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Options Greeks Explained Delta Gamma Theta Vega Rho
Most retail option traders in India do not know option Greeks or do not care for them. Option Greeks are very vital part of options trading. If you do not understand them, than it is very important to know about them. At least you should have an idea of what they are. Lets discuss them.
The Five Option Greeks:
1. Delta:
It is the amount an option price will move with every 1 point move in the Index/Stock. If expiry is not near, Delta movement is NOT 1 point increase with 1 point increase in the stock. Which means if the stock moves 1 point up, depending on the strike price of the option, the option will move less than 1.
The reason is that you buy option at a lesser price than the stock in cash for the same lot size, so why should you get profits equal to someone who bought the same stock in cash? Of course option buyer’s losses are also less than stock buyers’.
At the money (ATM) options usually have a delta of 0.5. If the stock moves up 1 point – the price of the ATM option will go up by 0.5. In The Money options have more Delta than out of the money options. Deep In The Money options move almost 1 to 1 with the stock. This is reason why some traders prefer buying deep ITM options. If right ITM options will make more than ATM or OTM options.
As an example. Lets assume Nifty at 8000. The 8000 strike price of calls and puts will have Delta of 0.5. Similarly 8100 CE (OTM) may have a delta of 0.4, 8200 CE (far OTM) may be 0.3, and 7900 CE (ITM) may have a Delta of 0.6. Note how they are changing. Deltas are assumed here not real – but you get the idea.
As expiry nears Delta of all in the money options will move very closely with the stock price as there is no time value left. When expiry is very near Delta of all ITM options move towards 1. Delta of all Out Of The Money (OTM) options will move towards zero. Therefore on expiry day the premium of all Out Of The Money options becomes zero and they expire worthless.
Note: Some traders think that a Delta of 0.1 means the option has 10% chance to expire In The Money. Or 0.5 means the option has a 50% chance of expiring In The Money. You got the idea. This is very important figure for option sellers. I do not have any data to prove this to be true. So please take precaution while selling option even if it has a Delta of 0.1.
2. Gamma:
With the movement of the stock someone has to change the price of Delta as the option moves from ATM to OTM and then back to ATM to ITM.
In the example that we described above, when Nifty moves to 8100 – the 8000 CE becomes In The Money and its Delta increased from 0.5 to 0.6, similarly Delta of 8100 CE increased from 0.4 to 0.5. Gamma is responsible for this change.
Gamma controls the Delta. It is the mathematical formulae (a software) that decides the change in Delta based on a 1 point change in the stock. If Nifty goes back to 8000 – the 8000 strike will again become Delta 0.5.
3. Theta:
This factor is known by most traders. Theta is the Time Factor in the option premium. This time factor moves towards zero as expiration approaches. Theta is the amount the premium will decrease for a oneday change in the time to expiration. Theta works on holidays and nontrading days too. Theta behaves differently for different strike prices.

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One important thing that needs mention. Considering options expiring in 30 days – Theta for deep OTM (Out Of The Money) and Deep ITM (In The Money) options decrease faster in the first 15 days and almost nothing is left for the last 10 days. However ATM options (and the near strike prices) behave exactly opposite. The speed of decrease in Theta is almost constant till the last 5 days – after this the speed increases rapidly. In the last 1 hour it is the maximum.
Compare Theta to the melting of ice. If you take some ice out of your freezer and observe, you will see that for the next few minutes almost nothing happens, then slowly the ice starts melting. After 1015 minutes the speed if ice melting increases. The last 2 minutes are pretty fast when the ice totally melts. Theta behaves the same way especially for ATM options.
Option sellers are the one who love to see the Theta of options decreasing – because this is what makes money for them. Most option sellers sell out of the money options – which means they are only selling Theta. They buy back the options when Theta decreases in value significantly to make a profit.
4. Vega:
Is the volatility factor. Vega is the amount option prices will change for one point change in implied volatility. It is a measure of fear or uncertainty in the markets. When a big news is expected – there is uncertainty in the markets – so the volatility too increases. When volatility increases option prices for both calls and puts also increases. When volatility decreases option prices for both calls and puts also decreases. Vega only effects the time value of the options not its intrinsic value.
