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In order to truly understand the leverage of an option, we must compare “dollar equivalent” exposure. For example, let’s assume the $75 call trading for $8.30 has a delta of 0.60. For the next one-dollar move, this option’s price will rise by the delta, or 60 cents, from $8.30 to $8.90. This 60-cent move is equivalent to $60 per contract. Now let’s see what a stock investor must spend to get this same $60 gain from a one-dollar move in the stock. A stock buyer must buy the delta equivalent number of shares, which is 60 shares of stock that would cost 60 * $79.46 = $4,767.60. So if an option trader buys the $75 call and a stock trader buys 60 shares of stock, then both will capture a $60 profit on the next one-dollar move in the stock. Now we just need to compare the costs of these dollar equivalent exposures. The stock trader spends $4,767.60 while the option trader spends $830, which means there is $4,767.60/$830 = 5.7 times as much leverage in the option as compared to the stock. (But keep in mind that this number will change as the delta of the option changes. We’re just saying this is how you’d need to calculate the leverage in the option at this point in time.)

*Other Views of Leverage*Although the above calculation is probably best for comparing the true leverage of an option there are other views we could take.

For example, say a stock is trading for $100 and a $100 call is trading for $5. One way to view the leverage is to realize that the option trader, in this example, has leveraged the returns by a factor of 20. That is, for every 100 shares the stock investor buys ($10,000 worth), the option buyer can buy 20 contracts ($10,000/$500 per option = 20).

Let’s assume that the stock now rises from $100 to $115. If the stock trader buys 100 shares then the total value would be 100 * $115 = $11,500, which leaves a profit of $1,500. With the stock at $115, the $100 call would be worth $15, or $15 * 2,000 = $30,000 for the 20 contracts.

If we multiply the $1,500

*profit*of the stock trader by 20 we end up with $30,000, which is the value of the option trader’s*total**position*. In this example, the option trader’s total value will always be worth 20 times the stock trader’s profit, assuming the $100 call option has intrinsic value. This is a somewhat awkward view of leverage since we’re comparing the profit of the stock trader to the total value of the option trader. Still, it is a very common use that you will encounter.It’s important to understand that this method only works in such a straightforward way if we compare at-the-money options. Using our IBM example, the stock is $79.46 and the $75 call is trading for $8.30, which means the option trader has leveraged the returns by a factor of $79.46/$8.30 = 9.5 times. Once again, this means that for every 100 shares the stock investor buys ($7,946 worth of stock) the option buyer can control 950 shares since $7,946/$830 = 9.5 contracts, or 950 shares. (You cannot buy fractional contracts but we must assume this to make the comparisons.) Now we should expect that for any given gain in the stock’s price, the option’s

*total**value*would be 9.5 times as great as the stock trader’s*profits*.Let’s see if it works. Assume the stock rises from $79.46 to $85 by expiration. The stock trader invested $7,946 and can sell for $8,500 for a total

*profit*of $554. With the stock at $85, the $75 call would be worth $10, or a total value of 950 shares * $10 = $9,500. However, we see that $9,500/$554 = 17 times. Why does it not equal 9.5 times? The reason is that this option is not at-the-money. The stock is $79.46, which means this $75 call has $4.46 worth of intrinsic value that we must take back out. The value that must be subtracted from the $9,500 total option value is then 950 shares * $4.46 = $4,237, which leaves us $5,263. If we take $5,263/$554 we get leverage of 9.5!

*Gearing*The leverage described above is known as

*gearing*and is actually just an old British term that means leverage. It is not uniquely defined but the two most common definitions are (1) The*stock*price divided by the option price or (2) The*strike*price divided by the option price.Using definition 1, the way to find gearing is to simply divide the stock price by the option price:

**Gearing = Stock price / option price**

In our first example, the stock was $100 and the option was $5 so $100/$5 = 20. This is just another way of saying the stock trader required 20 times the amount of capital to control the same amount of shares.

