Meet the Greeks

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Because the price of options depends on the price of the underlying asset and because options are a wasting asset due to their limited lifetimes, option premiums vary with the price and volatility of the underlying asset and time to expiration of the options contract. Several ratios have been developed to measure this change in price with respect to the price or volatility of the underlying, and the effect of time decay.

Since most of these ratios are represented by Greek letters—delta, gamma, theta, and rho—the group is often referred to simply as the greeks. Vega is also a commonly used ratio and is also considered a greek, although it is not actually a Greek letter some purists prefer to use the Greek letter tau for vega.

These ratios are used to measure potential changes in the value of an actual portfolio or of test portfolios of options from potential changes in the underlying stock price, volatility, or time until expiration. The delta ratio is the percentage change in the option premium for each dollar change in the underlying.

Note that a put option with the same strike price will decline in price by almost the same theta and vega options trading delta gamma, and will therefore have a negative delta. Options are frequently used to hedge risk.

But what if earnings are less than what the market expected. Then the price may drop a few dollars, resulting in a loss. Therefore, you would want to buy 2 put contracts to cover or hedge your position. Since the value of the portfolio doesn't change within a narrow range, it is said to be delta neutral. This technique is also called delta hedging. The delta of a portfolio, which is calculated by summing the deltas of each option in the portfolio, is sometimes called its position delta.

Delta is also used as a proxy for the probability that theta and vega options trading delta gamma call will expire in the money. However, delta does not measure probability per se.

Delta can serve as a proxy for the probability only because both delta and the probability that a call will go or stay in the money increases as the option goes further into the money. However, delta is not a direct measure of the probability. As an example of where delta and probability will diverge is on the last trading day of the option. Most of the value of a call will depend on the intrinsic value, which is the amount that the underlying price exceeds the strike price of the call.

The above example will not work out perfectly in the real world. You may even ask, why adopt a delta neutral portfolio when your objective is to make a profit? A delta neutral portfolio is only delta neutral within a narrow price range of the underlying. Delta itself changes as the price of the underlying changes. Then you would profit from the puts, but lose on the stock. So would the profit from the puts completely neutralize the loss on the stock. Actually, you would do better. This results because delta itself changed.

Gamma is the change in delta for each unit change in the price of the underlying. The absolute magnitude of delta increases as the time to expiration of the option decreases, and as its intrinsic value increases.

Gamma changes in predictable ways. As an option goes more into the money, delta will increase until it tracks the underlying dollar for dollar; however, delta can never be greater than 1 or less than When delta is close to 1 or -1, then gamma is near zero, because delta doesn't change much with the price of the underlying.

Gamma and delta are greatest when an option is at the money—when the strike price is equal to the price of the underlying. The change in delta is greatest for options at the money, and decreases as the option goes more into the money or out of the money. Both gamma and delta tend to zero as the option moves further out of the money. The total gamma of a portfolio is called the position gamma. Options are a wasting asset. The option premium consists of a time value that continuously declines as time to expiration nears, with most of the decline occurring near expiration.

Theta is theta and vega options trading delta gamma measure of this time decay, and is expressed as the loss of time value per day. Thus, a theta of. Theta is minimal for a long-term option because the time value decays only slowly, but increases as expiration nears, since each day represents a greater percentage of the remaining time.

Theta is also greatest when the option is at the money, because this is the price where the time value is greatest, and, thus, has a greater potential to decay. For the same reason, theta is greater for more volatile assets, because volatility increases the option premium by increasing the time value of the premium.

With higher volatility, an option has a greater probability of going into the money for any given unit of time. For the option writer, theta is positive, because options are more likely to expire worthless with less time until expiration. Theta measures changes in value of options or a portfolio that is due to the passage of time.

The holding of options has a negative position theta because the value of options continuously declines with time. Because time decay favors the option writer, a short position in options is said to have positive position theta. The net of the positive and negative position thetas is the total position theta of the portfolio. Volatility is the variability in the price of the underlying over a given unit of time. The Black-Scholes equation includes volatility as a variable because it affects the probability of the option going into the money: Historical volatility is easily measured, but current volatility cannot be measured because the unit of time is reduced to now.

On the other hand, the price of the underlying, the option premium, time until expiration, and the other factors, except volatility, are known. Therefore, volatility can be measured by rearranging the Black-Scholes equation to solve for volatility in terms of the other known factors.

This is referred to as implied volatilitybecause the volatility is implied by the other known variables to the Black-Scholes equation. Consequently, vega is often used to measure the change in implied volatility. Vega measures the change in the option premium due to changes in the volatility of the underlying, and is always expressed as a positive number. Because volatility only affects time value, vega tends to vary like the time value of an option—greatest when the option theta and vega options trading delta gamma at the money theta and vega options trading delta gamma least when the option is far out of the money or in the money.

Theta and vega options trading delta gamma position vega measures the change in option or portfolio values with changes in the volatility of the underlying.

Higher interest rates generally result in higher call premiums, according to option pricing modelsbecause the present value of the strike price is subtracted in these models. Hence, higher interest rates correspond to lower present values, so less is subtracted, leading to higher call prices. A more intuitive way to understand why higher interest rates increases call prices is to understand that a call is like a forward contract, in that it allows the holder to buy the stock at a specified price before the expiration date, so the money that would have been used to otherwise buy theta and vega options trading delta gamma stock can, instead, be invested in Treasuries to earn a risk-free interest rate until the date in which the stock is purchased.

