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. It works in the following way:
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 while 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 September 2015 the prevailing rate is 7.4769% per annum.
Dividend – The expected dividend per share of the stock is taken into consideration when calculating the Option Greeks if the stock goes ex dividend within the contract’s expiry period. To illustrate, on 11th September an individual would need to calculate the Option Greeks of ICICI Bank option contract, which had a scheduled ex-dividend date of 18th Sept and an expiry date of 24th September 2015. The expected dividend in this example would be Rs.4.
Number of days to expiry – This the number of calendar days left to expiry
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 ICICI Bank 280 CE, its IV is 43.55%.
Let us use this information to calculate the option Greeks for ICICI 280 CE.
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 –
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.
– 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 –
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’.
Realize 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.
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 = Infosys
Strike = 1200
Spot = 1200
Trader A holds = 1200 PE + 1 share of Infy at 1200
Trader B holds = 1200 CE + Cash equivalent to strike i.e 1200
When Infosys’ expiration arrives at 1100, what do you anticipate will transpire?
Trader A earns Rs.100 through his put option, offset by the loss he takes on the stock he has. In the end, he makes a net gain of Rs.1200.
Trader B’s Call option drops to zero, producing a cash equivalent of 1200. His account value subsequently totals at 1200.
Let’s consider a hypothetical example. Suppose Infy reaches 1350 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 stock will rise to 1350.
Trader B’s call value goes to 150 and an additional sum of 1200 in cash, amounting to a total of 1350.
It is evident that this equation holds regardless of where the stock finishes, meaning both trader A and trader B will earn the same amount.
We’ll soon learn how to utilize 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.
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