Using the second definition, gearing would be:

**Gearing = Strike price / option price**

This gives the same answer of 20. But if the strike was $110 then the gearing is $110/$5 = 22. In this way, the option trader may pay $110 for the stock but is controlling it for $5, so it is leveraged by a factor of 22. Many of the trading software you will encounter will have a column labeled “gearing” and it simply shows one of these definitions of leverage.

*Omega*There is another term you may see that describes leverage and is called

*omega*. Omega measures the relative percentage changes between the stock and the option, which is called an*elasticity*measure. For instance, assume the call in the above example has a delta of 0.50. With the stock at $100 and the call at $5, if the stock were to move $1 (a 1% move) the call will move roughly one half of a point from $5 to $5.50 for a 10% increase. Because the option moved 10 times faster relative to the stock (10% compared to 1%), the elasticity (omega) is 10.Omega =

__Delta / option price__ 1 / stock price

This numerator of this formula simply compares the “share equivalent” terms of the option to its price (delta / option price). The denominator just compares one share of stock to its price (1 / stock price). Omega simply finds the ratio of these two values.

Omega can also be written as (stock price / option price) * delta. Using the earlier example, we have a $100 stock price divided by a $5 call option with delta of 0.50 so $100/$5 * 0.50 = 10. Regardless of which measure you use, don’t forget the most the most important concept: The higher the leverage the more speculative the position.

The option’s leverage comes from the fact that the strike price is simply a partition of the stock price. In this example, if you buy shares of IBM at $79.46, you get all of the upside gains but are also exposed to all of the downside losses. That’s because a long stock position contains value for all stock prices above zero. In fact, Pricing Principle #5 from Chapter Two showed us that an option with a zero strike price and infinite time to expiration would be trading for the same price as the stock. A long stock position can therefore be thought of as an option with a zero strike price and no expiration date.

However, if you are holding the $75 call at expiration, it will not have value for all stock prices above zero. Instead, it will only have value for all stock prices above $75. The $75 strike simply splits the stock into two parts: All prices below $75 and all prices above $75. When you buy the $75 call, you’re only participating in the gains if the stock rises above $75 but not if it falls below, which is why long call options have an asymmetrical payoff to their profit and loss diagrams.

So option returns appear much higher because we’re partitioning the stock’s price. In this example, the stock buyer must pay $79.46 but the option trader only pays $8.30 to participate in the gains for all stock prices above $75. It is this difference in bases – $79.46 compared to $8.30 – that creates the leverage. A one-dollar gain to the stock trader produces a much smaller percentage gain than a one-dollar gain to the option trader. However, the total dollar gain to the stock trader will be larger than the total dollar gain to the option trader since the option trader loses out on the time premium.

Many investors get attracted to options because they hear about the high leverage and think they will make more money by trading options rather than stocks. It would be easy to think you would have done much better with options since you would have earned 21% on your money rather than 7% in our example. However, this is really a misperception and comes from the fact that option traders have a much smaller dollar amount of money invested if they are trading an equivalent number of shares in the options. True, their percent returns are higher but the investments are smaller but the total dollars earned will be less than those investing in stocks. In this example, the stock buyer earned 7% on a $16,000 investment while the call buyer earned 21% on a $1,660 investment. The important point to understand is that if you trade the contract equivalent number of shares with options that you will have higher percentage returns but lower total dollar returns (assuming that both the stock and options are profitable).

Couldn’t we get a 21% return on our investment if we had purchased $16,000 worth of options? The answer is yes; however, that is a very dangerous (although quite common!) way to use leverage.

There are actually two definitions of leverage that you need to understand:

- Control more shares with the same amount of money (risky use)

- Control the same amount of shares with less money (conservative use)

The great mathematician, Archimedes, once said, "Give me but one firm spot on which to stand, and I will move the earth.” He was, of course, talking about the enormous power of the lever. Investors who do not understand the difference between the above two definitions will eventually find out just how powerful a force it is.

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