Options (Part I)

           The public's interest in stock options trading is reaching new heights (based on CBOE volume). I am a somewhat experienced stock option trader for my own brokerage accounts. I have no other special credentials, so take all this information with a grain of salt.

           To begin, I will give some of the basic definitions involved in the option business. After that I will give an introduction to how options are structured and some intuition as to how they are valued. Finally, I will give sources of more information.

           Options are a type of derivative security (a security that, rather than representing financial ownership of something, is merely a fluctuating asset depending on the value of some other security). Options on US stocks are traded on several exchanges and come in two basic categories: call and put options. Because selling a stock was once referred to as "putting" it to someone else, and because "calling" a stock is a floor-trader firm for a buy, a put option is the option to sell, and a call is the option to buy, both at a specific price. That price is referred to as the "strike price", or more informally just the strike.

           Let's explore the basic idea of why an option has value. First of all, an option is a right, not an obligation. The latter is easy to price. An obligation to buy is essentially a forward contract on a stock, which has value equal to (S-x), where S is the stock price and X is the agreed-upon price. (There should also be a discounting applied to better reflect the carrying cost of the stock, but that's going a little far afield.) Note that this can very easily be a negative number.

           Since an option carries no obligation, however, it can never have a negative value. The purchaser of a call or put is not bound to do anything with it, so it is worth no less than zero. Since (S-x) can very easily take either arithmetic sign, it obviously doesn't tell us the whole story about option values. When (S-x) is negative, the call option is said to be "out of the money" and the put option is "in the money". If (S-x) changes sign, the designations are reversed, and "at the money" generally refers to the closest available call or put option contract to where the stock is currently trading. (There are enough options available on most stocks to make this fairly unambiguous.)

           Question: How can the option to buy ABC at 100 be worth anything if ABC is trading at 90? This is a lot tougher to answer than it looks, but I will try to give some of the intuition behind it. The answer is apparent when we think of the counterparty who would sell this call option contract. Suppose you gave away the option to buy 1000 shares of ABC, by some specific date in the future, at 100. Unless you're the gambling type, you need to mitigate your risk by owning a block of ABC. First of all, this is going to cost money, and that money could be earning the (essentially) riskfree Treasury rate, so you're losing that interest payment immediately. But let's suppose you have access to free money, so this isn't the problem (the interest-rate component of most option prices is very small, in fact). The big deal is deciding how much ABC to own.

           For now, let's assume your broker is free (hook me up if you know one like that). If it goes up to 100, you need to be long the full 1000 so you can deliver it at expiration and not lose a thousand bucks per point it goes above 100. However, if it's below 100, you need to own no ABC at all. Already you can see that giving away that option means you're going to have to through considerable pain. Still, though, it seems like we can just use a stop order at 100 to buy 1000 ABC, and enter a stop order at 100 to sell after we get filled on the purchase, and repeat that forever.

           This is where the heavy math comes in. Basically, the way stock prices jump around means it's pretty unlikely you can establish that perfect hedge at $100. Even if you take out bid-ask spread costs, and don't allow the market to ever close, the math still comes out against you. The more volatile ABC is, the harder it is to take this position. To get an idea of this cost, bring up an intraday stock chart, pick some imaginary strike price, and try to imagine where you would have made your trades to keep yourself hedged. For an ordinary stock with volatility near that of the S&P 500, the call at $100 with a month to go might trade for around $1 in this case. (This is a good time to point out that options exchanges price option contracts in 100-option bundles and don't let you trade smaller units.)

           Mathematicians, relying on models for the volatility of stock prices, have come up with ways to compute fairly good theoretical option prices. None of them precisely reflects real-world conditions, however. You can find information on Black-Scholes (the simplest, most famous model) and its cousins in nearly any good book (but more on this later). For this discussion, I'd like to build on the intuition above and use a nice analogy I picked up: namely, that options are like a lease on stock.

           Consider the call-buyer again. He puts up $1000 instead of $90,000. This is his lease payment for the next month on 1000 shares of ABC. He can experience the thrill of owning 100 ABC (when it nears the strike price anyway), and if he likes, when the lease is over he can buy the ABC for $100 per share. He doesn't have to tie up any capital or margin to get the ABC, either. It's not exactly like a conventional lease, though, because the call-buyer can negotiate his way out of the lease at any time (corresponding to selling the call in the open market). If ABC goes to $80, maybe he can end the lease and still get $25-50 of his money back from the initial payment. If it's at $110, he'll get plenty of compensation, probably even more than $10,000. That's an awe-inspiring return on investment. Obviously the option price is telling us that the chances of such a windfall are rather slim!

           In the previous paragraph I've tried to hit on some of the more important parts of pricing an option. The option cost is composed of intrinsic value and time premium. Either can dominate the total price of the option. Before you learn about trading concepts, let's discuss the option delta, which is the most important thing your computational formula can tell you about an option.

           The delta is the sensitivity of the option price to the stock price, expressed as a number between -1 and +1. Calls always have delta larger than zero, and puts always have negative delta. If a call is deep-in-the-money, like a call on ABC at $50 would be, it will act just like owning ABC, for a delta of +1 (approached asymptotically). If a call is far out-of-the-money, however, it could have a delta of only +0.05 or so. Here's the really neat part: delta corresponds to the probability that the option finishes its life in-the-money.

           This is hard to see, though. I can't find a satisfying explanation that doesn't use any advanced concepts. Right now my best one is like this: If I could short the call option at 101, I would establish a spread position. Because it's very likely that if ABC is above 100 at expiration it also surpasses 101, the spread should be worth an amount close to $1 [the width of the spread] times the probability of profit. Because the delta is just the slope of the graph of option price vs. strike price, and the spread can be made as narrow as I want, the spread value also must be tending toward the delta of the 100 call, which is what I was trying to prove.

           The next most important thing about options is gamma. Gamma is what makes an option nonlinear and allows it to return over 1000% in certain cases (or even more I suppose). Gamma is simply the rate at which delta changes with respect to the price of the underlying. You may also hear the term 'convexity'. All gammas are positive (think about it). The options carrying the largest gammas are near-term, out-of-the-money options, because while they are unlikely to enter the money, the rise in delta will be very sharp when they manage to do it.