For example if Nifty is at 8000. Assuming the 7900 call option is available at a premium of 130, then the intrinsic value of this option is 100 (the difference between the spot price and the strike price of the option). The time value is 30. When Vega increases, only the time value is affected. This 30 can increase to say 32 depending on the volatility increase.
Similarly when the news is out and uncertainty dies down, volatility decreases. This in effect decreases the premium of the options. This is where most trades lose money. Vega has such a big effect that sometimes even if the direction is right, an option buyer loses money because Vega decreased.
This happens mostly with call option buyers. Usually when the markets go up, the volatility decreases. Call option buyers are up against Time (Theta), and Volatility (Vega). OTM call option buyers lose money even if the stock goes up because by the time it goes up a significant portion of the premium would already have been eaten by Theta and the decrease in Vega will also reduce the premium.
For them to make money the speed of the stock going up is very important. However put buyers are in slightly advantageous position because usually when the markets fall, the Vega increases and they can benefit. However they are also up against Theta. Here too speed matters, though not as important when the stock moves up.
In India volatility is called India VIX. NSE (National Stock Exchange) has allowed Vega Trading too. Right now only Futures trading is allowed. But I think too much capital is required for margin. Right now it is only for (HNIs) High Net worth Individuals. India VIX can be found here:
Lower end of VIX (when sellers get less premium): 1015
Average VIX: 1520
High VIX: 20 and above
As an example on 12May2020 when the India VIX hit a high of 39.3 because the election results were due, and spot Nifty closed at 7014 the ATM call was at 294 and the put at 244. Total premium a seller would have got: 294+244 = 538
Today is 13Nov2020 (almost same days left for expiry for the current month). India VIX is 13.80 (64.88% less than 12May2020). Nifty closed at: 8357. Here are closing prices for both 8300 and 8400 strike options.
8300 CE: 122
8300 PE: 42
Total: 122+42 = 164 (69.51% less than total of the ATM options on 12May2020)
8400 CE: 61
8400 PE: 79
Total: 61+79 = 140 (73.97% less than total of the ATM options on 12May2020)
As you can see for the same Theta left, when Vega is down 65%, the option premium also reduces by almost the same amount.
I hope now you understand how important Vega is for option traders.
Why does premium of options increase when Vega increases?
The reason is simple. When there is uncertainty in the markets no one knows exactly where the markets are heading. The risk during these times are more. The risk is much more for the option seller. Why? Because they are willing to take unlimited risk for a limited profit. When the markets are uncertain and the premium they are getting is not sufficient why would they sell an option and take unlimited risk for a small profit?
For example on 12May2020 if the total premium of the ATM strikes was just 150, do you think anyone would have sold these options? If there are no sellers, there can’t be any buyers. And if this happens – options trading will cease to exist. Therefore when Vega increases the option premium also increases to lure the sellers.
5. Rho:
Is the interest rate offered at the banks for a fixed deposit of 1 year. It is the amount an option value will change with one percentagepoint change in interest rates. As mostly the interest rates are same for a long period, Rho does not have a big impact on the option prices. Since interest rates are not important lets not discuss this further.
How you can get the Option Greeks while trading?
Your broker should be able to provide you with an option Greek calculator if you are approved to trade options. You may also look online. You need to put in the values like strike price, time left, interest rates etc and the calculator should return the Option Greeks.
How to trade with the help of Option Greeks?
This is a very big topic. You will find books written on this topic alone. However I will tell you what most retail traders do in the US. By default in the US all brokers show ALL the Greeks on a traders screen. They need not use any calculator. Unfortunately in India there is not a single broker that shows them by default. If you know any please mail me or write in the comments.
If they want to sell options, they sell option that has Delta of less than 0.15. Which means they sell deep Out Of The Money options. The idea behind this trade is that the chance of this option to expire worthless is 85%. (10.15 = 0.85 or 85%)
If they want to buy, they buy options that have delta of 0.5 or more. Which means they buy At The Money or slightly In The Money Options.
