Mastering Options Trading with the Greek Calculator

Background

Thus far in this module, the Option Greeks and their uses have been reviewed. Now, it’s time to learn how to compute them using the Black & Scholes Options pricing calculator. The BS model was originally fleshed out by Fisher Black & Myron Scholes in 1973, but Robert C Merton gave it a more thorough mathematical backing.

This pricing model is immensely respected in the financial sector, leading Robert C Merton and Myron Scholes to win the 1997 Noble Prize for Economic Sciences. The B&S options pricing module entails mathematical aspects such as partial differential equations, normal distribution, stochastic processes etc., but this tutorial does not guide you in depth through these mathematical concepts—instead you ought to check out this video from Khan Academy for clarification.

Overview of the Model

Consider the Black-Scholes calculator akin to a black box, which necessitates a variety of inputs, mostly made up of market data for an options contract and in exchange produces the Option Greeks as outputs.

This pricing model is based on a certain framework. Here’s how it works:

We used the model with Spot price, Strike price, Interest rate, Implied volatility, Dividend, and Days until maturity as inputs

The pricing model does the necessary mathematical calculations and produces a number of results

The output yields the values for the Option Greeks, as well as the theoretical price of both the call and put option for the specified strike

The diagram depicted below is of a usual options calculator.

On the input side:

Spot price – Spot price and futures price can both be used to determine the value of an option contract. Commodities and, in some cases, currencies may use the futures price whilst equity options contracts always require the spot price.

Interest Rates – The risk-free rate found in the economy is defined by the RBI 91 day Treasury bill rate. This can be accessed directly from the RBI website, presented on the landing page for your convenience.

As of a recent date the prevailing rate is approximately 7.2% per annum.

Dividend – The expected dividend per share of the equity is taken into consideration when calculating the Option Greeks if the equity goes ex dividend within the contract’s expiry period. To illustrate, on 11th September an individual would need to calculate the Option Greeks of HDFC Bank option contract, which had a scheduled ex dividend date of 18th Sept and an expiry date of 24th September. The expected dividend in this example would be Rs 3.50.

Number of days to expiry refers to the remaining number of calendar days until the expiration date of an option or contract.

Volatility. To determine the volatility of an option, you have to look at the option chain provided by NSE and extract the implied volatility data. For instance, in the case of HDFC Bank 1,650 CE, its IV is 38.25%.

Let us use this information to calculate the option Greeks for HDFC Bank 1,650 CE:

Spot Price = 1,625

Interest Rate = 7.2%

Dividend = 0

Number of days to expiry = 1 (today is 23rd of the month, and expiry is on 24th)

Volatility = 38.25%

Once we have this information, we need to feed this into a standard Black & Scholes Options calculator.

After you key in the necessary information on the calculator and press ‘calculate’, it will display the Option Greeks.

On the output side, notice the following:

The premium of 1,650 CE and 1,650 PE is mathematically determined. This theoretical price, as worked out by the B&S options calculator, should ideally equate to the current market value of the option

Below the premium, all the Options Greeks are detailed

I’m guessing you’ve now gained an understanding of what the Greeks are trying to say and how it applies.

A final point on option calculators, they are mainly used to calculate the Option Greeks and the theoretical option price. Differences may appear due to variations in input assumptions, which is why it’s wise to leave some leeway for potential modelling inaccuracies. On the whole however, the calculator is fairly accurate.

For those exploring equity investment opportunities through a stock broker or consulting with a financial advisor, understanding how to use the Black-Scholes calculator proves essential when navigating the stock market. Whether evaluating trading calls or utilising a stock screener to identify opportunities, comprehending how to calculate Option Greeks enables more precise options pricing analysis and better-informed trading decisions based on theoretical valuations versus market prices.

Put Call Parity

Let us consider the subject of Option pricing, and discuss ‘Put Call Parity’ (PCP). This is a straightforward mathematical equation which states:

Put Value + Spot Price = Present value of strike (invested to maturity) + Call Value

The equation above holds true assuming:

Both the Put and Call are ATM options

The options are European

They both expire at the same time

The options are held till expiry

If you know about existing value, the equation can be expressed as follows:

P + S = Ke(-rt) + C

Where, Ke(-rt) represents the present value of strike, with K being the strike itself. In mathematical terms, strike K is getting discounted continuously at rate of ‘r’ over time ‘t’

Realise that if you maintain the present value of the strike until maturity, you will receive the strike’s worth. Therefore, this can also be expressed as follows:

Put Option + Spot Price = Strike + Call options

So why should the equality hold? To help you understand this better think about two traders, Trader A and Trader B:

Trader A holds ATM Put option and 1 share of the underlying equity (left hand side of PCP equation)

Trader B holds a Call option and cash amount equivalent to the strike (right hand side of PCP equation)

This being the case, as per the PCP the amount of money both traders make (assuming they hold till expiry) should be the same. Let us put some numbers to evaluate the equation:

Underlying = Wipro

Strike = 450

Spot = 450

Trader A holds = 450 PE + 1 share of Wipro at 450

Trader B holds = 450 CE + Cash equivalent to strike i.e. 450

When Wipro’s expiration arrives at 410, what do you anticipate will transpire?

Trader A earns Rs 40 through his put option, offset by the loss he takes on the equity he has. In the end, he makes a net gain of Rs 450.

Trader B’s Call option drops to zero, producing a cash equivalent of 450. His account value subsequently totals at 450.

Let’s consider a hypothetical example. Suppose Wipro reaches 500 at expiry; let’s explore the implications for both traders’ positions.

Trader A predicts that the put option will reach zero, and as a result, the equity will rise to 500.

Trader B’s call value goes to 50 and an additional sum of 450 in cash, amounting to a total of 500.

It is evident that this equation holds regardless of where the equity finishes, meaning both trader A and trader B will earn the same amount.

We’ll soon learn how to utilise the PCP to construct a trading strategy. Before we move on to the section dedicated to “Option Strategies” though, we have two remaining chapters left in this part of the course.

For those exploring equity investment opportunities through a stock broker or consulting with a financial advisor, understanding Put Call Parity proves essential when navigating the stock market. Whether evaluating trading calls or utilising a stock screener to identify opportunities, comprehending this fundamental pricing relationship enables identification of arbitrage opportunities and better understanding of the mathematical relationships that govern options pricing.