The ‘Other’ Greeks: Theta, Vega and Rho
In the last two articles I covered delta and gamma, the two Greeks that most impact your option positions. As I pointed out, the Greeks measure the sensitivity of the option value to the various parameters that determine that value. The next three Greeks to discuss are theta, vega and rho.
These correspond to the remaining parameters that affect option value in the Black-Scholes model. Where delta measures the sensitivity of the option value to changes in the price of the underlying, these measure the sensitivity of the option value to changes in the time to expiration (measured by theta), volatility (measured by vega), and the riskless interest rate (measured by rho).
Theta
Theta is the Greek letter for “t,” which in the context of options stands for time. While delta measures the amount of change in the option value for a unit change in the underlying, theta measures the amount of value the option loses each day. Theta is thus the daily decay in option value at the current time to expiration and underlying price, and the cost of owning the option, which the option owner pays to the option writer in return for the possibility that the option will be worth much more at expiration.
Theta is usually quoted as a daily rate, and often as a negative number, highlighting the fact that the option loses value. The units are usually cents, so that a theta of “-3.5” means that the option loses 3.5 cents in value with each passing day at the present time. Like delta and the other Greeks, theta changes as time passes and also as the price of the underlying changes.
When the time to expiration is great, say four months, the option decay in small, but as time to expiration gets smaller, the rate of decay increases. Suppose a call option has 100 days to expiration, that it has a strike price of $50 and a value of $8, and that the underling has a current price of $53. The intrinsic value of this call option is $3, and the time value of the option is the remaining portion of its value, which is $5. You can estimate theta roughly by observing that the option stands to lose $5 in value over the next 100 days, or about 5 cents per day. This estimate will not be far off, although it will be a little high because of the lower present rate of decay and the higher rate of decay near expiration. This theta of 5 cents tells you that the option value will decay about 5 cents for each passing day.
Vega
The sensitivity of an option’s value to the volatility is commonly called vega. Vega is not a Greek letter and in the academic community you will often see this Greek referred to as “kappa,” which is a Greek letter. I will use vega, as this is the term usually used by traders.
Volatility is usually quoted as an annualized percentage, and vega is usually quoted in cents per 1% increase in this annualized volatility. For example, suppose a stock has annualized volatility of 40% and an option on the stock has vega equal to .20. Then an increase in the volatility to 41% will cause a 20 cent increase in the value of the option. So if the option value at a volatility of 40% is $4.50, then at a volatility of 39% the option will be worth $4.30 and at a volatility of 41% it will be worth $4.70.
Both calls and puts have positive vega and increase in value as volatility increases and, as you might expect, vega for a long term option is higher than the vega for a shorter term option with the same strike price. This is simply because a longer term gives the volatility more of an opportunity to lead the underlying into the option’s in-the-money territory, so the chance that the option will expire in-the-money will be higher and thus the option value will be higher.
Also, an at-the-money option (ATM) will have a greater vega than either an in-the-money option (ITM) or an out-of-the-money (OOTM) option. You can see this intuitively by considering deep OOTMs and deep ITMs. A deep OOTM will still have little chance of expiring in the money even if volatility increases. And a deep ITM option will have nearly 100% chance of expiring in the money and that chance will not change much even if volatility increases.
Rho
Rho is the Greek letter for “r,” which is the letter usually used to represent the interest rate in the Black-Scholes formulation. The standard log normal model for price behavior, which underlies the Black-Scholes formulation, assumes that prices rise on average at a rate equal to the risk-free interest rate. This assumption may be unrealistic in the short term, and it is only approximately correct in the long term, but this model contains it implicitly and it is arguably the “market-neutral” part of the model.
The discussion of rho is a little complicated by the fact that different types of options are affected in different ways by interest rates. For example, foreign currency options require delivery of the currency itself rather than of the futures contract, and therefore are affected by both the foreign interest rate and the interest rate of the currency in which the transaction is conducted, and thus have two rho’s. For options whose settlement is in futures, rho is zero. For stock options, an increase in the interest rate increases the value of a call but decreases the value of a put.
Because rho has the smallest effect on the value of an option, and sometimes no effect at all, many traders do not consider it. Certainly for short term options, small changes in the interest rate do not have much
effect on the value of an option. For leaps, which may expire in two years or even more, rho can be important. A complete discussion of rho and its effect on various options requires a full chapter, and would not be useful at this time.
Conclusion
This concludes the discussion of the basic Greeks. Aside from a few remarks about them, we have covered little more than their definitions, and I will at a later time discuss the relationships between the various Greeks, how you can use the Greeks to build positions, and other important topics. For example, positions that have high gamma also tend to have high vega; you might wish to construct positions that are both delta neutral and gamma neutral (and vega neutral too, perhaps); and so on. The basic Greeks allow you to find ways to protect yourself against virtually any parameter risk, although constructing a totally riskless position is general not possible.
In addition, there are other “Greeks” that impact option values, what I call the “cross Greeks.” For instance, just as delta is sensitive to changes in the price of the underlying, so it also is sensitive to the volatility that you use. And if you use a different volatility, you will get a different delta. And gamma can be quite sensitive to the time to expiration if the strike price is close to the price of the underlying. These situations all require that you look at the sensitivity of one Greek to changes in another.
The cross Greeks are important only in relatively delicate situations, and the ordinary options trader does not need to study them closely, but there are situations where they warn the trader against potential uncomfortable possibilities and a general awareness of these situations is useful.
Also, the Greeks are all part of the standard Black-Scholes option model, which is based upon the assumption that prices are log normally distributed. It is well known that prices are not in fact log normally distributed and that the tails of the actual distributions are heavier than the log normal predicts, meaning that large changes in price are somewhat more common than the log normal thinks they are. This has a definite impact on option values. Unfortunately, the mathematical techniques for evaluating options in this environment are not as straightforward as the mathematics of the Black-Scholes model. I will also discuss ways of dealing with this problem at a later time.