Some traders sell both call and put options – mostly out of the money options. These traders try to keep their trades at Delta Neutral. Again this is a very big topic out of scope of this article. Still I will try to explain in a simple way.
Delta Neutral means keeping Deltas of all the options they sold same with any movement of the stock.
For example if stock moves up – the Delta of the call option will increase. So they will move the position a few points up. Essentially booking profits in the puts and booking loss in the calls – but maintaining the Deltas of the calls and puts more or less same. The idea is to profit when expiry nears. The Theta will decrease rapidly of both the calls and puts when expiry nears. Ultimately both the options will expire worthless.
Profits or losses are known only on the expiry day, or when the trade is closed. If all the delta neutral adjustments resulted in profits then the trader is in profit, else he loses.
Some people write software to automate this trading. Yes you can write a piece of software to automate your trading and strategies. Benefit is that emotions are not involved. I have rarely come across a person who made money doing automated trading.
Frankly it is very challenging to keep a trade Delta neutral. You need to keep a watch on the stock every time and be ready to shift your trades with few points movement in the stock. Lots of trading is involved therefore lot of brokerage also needs to be paid. Profits or not, your broker will thank you for being a Delta Neutral trader. ��
If you are on a job or a busy person it is humanly impossible to keep changing your trades to be Delta Neutral. And there is no guarantee of profits either. We don’t trade to waste time and lose money too right? If you are busy and don’t want to trade too much and are happy with small profits month after month I recommend my course. Its much better than wasting money on losing trades that teach you nothing.
This came out to be a pretty long post, still this is tip of an iceberg as far as Option Greeks are concerned. Though, I hope after reading this article you will have some idea about them.
Do ask any questions on them in the comments section. Do you look at Option Greeks before taking your decision on option trading?
More information on Option Greeks can be found here:
Option Greeks
What are Option Greeks?
Option Greeks are financial measures of the sensitivity of an option’s price to its underlying determining parameters, such as volatility or the price of the underlying asset. The Greeks are utilized in the analysis of an options portfolio and in sensitivity analysis What is Sensitivity Analysis? Sensitivity Analysis is a tool used in financial modeling to analyze how the different values for a set of independent variables affect a dependent variable of an option or portfolio of options Options: Calls and Puts An option is a form of derivative contract which gives the holder the right, but not the obligation, to buy or sell an asset by a certain date (expiration date) at a specified price (strike price). There are two types of options: calls and puts. US options can be exercised at any time . The measures are considered essential by many investors for making informed decisions in options trading.
Delta, Gamma, Vega, Theta, and Rho are the key option Greeks. However, there are many other option Greeks that can be derived from those mentioned above.
Name  Dependent Variable  Independent Variable 

Delta  Option price  Value of underlying asset 
Gamma  Delta  Value of underlying asset 
Vega  Option price  Volatility 
Theta  Option price  Time to maturity 
Rho  Option price  Interest rate 
Option Greek Delta
Delta (Δ) is a measure of the sensitivity of an option’s price changes relative to the changes in the underlying asset’s price. In other words, if the price of the underlying asset Types of Assets Common types of assets include: current, noncurrent, physical, intangible, operating and nonoperating. Correctly identifying and classifying assets is critical to the survival of a company, specifically its solvency and risk. An asset is a resource, controlled by a company, with future economic benefits. increases by $1, the price of the option will change by Δ amount. Mathematically, the delta is found by:
 ∂ – the first derivative
 V – the option’s price (theoretical value)
 S – the underlying asset’s price
Keep in mind that we take the first derivative Futures and Forwards Future and forward contracts (more commonly referred to as futures and forwards) are contracts that are used by businesses and investors to hedge against risks or speculate. Futures and forwards are examples of derivative assets that derive their values from underlying assets. of the option and underlying asset prices because the derivative is a measure of the rate change of the variable at a given period.