Because the stockholder incurs a cost of holding the stock, which is the forfeited interest that could otherwise be earned, a higher price is charged for the call to compensate the stockholder for the forfeited interest. By the same reasoning, dividends decrease the price of calls because theta and vega options trading delta gamma the stockholder is entitled to receive the dividends, not the call holder.

On the other hand, the application of the put-call parity theorem to option pricing models yields lower put premiums due to higher interest rates. Thus, a rho of 0. The values are theoretical because it is market supply and demand that ultimately determines prices.

In fact, rho can be misleading because interest rates may have a larger effect on the price of the underlying, which is a more significant determinant of option prices. The demand for stocks, for instance, varies inversely with interest rates. When interest rates are low, investors buy stocks in an attempt to earn more income. When interest rates rise, risk-averse investors move their theta and vega options trading delta gamma from stocks to safer bonds and other interest-paying investments.

Thus, puts will tend to increase with interest rates while calls will decrease, because the price of the underlying will have a more significant effect on option premiums than the interest rate.

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The Greeks represent the consensus of the marketplace as to how the option will react to changes in certain variables associated with the pricing of an option contract. There is no guarantee that these forecasts will be correct. And as Plato would certainly tell you, in the real world things tend not to work quite as perfectly as in an ideal one.

The option costs much less than the stock. Why should you be able to reap even more benefit than if you owned the stock? Calls have positive delta, between 0 and 1. That means if the stock price goes up and no other pricing variables change, the price for the call will go up. If a call has a delta of. Puts have a negative delta, between 0 and That means if the stock goes up and no other pricing variables change, the price of the option will go down.

For example, if a put has a delta of -. As a general rule, in-the-money options will move more than out-of-the-money options , and short-term options will react more than longer-term options to the same price change in the stock. As expiration nears, the delta for in-the-money calls will approach 1, reflecting a one-to-one reaction to price changes in the stock. As expiration approaches, the delta for in-the-money puts will approach -1 and delta for out-of-the-money puts will approach 0.

Technically, this is not a valid definition because the actual math behind delta is not an advanced probability calculation. However, delta is frequently used synonymously with probability in the options world.

Usually, an at-the-money call option will have a delta of about. As an option gets further in-the-money, the probability it will be in-the-money at expiration increases as well. As an option gets further out-of-the-money, the probability it will be in-the-money at expiration decreases. There is now a higher probability that the option will end up in-the-money at expiration. So what will happen to delta?

So delta has increased from. So delta in this case would have gone down to. This decrease in delta reflects the lower probability the option will end up in-the-money at expiration. Like stock price, time until expiration will affect the probability that options will finish in- or out-of-the-money. Because probabilities are changing as expiration approaches, delta will react differently to changes in the stock price. If calls are in-the-money just prior to expiration, the delta will approach 1 and the option will move penny-for-penny with the stock.

In-the-money puts will approach -1 as expiration nears. If options are out-of-the-money, they will approach 0 more rapidly than they would further out in time and stop reacting altogether to movement in the stock. Again, the delta should be about. Of course it is. So delta will increase accordingly, making a dramatic move from. So as expiration approaches, changes in the stock value will cause more dramatic changes in delta, due to increased or decreased probability of finishing in-the-money.

But looking at delta as the probability an option will finish in-the-money is a pretty nifty way to think about it. As you can see, the price of at-the-money options will change more significantly than the price of in- or out-of-the-money options with the same expiration.

Also, the price of near-term at-the-money options will change more significantly than the price of longer-term at-the-money options. So what this talk about gamma boils down to is that the price of near-term at-the-money options will exhibit the most explosive response to price changes in the stock.

But if your forecast is wrong, it can come back to bite you by rapidly lowering your delta. But if your forecast is correct, high gamma is your friend since the value of the option you sold will lose value more rapidly. Time decay, or theta, is enemy number one for the option buyer.

Theta is the amount the price of calls and puts will decrease at least in theory for a one-day change in the time to expiration. Notice how time value melts away at an accelerated rate as expiration approaches. In the options market, the passage of time is similar to the effect of the hot summer sun on a block of ice. Check out figure 2. At-the-money options will experience more significant dollar losses over time than in- or out-of-the-money options with the same underlying stock and expiration date.

And the bigger the chunk of time value built into the price, the more there is to lose. Keep in mind that for out-of-the-money options, theta will be lower than it is for at-the-money options. However, the loss may be greater percentage-wise for out-of-the-money options because of the smaller time value. Obviously, as we go further out in time, there will be more time value built into the option contract. Since implied volatility only affects time value, longer-term options will have a higher vega than shorter-term options.

Vega is the amount call and put prices will change, in theory, for a corresponding one-point change in implied volatility. Typically, as implied volatility increases, the value of options will increase.

Vega for this option might be. Now, if you look at a day at-the-money XYZ option, vega might be as high as. Those of you who really get serious about options will eventually get to know this character better. Options involve risk and are not suitable for all investors. For more information, please review the Characteristics and Risks of Standardized Options brochure before you begin trading options. Options investors may lose the entire amount of their investment in a relatively short period of time.

Multiple leg options strategies involve additional risks , and may result in complex tax treatments. Please consult a tax professional prior to implementing these strategies.

Implied volatility represents the consensus of the marketplace as to the future level of stock price volatility or the probability of reaching a specific price point. There is no guarantee that the forecasts of implied volatility or the Greeks will be correct. Ally Invest provides self-directed investors with discount brokerage services, and does not make recommendations or offer investment, financial, legal or tax advice.

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Vega for the at-the-money options based on Stock XYZ Obviously, as we go further out in time, there will be more time value built into the option contract. Meet the Greeks What is an Index Option?