In the last couple of articles (Part 1 and Part 2), I’ve discussed two of the five greeks. The greeks are variables that measure the amount of option price change that is expected to result from changes in the forces that define option values. As a quick review, here are the five greeks:
Greek Name Measures option price change resulting from:
Delta Changes in the current price of the underlying asset
Vega Changes in expectations for future price change in the underlying asset
Theta Drop in option price due to the passage of time
Rho Changes in the risk-free interest rate
Gamma Change in Delta due to changes in the current price of the underlying asset
Last week we finished up Delta and started on Vega. Today we’ll continue with that.
As discussed before, each of the forces measured by the greeks acts separately on every option. So if a Call option has a Delta of .50, an increase of one dollar in the stock’s price will cause an upward push of fifty cents on the option price.
But that doesn’t mean that a one dollar increase guarantees that the option’s price will actually go up by fifty cents. The underlying price effect, measured by Delta, is not working in isolation. Delta’s plus-fifty-cent effect will have to be added to the sum of the effects measured by the other Greeks.
Let’s say that our option with a Delta of .50 has a Vega value of .15. As we discussed last week, that means that a one-percentage-point change in the option market’s expectation for future annualized stock volatility will change the option’s price by fifteen cents. For example, let’s say that yesterday the reading for implied volatility on this option was 30%. This means that the option-buying public, by being willing to pay the price at which this option was trading, were indicating that they expected this stock to move within a 30% annual range. If today they decide that they only expect further movement within a 28% range, then implied volatility has gone down by two percentage points, from 30 to 28. Multiplying that 2-point change by the Vega value of .15, the effect on the option’s price would be a decrease of thirty cents. So the net change would be plus fifty cents (from the Delta effect) minus thirty cents (from the Vega effect), for a running total of plus twenty cents.
Notice that above I said running total, not net change. That is because we’re not done yet.
Between yesterday and today, while the underlying stock was rising by a dollar and implied volatility was falling, another event occurred – the passage of one day’s time. Every option that contains any time value loses some of that time value every day.
Notice the qualification above – every option that contains any time value. This qualification is needed because not every option does. Those that are very deeply in the money and have a Delta of 1.0, have no time value, only a very large amount of intrinsic value. They are effected by underlying price movement only, and by nothing else. On the other hand, options that are very far out of the money, such that they have no value at all, clearly have no time value either. With no time value to lose, they too are unaffected by the passage of time.
But for those options that do have time value, that time value is a wasting asset. Every day that passes means less time for the stock to move into (or further into) the money. So all such options lose some time value every day (including weekends and holidays). We call this loss of value time decay.
The amount of time decay an option will incur for the next one-day passage of time is measured by the greek called Theta. It gives the amount of option price decrease, in dollars per share, for the next day. Notice I did not say per day. The amount of time value lost per day changes as time passes. So if an option’s Theta value is .02 today, it could be .03 tomorrow, and more the day after that, and so on until expiration.
Let’s say for our example, that the option we’re looking at had a Theta value of .02 yesterday. Then we’d expect the value lost due to time decay from yesterday to today to be two cents. Netting this with the fifty-cent increase measured by Delta and the thirty-cent decrease measured by Vega, we now have a running total of +$.50 -$.30 -.02 = + $.18.
Here’s our summary scorecard for Theta:
Greek name: Theta
Force acting on option price: Passage of time
Unit of measure: Pennies per share of option price change for a one-day time increment (multiply by 100 to get the amount of price change per option contract)
Sign: Negative for both puts and calls
Notes: Options with more time value have higher Theta values. For a given expiration date, at-the-money (ATM) options have the highest Theta values. For a given strike price, options with a nearer expiration date have higher Theta values. Theta generally increases each day, unless an option is far in or out of the money.
The next effect we have to consider is from a change in interest rates. This change is measured by the greek called Rho. In brief, higher interest rates make call options more valuable, and put options cheaper. This is because a call option is an alternative to buying stock, while a put option is an alternative to selling stock. If we bought stock, it’s assumed that we would borrow the money to do it. The amount of interest we would pay to borrow the money for the stock purchase is built into the call’s price. The higher the interest cost we’re avoiding by buying calls instead of stock, the more valuable the calls are.
For puts the shoe is on the other foot. If we sold stock, we would have the cash proceeds and could be earning interest on that amount. So the value of the interest income foregone by owning the put instead of selling the stock, is in effect subtracted from the value of the put.
The bottom line is that higher interest rates make calls more expensive while they make puts cheaper. And if the interest rate changes in the course of an option’s life, its price will change too. For an increase in interest rates, call prices go up while put prices go down. For an interest rate decrease, the effects will be reversed.
The interest rate used for these calculations is the risk-free rate, or the rate we could earn on money without taking any risk. In the U.S., the rate used is usually the rate on U.S. Treasury securities that mature at the same time as the option expires. Three-month T-bills for a 3-month option, etc.
The greek called Rho gives the option price change, in dollars per share, for a one percentage point increase in the relevant interest rate. Calls have positive Rho values, while puts have negative Rho values.
The risk-free interest rate does not change very often. But for our example, let’s say that today happened to be the day, and the risk free rate increased by one percent. This would be a huge change, by the way – these rates usually change in much smaller increments.
If our example option had a Rho value of .14, then a one percent change in the rate would have a 14-cent positive effect on our call options.
Taken together, the effects of the changes we have so far are:
Vega: – $.30
Theta: – $.02
As we can now see, the one-dollar change in the underlying stock price does not completely explain the total price change of an option. Understanding this will help us to create option positions that will either take advantage of, or at least not be hurt by, each of the Greeks.
We are not quite finished with the Greeks, but we’re getting there. Next week we’ll include the effects of dividends, talk about gamma, the last of the greeks, and wrap it up.
For questions or comments on this article, contact me at firstname.lastname@example.org.