The delta is usually calculated as a decimal number from 1 to 1. Call options Call Option A call option, commonly referred to as a “call,” is a form of a derivatives contract that gives the call option buyer the right, but not the obligation, to buy a stock or other financial instrument at a specific price – the strike price of the option – within a specified time frame. can have a delta from 0 to 1, while puts have a delta from 1 to 0. The closer the option’s delta to 1 or 1, the deeper inthemoney is the option.
The delta of an option’s portfolio is the weighted average of the deltas of all options in the portfolio.
Delta is also known as a hedge ratio. If a trader knows the delta of the option, he can hedge his position by buying or shorting the number of underlying assets multiplied by delta.
Gamma
Gamma (Γ) is a measure of the delta’s change relative to the changes in the price of the underlying asset. If the price of the underlying asset increases by $1, the option’s delta will change by the gamma amount. The main application of gamma is the assessment of the option’s delta.
Long options have a positive gamma. An option has a maximum gamma when it is atthemoney (option strike price equals the price of the underlying asset). However, gamma decreases when an option is deepinthemoney or outthemoney.
Option Greek Vega
Vega (ν) is an option Greek that measures the sensitivity of an option price relative to the volatility of the underlying asset. If the volatility of the underlying asses increases by 1%, the option price will change by the vega amount.
 ∂ – the first derivative
 V – the option’s price (theoretical value)
 σ – the volatility of the underlying asset
The vega is expressed as a money amount rather than as a decimal number. An increase in vega generally corresponds to an increase in the option value (both calls and puts).
Theta
Theta (θ) is a measure of the sensitivity of the option price relative to the option’s time to maturity. If the option’s time to maturity decreases by one day, the option’s price will change by the theta amount. The Theta option Greek is also referred to as time decay.
 ∂ – the first derivative
 V – the option’s price (theoretical value)
 τ – the option’s time to maturity
In most cases, theta is negative for options. However, it may be positive for some European options. Theta shows the most negative amount when the option is atthemoney.
Rho (ρ) measures the sensitivity of the option price relative to interest rates. If a benchmark interest rate increases by 1%, the option price will change by the rho amount. The rho is considered the least significant among other option Greeks because option prices are generally less sensitive to interest rate changes than to changes in other parameters.
 ∂ – the first derivative
 V – the option’s price (theoretical value)
 r – interest rate
Generally, call options have a positive rho, while the rho for put options is negative.
Basics Of Options Trading Explained
Before we delve deep into the world of options trading, let’s take a moment to understand why do we need options at all. If you are thinking it is just another way to make money and was created by some fancy guys in suits working in Wall Street, well, you are wrong. The options world predates the modern stock exchanges by a large margin.
While some credit the Samurai for giving us the foundation on which options contracts were based, some actually acknowledge the Greeks for giving us an idea on how to speculate on a commodity, in this case, the harvest of olives. In both cases, humans were trying to guess the price of a food item and trade accordingly (rice in the case of samurais), long before the modern world put in various rules and set up exchanges.
With this in mind, let us try to answer the first question in your mind.
What is options trading?
Let’s take a very simple example to understand options trading. Consider that you are buying a stock for Rs. 3000. But the broker tells you about an exciting offer, that you can buy it now for Rs. 3000 or you can give a token amount of Rs. 30 and reserve the right to buy it at Rs. 3000 after a month, even if the stock increases in value at that time. But that token amount is nonrefundable!
You realise that there is a high chance that the stock would cross Rs. 3030 and thus, you can breakeven at least. Since you have to pay only Rs. 30 now, the remaining amount can be used elsewhere for a month. You wait for a month and then look at the stock price.
Now, depending on the stock price, you have the option to buy the stock from the broker or not. Of course, this is an oversimplification but this is options trading in a gist.In the world of trading, options are instruments that belong to the derivatives family, which means its price is derived from something else, mostly stocks. The price of an option is intrinsically linked to the price of the underlying stock.
A formal definition is given below:
A stock option is a contract between two parties in which the stock option buyer (holder) purchases the right (but not the obligation) to buy/sell shares of an underlying stock at a predetermined price from/to the option seller (writer) within a fixed period of time.
