. How much more? The $10 coupon gives you a $5 advantage over the $15 coupon so that is the maximum additional value it could ever have when compared to the $10 coupon. If the $15 coupon is worth $1, the maximum the $10 coupon could be worth is $6.

To understand why, assume pizzas are selling for $20. The holder of the $15 coupon has a $5 advantage while the $10 coupon holder has a $10 advantage. The difference in these two advantages is $5. Try any price for the pizza and you will see that there is always an exact $5 advantage.

Therefore, it just wouldn’t make sense to bid the $10 coupon more than $5 above the price of the $15 coupon. Now, it is certainly possible that the market places less than a $5 difference between these two coupons. That would happen if the market didn’t see any advantage in holding either one (the coupons are out-of-the-money).

For instance, assume pizzas are selling for $6 and the market just doesn’t think there’s going to be much of a chance for a price hike. You may see a value of only 5 cents on the $15 coupon and 10 cents on the $10 coupon, which is only a five-cent difference.

Likewise, options must obey a similar principle. If you think about it, there really is no difference in owning a $50 call versus a $55 call other than the fact that the person with the $50 strike can pay $50 for the stock, while the person with the $55 strike can pay $55.

The maximum difference in value between holding the $50 call and the $55 call therefore cannot be more than $5. The market will never give you more than the difference in strikes for any option whether calls or puts (assuming the same underlying stock and time to expiration).

So let’s go back to our quotes that we said could not happen:

$50 Call = $10
$55 Call = $4

If these quotes were to appear, how would the markets correct for this? As usual, arbitrageurs will buy the cheap asset and sell the expensive one. In this case, they will buy the relatively cheap $55 call and sell the relatively expensive $50 call for a net credit of $6:

Buy $55 =  -$4
Sell $50  = $10
Net credit  $6

Now check the rights and obligations. The arbitrageur has the right to buy stock for $55 and the potential obligation to sell for $50, which would create a $5 loss as a worst case scenario. However, he was paid $6, which more than covers the potential loss and still leaves a $1 arbitrage profit. This profit would occur for any stock price above $55. With the stock above $55, the arbitrageur would exercise the call and pay $55 and also be assigned and receive $50 thus creating the $5 loss.

If the stock falls below $50 at expiration, both options expire worthless and the arbitrageur keeps the full $6 credit. If the stock closes between $50 and $55 the arbitrageur starts to lose money on the short call but can only lose a maximum of $5. So for stock prices between $50 and $55 at expiration, the arbitrageur will earn more than $1 but less than $6.

For example, with the stock at $52, the arbitrageur will be assigned on the short $50 call and be required to deliver stock worth $52 and receive only $50 thus creating a $2 loss. He can pay for this loss out of his $6 initial credit, which leaves him a net gain of $4. 

No matter where the stock price may be at expiration, the arbitrageur will make at least $1 and as much as $6.

Does this really happen in the real world of options? Take a look at the Cyberonics (CYBX) quotes in Table 6, which were taken the day of expiration:

Table 6: CYBX Option Quotes

Look at the asking prices of the first two listed calls, the June $12.50 and $15 strikes. The asking prices are $25.50 and $23, respectively, which is a $2.50 difference. And that’s exactly the difference in their strikes. Once the price of the $15 call is established by the market, the market will pay a maximum of $2.50 above that price for the $12.50 strike.

Let’s pick two different strikes. What’s the difference in prices between the $15 call and the $20 call? Their prices are $23 and $18, which is exactly $5 and, again, the difference in strikes. Once the price of the $20 call is established, the market will not pay more than $5 above that price for the $15 call.

These prices expanded to the full difference in strikes because the stock price was so far above them at expiration. In other words, with the stock at $37.55, the $12.50, $15, and $20 strikes were so far in-the-money that the market didn’t see a chance for any of them to close out-of-the-money so their prices converged to the exact differences in strikes.

(You may have noticed that the difference between the $30 and $35 strikes is $5.10 and this is simply a fluke. These quotes are 20-minutes delayed and were probably in the process of being updated and you can be sure this fall to exactly a $5 difference in strikes.)

Now take a look at the $35 and $40 strikes. Their asking prices are $3 and 15-cents respectively, which is only a $2.85 difference. Here we have a five-dollar difference in strikes but only a $2.85 difference in price. Remember, this principle states that the difference in prices cannot exceed the difference in strikes. It does not say that it cannot be less.

Because CYBX was trading for $37.55, neither the $35 strike or $40 strike are seen as being “guaranteed” to finish with intrinsic value even though this is expiration day. Notice in the upper left hand corner of Table 6 that the time is 10:06am so there are still about 6.5 hours of trading left in the day. With a stock like Cyberonics, the $35 and $40 strikes are far from being assured of finishing with intrinsic value and the market is pricing that risk accordingly. That’s why the market is not pricing a full five-dollar difference between the $35 and $40 strikes  

There are two conditions under which you’d see a $5 difference between the $35 and $40 strikes. First, if the stock’s price was sufficiently higher than these strikes, say $43 or higher, then you’d see a five-dollar difference between the $35 and $40 calls. The more time remaining until expiration (or the more volatile the stock) the higher that stock’s price needs to be before you’d see a $5 difference between these two strikes.

The second condition under which we’d see a $5 difference is if these quotes were taken in the final seconds of trading and the stock was $40.01 or higher. The sole determining factor is the market’s perception as to whether both of these options will expire in-the-money. If there is time remaining, then the stock’s price needs to be well above both strikes. If there is little time, then the stock’s price only needs to be just slightly above the higher strike in order for the difference in prices to be equal to the difference in strikes.

Now you should have a basic understanding of why this principle is true for any set of option quotes. If the option is deep enough in-the-money, the markets will view them as guaranteed to expire with intrinsic value, in which case the difference in strikes will equal the difference in price. If there is a risk of that not occurring (that is, a perception that the stock price may fall below the strike), the difference in prices will be reduced to something less than the difference in strikes.

While option prices are free to fluctuate, there are some invisible fences surrounding their prices. These boundaries are not set by exchanges, any trader, or market maker. Rather, they are economic and financial principles at work. Traders and investors who understand option pricing principles such as these will quickly climb through higher levels of options trading and strategies.