# Professional Trading through Option Theory

## 12.1 - The opposite side of the mountain

Do you still remember high school calculus? Do you remember the words integration and differentiation? Back then, the word "Derivatives" meant something different to everyone. It simply meant solving long-term differentiation and integration issues.

I will try to refresh your memories. The idea is to get the point across, not to get into the details of solving a calculus question. The following discussion is relevant to options. Please continue reading.

Take a look at this:

The car is put into motion. It travels 10 minutes from 0 km and reaches the 3 mile mark. The car continues on for 5 minutes to reach the 7 th km mark.

Let's focus on the 3 rd and the 7 th Kilometer.

1. Let's assume 'x" to be distance and the  change in distance be dx.
2. Change in distance i.e. The distance between two points is called 'dx'.
3. Let 't = time and 'dt= change in time
4. Time change i.e. The time of change, i.e.

If we divide dx over dt i.e.we will get 'Velocity' (V) through change in distance over change in time.

V = dx/dt

= 4/5

The car travels 4Kms per 5 minutes. This velocity is expressed in Kms travelled each minute. This is clearly not a convention that we use in everyday conversation. We are more used to expressing speed or velocity in Kms travelled an hour (KMPH).

A simple mathematical adjustment can convert 4/5 KMPH to KMPH.

5 minutes is 5/60 hours when expressed in hours. Plugging this back into the equation above will result in

= 4/ (5/ 60).

= (4*60)./5

= 48 Kmph

The car moves at 48 kmph (kilometers an hour).

Remember that Velocity is the change of distance traveled divided by change in time . The Speed or Velocity in calculus is the distance traveled.

Let's take the following example: After 15 minutes, the car reached the 7 th km in the first leg. Assume that the second nd leg is also completed. The car continues on for 5 more minutes to reach the 15 th km mark.

We know that the speed of the car on the first leg was 48 km/h. However, we can calculate the velocity for the second leg as 96 km/h (here dx = 8, and dt = 5)

It is obvious that the car traveled twice as fast on the 2 and legs of the journey.

We will refer to the change in velocity as "dv".The change in velocity is acceleration,we already know that.

We know that velocity changes are inevitable.

= 96KMPH-48 KMPH

= 48 KMPH/? = 48 KMPH /?

This suggests that 48 KMPH .... is the velocity change. But over what? Are you confused?

Let me explain -

** This explanation may seem like a side-topic from the main topic about Gamma. But it isn't. Please continue reading, if you don't mind it refreshing your high school Physics**.

The first thing a salesman will tell you when you are looking to purchase a car is that it is fast. It can accelerate from 0 to 60 in just 5 seconds. He is basically telling you that the car's velocity can change from 0 to 60 km/h (from a state of complete rest) in just 5 seconds. This is 60KMPH (60-0) in 5 seconds.

Similar to the previous example, we know that the velocity change is 48KMPH. But over what? We would not be able to determine the acceleration if we didn't answer the "over what" question.

We can make some assumptions to determine the acceleration in this case.

1. The rate of acceleration is constant
2. For now, we can ignore the 7th km mark. We also consider the fact the car reached the 15th km mark at the 20th minute.
3. We can use the information above to deduce additional information (these are the 'initial circumstances' in calculus).
• Velocity at the 10 th minute (or the 3 rd km mark) = 0KMPS. This is the initial velocity.
• Time lapsed at the 3 km mark = 10 minutes
• Acceleration is constant between 3 rd, and 15 _ th km mark
• Time at the 15 th Kilometer mark = 20 minutes
• Velocity at 20 the minute or 15 the km marks is known as "Final Velocity".
• We know that the initial velocity was 0 km/h, but we don't know the final velocity.
• Distance total covered = 15 - 3 = 12kms
• Driving time total = 20 -10 = 10 min
• Average speed (velocity), 12/10 = 1.2 km/min or 72 kmph in hours.

Think about it, now we know.