We are going to make sure that by the end of this article you are well versed with the options trading world along with trying out a few options trading strategies as well. We will cover the following points in this article. If you feel that you want to skip the basics of options, then head straight to the options trading strategies.
Let’s start now, shall we!
Options trading vs. Stock trading
There must be a doubt in your mind that why do we even have options trading if it is just another way of trading. Well, here are a few points which make it different from trading stocks
 The Options contract has an expiration date, unlike stocks. The expiration can vary from weeks, months to years depending upon the regulations and the type of Options that you are practising. Stocks, on the other hand, do not have an expiration date.
 Unlike Stocks, Options derive their value from something else and that’s why they fall under the derivatives category
 Options are not definite by numbers like Stocks
 Options owners have no right (voting or dividend) in a company unlike Stock owners
It is quite often that some people find the Option’s concept difficult to understand though they have already followed it in their other transactions, for e.g. car insurance or mortgages. In this part of the article, we will take you through some of the most important aspects of Options trading before we get down to the world of options trading.
Options terminologies
Strike Price
The Strike Price is the price at which the underlying stocks can be bought or sold as per the contract. In options trading, the Strike Price for a Call Option indicates the price at which the Stock can be bought (on or before its expiration) and for Put Options trading it refers to the price at which the seller can exercise its right to sell the underlying stocks (on or before its expiration)
Premium
Since the Options themselves don’t have an underlying value, the Options premium is the price that you have to pay in order to purchase an Option. The premium is determined by multiple factors including the underlying stock price, volatility in the market and the days until the Option’s expiration. In options trading, choosing the premium is one of the most important components.
Underlying Asset
In options trading, the underlying asset can be stocks, futures, index, commodity or currency. The price of Options is derived from its underlying asset. For the purpose of this article, we will be considering the underlying asset as the stock. The Option of stock gives the right to buy or sell the stock at a specific price and date to the holder. Hence its all about the underlying asset or stocks when it comes to Stock in Options Trading.
Expiration Date
In options trading, all stock options have an expiration date. The expiration date is also the last date on which the Options holder can exercise the right to buy or sell the Options that are in holding. In Options Trading, the expiration of Options can vary from weeks to months to years depending upon the market and the regulations.
Options Style
There are two major types of Options that are practised in most of the options trading markets.
 American Options which can be exercised anytime before its expiration date
 European Options which can only be exercised on the day of its expiration
Moneyness (ITM, OTM & ATM)
It is very important to understand the Options Moneyness before you start trading in Stock Options. A lot of options trading strategies are played around the Moneyness of an Option.
It basically defines the relationship between the strike price of an Option and the current price of the underlying Stocks. We will examine each term in detail below.
When is an Option inthemoney?
 Call Option – when the underlying stock price is higher than the strike price
 Put Option – when the underlying stock price is lower than the strike price
When is an Option outofthemoney?
 Call Option – when the underlying stock price is lower than the strike price
 Put Option – when the underlying stock price is higher than the strike price
When is an Option atthemoney?
 When the underlying stock price is equal to the strike price.
Take a break here to ponder over the different terms as we will find it extremely useful later when we go through the types of options as well as a few options trading strategies.
Type of options
In the true sense, there are only two types of Options i.e Call & Put Options. We will understand them in more detail.
To Call or Put
A Call Option is an option to buy an underlying Stock on or before its expiration date. At the time of buying a Call Option, you pay a certain amount of premium to the seller which grants you the right (but not the obligation) to buy the underlying stock at a specified price (strike price).
Purchasing a call option means that you are bullish about the market and hoping that the price of the underlying stock may go up. In order for you to make a profit, the price of the stock should go higher than the strike price plus the premium of the call option that you have purchased before or at the time of its expiration.
In contrast, a Put Option is an option to sell an underlying Stock on or before its expiration date. Purchasing a Put Option means that you are bearish about the market and hoping that the price of the underlying stock may go down. In order for you to make a profit, the price of the stock should go down from the strike price plus the premium of the Put Option that you have purchased before or at the time of its expiration.
