# Three "Greek" Variables in Options*

## Summary

Summary The so-called “Greeks” (referring to their symbols in the Greek alphabet) help predetermine premiums on call and put options (an example is shown here) and modulate portfolio risks. There are currently 14 “Greek” variables, and their use is mainly institutional. However, the three most important ones are of interest to all investors.

* Options are not suitable for all type of investors

If I stand in front of a light, my body creates a shadow behind me. If I move, the shadow follows passively. “Derivative” financial instruments are like the shadow: they have no independent life. Their existence and behaviour are derived exclusively from a real product (for example, equities) of which they are a mathematical projection. Call options and put options are examples: they are the shadows of the stocks they are derived from. Following precise criteria, when a stock price changes over time and when its volatility fluctuates, the option price (premium) also moves. Analysis of these criteria provides measures, known as variables.

A fourth basic factor, the influence of Treasury Bill interest rates on the premium, is of minor importance compared to the first three.

From these factors, others are derived, providing a total of 14 variables. Each of these measurements is highly useful in the financial institutions world. Only three of these are practical for individual investors (see Table 1).

These variables are called “The Greeks” because they are designated by letters from the Greek alphabet (except in one case): this is common in the academic world where these functions were created. These variables are the quantitative expression of the three basic factors mentioned above:

- An option’s sensitivity to changes in its value when the price of the security (generically referred to as the
*support*) changes, is called.*Delta* - An option’s sensitivity to changes in its value over the passage of time is called
.*Theta* - An option’s sensitivity to changes in its value when volatility changes, is called
This is the name of the brightest star in the constellation Lyra. This name has most likely been used in the world of finance because volatility was a rather extraordinary concept at the time options were created, in the 1970s. The Greek letter,*Vega.*or*Kappa*, is used in the academic world.*Tau*

Table 1 provides the quantitative definition of these three factors:

## THE ESSENTIAL “GREEK” VARIABLES

- Delta
- sensitivity of the option premium to the support price
- (a change in the premium when the support price changes by $1)
- Thêta
- sensitivity of the option premium over time
- (a change in the premium each calendar day)
- Véga
- sensitivity of the option premium to volatility
- (a change in the premium when volatility changes by one % point)

Here is an example of using the three variables in determining an option premium:

## Question

On Friday morning, the October/115 call option on ABC (a fictional security) is at $1.45, and the stock is at $114. The option evaluation program gives the following information:

- Delta = $0.44 for each $1 change in the stock price.
- Theta = $0.06 per calendar day.
- Vega = $0.10 for each percentage point variation in volatility.

At approximately what level will the premium be on Monday morning (after three calendar days) if the support price is at $115 and volatility has declined by two percentage points?

## Answer

On Monday morning, the support is up $1.00 compared to Friday: the option premium will have gained $0.44 due to Delta. However, three calendar days have gone by. Each day the option premium loses $0.06 (Theta), multiplied by three, the total loss due to Theta will be $0.18. On Monday morning, volatility is down by two percentage points (for example, decreasing from 23% to 21%). This results in a $0.20 loss due to Vega. On Monday morning, the option premium is thus likely to be $1.51 ($1.45 + $0.44 – $0.18 – $0.20).

## Conclusion

What this example shows is that the three factors do not necessarily move in the same direction: while Delta helped raise the premium, Theta and Vega played against it. The result is that, despite a $1 rise in the support, the premium is up by just $0.06. Naturally, if volatility had instead increased, the premium would have been greater. Furthermore, if the investor had not purchased the option on the eve of the weekend, he would not have lost three times the daily Theta between two consecutive stock market sessions.