• Initial velocity = 0 km/h
• Average velocity = 72 km/h
• Final velocity = Final velocity =?

We can reverse engineer the equation to determine that the final velocity should be at least 144 Kmph, as 72 is the average speed between 0 and 144.

We also know that acceleration can be calculated as = Final Velocity/time (provided acceleration remains constant).

The acceleration is therefore -

= 144 kmph/10 minutes

Converting 10 minutes to hours is (10/60), plugging this back into the equation

= 144 kmph/ (10/60), hour

= 864 km an hour.

This is the car's speed at 864 km/h. If a salesperson was selling this car, he would say that the car accelerate from 0 to 72kmph in just 5 seconds.

This problem was simplified by assuming that acceleration is constant. Acceleration is not constant. You accelerate at different speeds due to obvious reasons. To calculate problems that involve change in one variable because of the change in another, one would need to dig into derivative calculus. More precisely, one needs to use the concept 'differential equations.

Just think about it for a second.

We know that change in distance traveled (position) = Velocity. This is also known as the 1 st derivative of distance position.

Acceleration = Change in Velocity

Acceleration is the change in velocity over time. This in turn corresponds to the change in position.

It is therefore appropriate to refer to Acceleration as either the 2 nd order derivatives of the position, or the 1 st derivatives of Velocity.

As we move on to the Gamma, keep this in mind: 1 st order dependent and 2 second order derivative.

## 12.2 - Drawing Comparision

We have seen how Delta of an option works over the past few chapters. Delta, as we all know, is the premium that's paid for the given change of the underlying price.

If the Nifty spot price is 8000, then the 8200 CE option can be OTM. Therefore, its delta could range from 0 to 0.5. For the sake of discussion, let's fix it at 0.2

Let's say that the Nifty spot rises 300 points in one day. This means that the 8200 CE becomes slightly ITM.

One thing is clear with this change in the underlying: the Delta itself changes. Delta is a variable whose value changes depending on changes in the premium and the underlying. Delta is very similar in meaning to velocity, whose value changes with time and distance traveled.

Gamma is the measure of the delta change for the change in the underlying. The Gamma of an Option helps answer the question: "For a given change to the underlying, what is the corresponding change to the delta of this option?"

Let's now plug the velocity and acceleration example again and make some parallels with Delta and Gamma.

1st order Derivative

• The velocity measures the change in distance traveled (position) relative to time. It is also known as the 1 first order derivative position.
• Delta captures the change in premium relative to the change in underlying. Therefore, delta is known as the 1 first order derivative premium

2nd order Derivative

• Acceleration is a method that captures velocity change relative to time change. It is also known as the 2 nd order derived of position
• Gamma captures the change in delta with respect to the change in the underlying value. Gamma is therefore called the 2 nd order derivative premium

Calculating the Delta and Gamma values (and all other Option Greeks) requires a lot of number crunching as well as heavy use of calculus (differential equations, stochastic calculus).

Let me tell you something trivial: derivatives are known as derivatives, because they derive their value from the respective underlying.

The derivatives contract's value is calculated using "Derivatives", a mathematical concept. Futures and Options are referred as "Derivatives" because they use the same method to measure their respective underlying.

It may interest you to know that there is a parallel trading universe where traders use derivative calculus to find trading opportunities every day. These traders are commonly called "Quants" in the trading world. It's quite fancy, I have to say. Quantitative trading, on the other side is called 'Markets.

My experience shows that understanding the 2 nd order derivatives such as Gamma can be difficult. However, we will attempt to simplify it in the following chapters.

### Keypoints

1. Because of the dependence on calculus, differential equations and other financial derivatives (also known as Derivatives), they are called Financial derivatives.
2. The delta of an option can change for every change in premium and underlying.
3. Gamma measures the rate at which delta changes. It helps us answer questions such as "What's the expected value for delta given a change in the underlying?"
4. Delta is the 1 first order derivative premium
5. Gamma is 2 nd order derivative of premium

4.