In this manner, both Put and Call option buyer’s loss is limited to the premium paid but profit is unlimited. The above explanations were from the buyer’s point of view. We will now understand the putcall options from the seller’s point of view, ie options writers. The Put option seller, in return for the premium charged, is obligated to buy the underlying asset at the strike price.
Similarly, the Call option seller, in return for the premium charged, is obligated to sell the underlying asset at the strike price. Is there a way to visualise the potential profit/loss of an option buyer or seller? Actually, there is. An option payoff diagram is a graphical representation of the net Profit/Loss made by the option buyers and sellers.
Before we go through the diagrams, let’s understand what the four terms mean. As we know that going short means selling and going long means buying the asset, the same principle applies to options. Keeping this in mind, we will go through the four terms.
 Short call – Here we are betting that the prices will fall and hence, a short call means you are selling calls.
 Short put – Here the short put means we are selling a put option
 Long call – it means that we are buying a call option since we are optimistic about the underlying asset’s share price
 Long put – Here we are buying a put option.
S = Underlying Price
X = Strike Price
Breakeven point is that point at which you make no profit or no loss.
The long call holder makes a profit equal to the stock price at expiration minus strike price minus premium if the option is in the money. Call option holder makes a loss equal to the amount of premium if the option expires out of money and the writer of the option makes a flat profit equal to the option premium.
Similarly, for the put option buyer, profit is made when the option is in the money and is equal to the strike price minus the stock price at expiration minus premium. And, the put writer makes a profit equal to the premium for the option.
All right, until now we have been going through a lot of theory. Let’s switch gears for a minute and come to the real world. How do options look like? Well, let’s find out.
What does an options trading quote consist of?
If you were to look for an options quote on Apple stock, it would look something like this:
When this was recorded, the stock price of Apple Inc. was $196. Now let’s take one line from the list and break it down further.
Eg.
In a typical options chain, you will have a list of call and put options with different strike prices and corresponding premiums. The call option details are on the left and the put option details are on the right with the strike price in the middle.
 The symbol and option number is the first column.
 The “last” column signifies the amount at which the last time the option was bought.
 “Change” indicates the variance between the last two trades of the said options
 “Bid” column indicates the bid submitted for the option.
 “Ask” indicates the asking price sought by the option seller.
 “Volume” indicates the number of options traded. Here the volume is 0.
 “Open Interest” indicates the number of options which can be bought for that strike price.
The columns are the same for the put options as well. In some cases, the data provider signifies whether the option is in the money, at the money or out of money as well. Of course, we need an example to really help our understanding of options trading. Thus, let’s go through one now.
Options Trading Example
We will go through two cases to better understand the call and put options.
For simplicity’s sake, let us assume the following:
 Price of Stock when the option is written: $100
 Premium: $5
 Expiration date: 1 month after the option is bought
Case 1:
The current price of stock: $110. Strike price: $120
Case 2:
The current price of stock: $120. Strike price: $110
Considering that we have gone through the detailed scenario of each option, how about we combine a few options together. Let’s understand an important concept which many professionals use in options trading.
What is PutCall Parity In Python?
Putcall parity is a concept that anyone who is interested in options trading needs to understand. By gaining an understanding of putcall parity you can understand how the value of call option, put option and the stock are related to each other. This enables you to create other synthetic position using various option and stock combination.
The principle of putcall parity
Putcall parity principle defines the relationship between the price of a European Put option and European Call option, both having the same underlying asset, strike price and expiration date. If there is a deviation from putcall parity, then it would result in an arbitrage opportunity. Traders would take advantage of this opportunity to make riskless profits till the time the putcall parity is established again.
The putcall parity principle can be used to validate an option pricing model. If the option prices as computed by the model violate the putcall parity rule, such a model can be considered to be incorrect.
Understanding PutCall Parity
To understand putcall parity, consider a portfolio “A” comprising of a call option and cash. The amount of cash held equals the call strike price. Consider another portfolio “B” comprising of a put option and the underlying asset.
