# Keeping Up With the Greeks

Last week, I discussed delta and gamma, two of the five Greek letters in option trading. Today, let’s continue with the other three: theta, vega, and rho.

## Time Value

Theta measures the rate of decline for the time value of an option.

An option premium—what you pay for the option—consists intrinsic value and time value. Intrinsic value is what the value of the option would be if you exercised it today.

For example, an in-the-money call (ITM) option on stock XYZ with a strike price of \$20 is trading at \$5.70 when XYZ is trading at \$25. In this case the option has an intrinsic value of \$5 (\$25 – \$20). The leftover \$0.70 is the time value.

In the case of an out-of-the-money (OTM) option, the intrinsic value is zero. Therefore, whatever price of that option reflects its time value. You are paying for time remaining to expiration.

## One Way to Go

Therefore, there’s only one way for time value to go—down.

You can verify this yourself. Look at an option chain. For option contracts for the same stock with the same strike price, the longer dated ones are typically priced higher than the shorter dated ones. Any exception is very likely caused by the lack of liquidity or just simply mis-pricing.

As a result, theta is expressed as a negative number to represent the decline in time value. A theta of -0.1 would indicate that at that moment, a passage of one day can be expected to lead to a decline of \$0.10 in the option value (holding everything else constant).

## Theta Changes

Theta is not a static number. If there’s a long time until expiration, the theta is usually quite small. But as expiration approaches, theta will go up. (To be clear, “go up” means the absolute value of the theta increases, such as from -0.1 to -0.5.) Theta for at-the-money (ATM) options typically accelerate the fastest as expiration approaches. This means time value is decaying faster.

Think of it this way, an ATM option has an intrinsic value of zero, so its entire price consists of time value. Thus, as expiration approaches, and the option price falls, everything lost is time value. But for an OTM option, especially if it is very out of the money, it should already be priced close to zero, so there’s not much left to lose.

All things equal, if you are long an option (you are the buyer), a low theta is good for you. But if you are short the option (you are the seller), a high theta is good for you. (Again, “low” and “high” refer to the absolute number.)

## V For Volatility

Vega is a measure of the change in option value for a change in implied volatility (again, holding everything else equal).

Unlike theta, vega is a positive number. An increase in volatility is good for both calls and puts. High volatility is thus good for longs and bad for shorts.

At expiration, it doesn’t matter if an option is OTM by \$1 or \$10, it’s still worth zero. Therefore, if you are long an OTM option, the worst thing for you is if the underlying stock barely moves day to day. In such a case, the probability of the stock moving ITM by expiration is low.

In the case of a volatile stock, even if it moves against you on some days, if the stock moves a lot, the chances of it ending up ITM is higher than that of a stock that doesn’t move much either way. In other words, if you are choosing between buying an option on a marijuana stock or a utility stock, chances are you would have a better shot of making big money on the marijuana stock option.

Vega tends to be highest for options that are either ATM or very near the money and for long-dated options. And if you are going for the home run, all things equal, you will want to seek out options with relatively high vega that will give you the biggest reaction to increases in implied volatility.

## The Role of Interest Rates

Lastly, we have rho.

Rho measures the sensitivity of option value to changes in the risk-free interest rate, typically considered to be the Treasury bill yield.

The rho is positive for long calls and negative for long puts. And the opposite is true for short option positions.

Since options are priced lower than the stocks themselves. Buying a call gives exposure to potential upside and leaves you more cash to earn interest with.

Let’s say XYZ trades for \$25 a share and an ATM call expiring in six months trades for \$2. To buy 100 shares of XYZ, you would need \$2,500. But you would need only \$200 to buy one contract of the call, which represents 100 shares of XYZ. You would still have \$2,300 left over to invest. If you only wanted to put that \$2,300 in a “riskless” investment, you could buy a T-bill at the “risk-free rate.” Thus, the higher the “risk-free rate,” the more money you would earn with that \$2,300.

On the other hand, buying a put keeps money out of your hands. Using the same example above, if you sold or shorted 100 shares of XYZ, you would have \$2,500 to invest. But if you instead bought a put, you actually spent \$200 and have \$200 less to earn interest with.

The higher the risk-free interest, the bigger the rho difference between calls and puts. In practice, however, the impact of interest rate changes on option value is quite small.

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