21.1 -Framework

This module has covered all of the important Option Greeks as well as their applications. Now it is time to learn how to calculate these Greeks with the Black & Scholes Options pricing calculator. The BS options pricing calculator uses the Black and Scholes options pricing modeling model. It was published first by Fisher Black (hence the name Black & Scholes), in 1973. However, Robert C Merton created the model and provided a complete mathematical understanding of the pricing formula.

This pricing model is very well-respected in the financial markets. Myron Scholes and Robert C Merton were awarded the 1997 Noble Prize for Economic Sciences. B&S option pricing models include mathematical concepts such as partial differential and stochastic equations. This module is not intended to teach you the math behind B&S model.

My goal is to guide you through the practical application and pricing of Black & Scholes options pricing.

21.2 -An outlook of the model

The BS calculator is a black box that takes in a lot of inputs and produces a lot of outputs. The options contract market data is the most important input, while the Option Greeks are the outputs.

This is how the pricing framework works:

1. We enter the following information into the model: Spot price and Strike price, Interest rate and Implied volatility, Dividend and Number of days until expiry
2. The pricing model generates the necessary mathematical calculations and produces a variety of outputs.
3. This output includes all Option Greeks as well as the theoretical price for the call or put option for the strike chosen.

This illustration shows the structure of an option calculator.

The input side:

Spot Price - This price is at which the underlying is traded. You can also replace the spot price by the futures price. When an option contract is based upon futures, we use the futures prices. The commodity options and, in certain cases, the currency options are usually based on futures. Always use the spot price for equity option contacts.

Interest rate - This rate is risk-free and prevailing in the economy. This rate is the RBI 91-day Treasury bill rate. The rate can be found on the RBI website.
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The prevailing rate was 7.4769% per year as of September 2015.

Dividend - This is the expected dividend per share in the stock, provided that the stock expires before the end of the expiry period. Let's say that today is 11 September. You want to calculate the Option Greeces for the ICICI Bank option contracts. Let's say that ICICI Bank will pay a dividend on the 18 th September with a Rs.4 dividend. The September series expires on September 24, th September 2015. Therefore, the dividend would be Rs.4. In this instance,

Expiry date - This is the remaining calendar day.

Volatility - Here you can enter the option's implied volatility. To see the implied volatility data, you can always refer to the NSE option chain. Here is an example of ICICI Bank's 280 CE. As you can see, the IV is 43.55%.
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Let's use this information for the calculation of the option Greeks to ICICI 280CE.

• Spot Price = 272.7
• Interest Rate = 7.4769%
• Dividend = 0
• Number of days until expiry = 1. (Today, it is 23 rd September. Expiry is on the 24 th September).
• Volatility = 43.55%

Once you have all this information, you can use it to input the data into a Black & Scholes Options calculator.
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After you have entered the data and clicked on "calculate", the calculator will display the Option Greeks.
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The output side of the equation shows the following:

• Calculated is the premium of 280 CE & 280 PE. This is the theoretical option pricing as calculated by the B&S options calculator. This should be equal to the market option price.
• All the Options Greeks listed below the premium values are available.

Assuming you're familiar with the meanings of each Greek word and how they are applied, I assume that you now know the basics of what each one means.

Last note about option calculators: The option calculator is used primarily to calculate Option Greeks as well as the theoretical option price. Variations in input assumptions can sometimes cause small differences. It is important to allow for modeling errors. The option calculator is generally accurate.

21.3 -Understanding the ' Put Call Parity'

As we discuss Option pricing, it might be a good idea to talk about 'Put Call Parity (PCP) while we are still discussing Option pricing. The PCP equation is a simple mathematical formula that states:

Spot Price + Put Value = Present Value of Strike (invested until maturity) + Call Valu.

If - is used, the equation holds true.

1. Both the Call and Put are available ATM options
2. There are many options available.
3. Both expire simultaneously
4. The options can be held until expiry

5. If you're familiar with the concept Present value, the following equation can be rewritten as:

P + S = (-rt + C)

Where Ke (or-rt) is the present value of strike. K is the strike itself. Strike K is being discounted at a rate of 'r over time't in mathematical terms.

You should also remember that if you have the strike's present value and you keep it until maturity, you will receive the strike's value.

Spot Price + Option to Put + Strike + Call Options

Why should equality be maintained? This is why you need to look at two traders: Trader A, and Trader B.

• ATM Put option held by Trader A and 1 share of the underlying Stock (left side of PCP equation).
• Trader B has a call option and a cash amount equal to the strike (right side of PCP equation).

As such, the PCP should dictate that the money made by both traders (assuming they keep it until expiry) should be equal. Let's look at the equation from a number of numbers.

Infosys = Underlying
Strike = 1200
Spot = 1200

Trader A has 1200 PE and 1 share of Infy at 11200
Trader B holds = 1200 CE + cash equivalent to strike, i.e. 1200

What do you think will happen if Infosys expires after 1100?

The Put option of Trader A becomes profitable. He makes Rs.100, but he loses 100 on his stock. His net payoff is 100 + 1100 = 1200.

Trader B's call option is worthless and the option's value drops to 0. However, he has 1200 cash equivalent so his account value is 0 + 1200 = 1.200.

Let's look at another example. Let's say Infy reaches 1350 after expiry. Let's see what happens to both trader's accounts.

Trader A = Stock goes to 1350/-1, put goes to zero
Trader B = Call Value goes to 150 + 1200 cash = 1350/

It is clear that regardless of the expiration date of the stock, the equations still hold true. This means that both trader A (or B) end up with the same amount of money.

It's all good but how do you use the PCP for a trading strategy development? You will need to wait until the next module, which is devoted to "Option Strategies" J. We have two chapters left to cover before we move on to the next module about Option Strategies.

Keypoints

1. Based on the Black & Scholes model, the options calculator
2. Black & Scholes' model is used to calculate the option's hypothetical price and its Greek.
3. The B&S calculator calculates the risk-free rate that is available on the RBI website.
4. You can find the option chain on the NSE website to calculate implied volatility
5. The put call parity is the sum of the payoffs from a spot and a put option, plus the strike.