# Deconstructing Delta: It's Not All Greek to Options Traders

Option markets are populated by so many Greek speakers that the average investor often founders amid the babble about delta, theta and other arcane alphabetics.

These symbols simply refer to an options trader's exposure to risks such as the directionality of market movement or the passage of time. All the Greek risk factors are products of option pricing algorithms. One which serves well to illustrate the underpinnings of pricing and the "Greeks" is the binomial model.

The most basic Greek is delta, which measures the expected change in an option's premium for a given change in the price of the underlying stock. Simply put, delta is a measure of an option's bullishness or bearishness or the probability of the option expiring in the money.

Suppose, for example, you believe that

**Sun Microsystems'**

(**SUNW**) - Get Report

common stock, currently trading at about $100, will challenge its previous highs in the upcoming quarter. You begin to weigh the prospect of buying Sun Micro calls. Let's say you're particularly interested in the at-the-money ($100) four-month call. Part of your consideration, perhaps, is how the call is likely to respond price-wise to subsequent changes in the value of underlying shares.

A simplified model can be created to illustrate an option's price behavior. The "expected value" of the call can be found as the product of each potential outcome's probability and its intrinsic value. For example, if there's a 50% probability the shares will move up or down $1 by day's end, we can forecast the theoretical premium of a Sun 100 call as the sum of each outcome's expected value like this:

Widening or narrowing the range of expected stock prices in the time period being examined adjusts the model to different volatility expectations. So, if Sun Micro is expected to be

*twice*

as volatile as before, the call should theoretically be priced higher as a result:

When you widen the horizon to examine the potential for the four-month call, you'll note that the probability of each final outcome changes. As seen below, there's only one path for Sun Micro to follow to get from $100 to $116 in a month's time. There are, however, six possible routes for Sun Micro to end up back at its $100 starting point within the same period. Obviously, the extreme price outcomes are statistical outliers -- there's just a greater likelihood that the stock will end up "somewhere in the middle."

Suppose by the end of the first month Sun shares have climbed to $108. Assuming volatility hasn't changed, what should the call be worth now? And, if the call originally had a delta of .50, what do we now know of the call's present sensitivity to Sun's price?

Intrinsically, the call is now worth $8. Note the expected value accounts for a half-dollar's worth of "time value." And as the call moved in the money, its delta rose. In fact, the call began to lose leverage and started to behave more like the underlying stock as intrinsic value developed.

Delta is not a constant, but can change in response to movements in the price of the underlying stock. Stock price movement isn't the only influence on delta, though. We've seen that widening or narrowing the range of expected stock prices alters a call's expected value. That means that shifts in volatility, as a consequence, alter delta as well. The passage of time also impacts delta as the option moves inexorably toward expiration.

Let's see what could happen to your call if Sun in fact tests its highs in the subsequent month.

With Sun now at 116, your call has moved 16 points in the money. You'll also note the erosion of the time value premium. Decay of time premium accelerates with the approach of expiration. More important, though, is the call's delta moving closer to its maximum potential value.

The utmost delta for an option is 1. If an option's in the money at expiration, its delta will have advanced to the loftiest level as the option's price behavior now mirrors the underlying stock's. Should an option end up at the money or out of the money (without intrinsic value) by its expiry, delta will sink to zero, reflecting the call's absolute insensitivity to changes in the price of the underlying stock.

This gives rise to an anomaly of option pricing: an unfavorable stock price movement accompanied by an actual increase in delta sensitivity. Suppose, for example, Sun ends up backpedaling $8 from the $116 within a month of the $100 call's expiration. What could you now reasonably expect of the call's premium and delta?

You'll note that even though the stock dropped precipitously, the call delta actually rose. And since the last discrete time period before expiration has been reached, the call's delta can be expected to remain at 1 -- at least in this greatly simplified model, unless Sun drops below the call strike price.

Hopefully, this will be just the first of many Greek lessons. By the time we're done with the Greek alphabet you'll be able to toss off Hellenic risk parameters like an Athenian cab driver bellows epithets.

*Opa!*

Brad Zigler is director of options marketing, research and education for the Pacific Exchange in San Francisco.