Γειά σας! Welcome to the second installment of our discussion of options! By now, I'm sure that all of you have had a chance to read the first section, which discusses the basics of options trading, so we'll jump right back into The Ballad of Billy the Buyer and Walter the Writer:

When we last left Billy and Walter, we had discussed the types of transactions that can be made in "options-land," and what those transactions mean for each party involved. The key takeaway for all options trades is that both sides are strictly at odds with each other. That is: Walter only profits when Billy loses money, and Billy only profits when Walter loses money. The cliff-hanger we were left with at the end of the last article was how does either of them understand the implications of each individual transaction in order to make a profit?

The answer? Minding the Greeks. But, what are they?

"The Greeks," as a collective, are **various measures of risk involved in taking a position in a derivative transaction.** For our purposes, we will examine each of them as they apply to options (but, who knows, we might talk about other super-slick derivatives in the future...). Without further ado, here they are:

Delta ($latex \Delta$)

Good ole Delta , what a pleasure. For those of you who do not recall, DELTA is used in mathematics and science as a measure of change in a system. In this system, DELTA is the rate the price of the option contract changes as the price of the underlying stock changes by $1. When it comes to the Delta of a purchased call option: $latex 0 \leq \Delta \geq 1$, and for a purchased put option: $latex -1 \leq \Delta \geq 0$; however, this is reversed when it comes to writing options. When you consider the fact that puts and calls are always opposing transactions, it makes sense that the Delta measure for bullish transactions are positive and the Delta measure for bearish transactions. To further demonstrate, suppose Billy buys an XYZ Dec 30 Call at 3, where XYZ is at $30 and lets say that the $latex \Delta=.5$. Once Billy buys the call option, XYZ's stock price rises to $31, which then raises the cost of the call option will rise to $3.50, should Billy choose to resell it.

If Billy or Walter wanted to use Delta to their advantage, they would have to understand the concept of "Delta Hedging." In order to hedge the Delta of one's option trade, they must trade a corresponding amount of the underlying security. In the above example, Billy would have to short sell 50 shares of XYZ stock in order to make $latex \Delta=0$. This is based on his bullish option (hence the short sale) and the number of shares he must short is equal to $latex \Delta$*Shares he controls via the option contract. There is one catch to trading based on delta: it increases as the time-til-expiration decreases (A.K.A.- as time passes), and changes as the price of the underlying security changes. The latter is defined as Gamma. As a bit of a foray from the complexity of the other Greeks (being a math-junkie and all), let's explore Gamma a bit.

Gamma ($latex \Gamma$)

Alright, I'm going to use what most consider to be a dirty word, but here it goes: Gamma is the second derivative of option price sensitivity. One can think of Gamma as a derivative measure of a derivative measure: it is the amount that delta will change based on a $1 change in the underlying security's price. Since an option's Gamma is the derivative of its Delta, we have that $latex \Gamma$ of long options is between zero and one and $latex \Gamma$ of short options is between negative one and zero. Returning to Billy's XYZ Dec 30 Call, we know that his $latex \Delta=.5$, so his option price will rise to $3.50 when the stock price rises to $31. If its $latex \Gamma=.1$, then his Delta will increase to .55, or $latex (1+.1) \cdot .5$.

Now, Gamma can get even more complex, since you can take its derivative and measure the degree to which it changes as time passes. **Color**, as this measure is known, is particularly useful for very large portfolios, but is a *bit overkill in smaller portfolios* (as are most of the second- and third-order derivatives for risk measures).

Theta ($latex \Theta$)

Getting back to the first-order Greeks, remember Theta from Geometry!? Yes, I know you loved the day when your teacher said that you were throwing $latex x$ out the window for a shiny, new $latex \Theta$, and here it is again, this time as a measure of an option's time sensitivity. That is to say: how much will the price of the option change as the time-til-expiration decreases? Since the value of an option always decreases as time passes, due to the decreasing window during which you can execute them, an option's Theta will always be negative (A.K.A.-$latex \Theta<0$). This is like assigning -$20,000 to the purchase of a car on your personal balance sheet; we all know that buying a car (or anything) is an expense, but you put it there for good measure. So, returning to our friend Billy, his December call is on a short clock until expiration. Suppose $latex \Theta=.01$. This means that, as each day passes, his price will fall by $0.01. So in 5 days, his option price will fall to $2.95.

In order for Billy or Walter to make decisions based on Theta, they must first understand the mechanics behind an option's premium. An option's premium is the sum of it's time value and its intrinsic value, or that Premium = Time Value + Intrinsic value, where time value is function of how much time is let until expiration and intrinsic value is how much the option is worth at the moment. Taken further, for a call option: Intrinsic Value = Underlying Stock Price - Strike Price; for a put option: Intrinsic Value = Strike Price - Underlying Stock Price. An option is considered "in-the-money" when its intrinsic value is greater than zero, "out-of-the-money" when the intrinsic value is less than zero, and "at-the-money" when intrinsic value equals zero. As previously mentioned, time-til-expiration is vital because it allows the option more time to become in-the-money. Therefore, if an option has high Theta, then Billy and Walter know that the value of the option is highly correlated to time-til-expiration, since time works against Billy and in favor of Walter.

Vega ($latex \nu$)

Okay, so *Vega isn't actually Greek* (well, except that it's actually nu...but yeah), the world puts it into the basket of Greeks anyway, so here it is (sorry?). Vega (which is actually a star) is a measure of an option's price in relation to implied volatility; that is: the increase or decrease in the option price as implied volatility increases or decreases by 1%, respectively. As the saying goes: *"rising tide raises all ships,"* well so does volatility: when volatility increases both calls and puts increase in value. This is especially true when considering a long straddle (which we will go into more depth in a future piece), which is in the money when the underlying price changes to a large degree. Intuitively, Billy wants the price to fluctuate to a large degree (in one way or another) since he will benefit only when the option is in the money. Walter, on the other hand, wants the stock price to stay relatively stable so that it doesn't cross the threshold to in-the-money.

Rho ($latex \rho$)

**Rho is sort of the ugly stepchild of the Greeks**: most don't care about it very much because it isn't widely relevant since it is the change in price given a 1% change in interest rates, which can be by the market demand or by the Federal Open Market Committee (the Fed). For those of you paying attention to the Fed, you know that an interest rate hike is in the cards (if you know when, please please let us know), so Rho isn't all that bad right now. From a pricing standpoint, bullish options are positively correlated to interest rates, while bearish options are inversely related. That is to say, if Billy owns his December call, which has $latex \rho=.01$, and interest rates increase by 1%, he will see the value increase to $3.05. Illustrated in this example, however, is the fact that an option is less sensitive to a change in interest than a change in underlying price or time-til-expiration, it has a small effect on the price.

So there you have it, all the Greeks in a row, but how do they come together for a comprehensive, butt-kicking options strategy for Billy or Walter? Well, it will take some clever strategies to successfully utilize their knowledge of the Greeks, but before we dive into those, **take some time to really understand what each of these mean**--it will pay dividends when executing a strategy (see what I did there?).