S0 is the initial price of the underlying asset and ST is its price at expiration.
Let “r” be the riskfree rate and “T” be the time for expiration.
In time “T” the Zerocoupon bond will be worth K (strike price) given the riskfree rate of “r”.
Portfolio A = Call option + Zerocoupon bond
Portfolio B = Put option + Underlying Asset
If the share price is higher than X the call option will be exercised. Else, cash will be retained. Hence, at “T” portfolio A’s worth will be given by max(ST, X).
If the share price is lower than X, the put option will be exercised. Else, the underlying asset will be retained. Hence, at “T”, portfolio B’s worth will be given by max (ST, X).
If the two portfolios are equal at time, “T”, then they should be equal at any time. This gives us the putcall parity equation.
Equation for putcall parity:
C + XerT = P + S0
In this equation,
 C is the premium on European Call Option
 P is the premium of European Put Option
 S0 is the spot price of the underlying stock
 And, XerT is the current value (discounted value) of Zerocoupon bond (X)
We can summarize the payoffs of both the portfolios under different conditions as shown in the table below.
From the above table, we can see that under both scenarios, the payoffs from both the portfolios are equal.
Required Conditions For Putcall Parity
For putcall parity to hold, the following conditions should be met. However, in the real world, they hardly hold true and putcall parity equation may need some modifications accordingly. For the purpose of this blog, we have assumed that these conditions are met.
 The underlying stock doesn’t pay any dividend during the life of the European options
 There are no transaction costs
 There are no taxes
 Shorting is allowed and there are no borrow charges
Hence, putcall parity will hold in a frictionless market with the underlying stock paying no dividends.
Arbitrage Opportunity
In options trading, when the putcall parity principle gets violated, traders will try to take advantage of the arbitrage opportunity. An arbitrage trader will go long on the undervalued portfolio and short the overvalued portfolio to make a riskfree profit.
How to take advantage of arbitrage opportunity
Let us now consider an example with some numbers to see how trade can take advantage of arbitrage opportunities. Let’s assume that the spot price of a stock is $31, the riskfree interest rate is 10% per annum, the premium on threemonth European call and put are $3 and $2.25 respectively and the exercise price is $30.
In this case, the value of portfolio A will be,
C+XerT = 3+30e0.1 * 3/12 = $32.26
The value of portfolio B will be,
P + S0 = 2.25 + 31 = $33.25
Portfolio B is overvalued and hence an arbitrageur can earn by going long on portfolio A and short on portfolio B. The following steps can be followed to earn arbitrage profits.
 Short the stock. This will generate a cash inflow of $31.
 Short the put option. This will generate a cash inflow of $2.25.
 Purchase the call option. This will generate cash outflow of $3.
 Total cash inflow is 3 + 2.25 + 31 = $30.25.
 Invest $30.25 in a zerocoupon bond with 3 months maturity with a yield of 10% per annum.
Return from the zero coupon bond after three months will be 30.25e 0.1 * 3/12 = $31.02.
If the stock price at maturity is above $30, the call option will be exercised and if the stock price is less than $30, the put option will be exercised. In both the scenarios, the arbitrageur will buy one stock at $30. This stock will be used to cover the short.
Total profit from the arbitrage = $31.02 – $30 = $1.02
Well, shouldn’t we look at some codes now?
Python Codes Used For Plotting The Charts
The below code can be used to plot the payoffs of the portfolios.
So far, we have gone through the basic concepts in options trading and looked at an options trading strategy as well. At this juncture, let’s look at the world of options trading and try to answer a simple question.
Why is Options Trading attractive?
Options are attractive instruments to trade in because of the higher returns. An option gives the right to the holder to do something, with the ‘option’ of not to exercise that right. This way, the holder can restrict his losses and multiply his returns.
While it is true that one options contract is for 100 shares, it is thus less risky to pay the premium and not risk the total amount which would have to be used if we had bought the shares instead. Thus your risk exposure is significantly reduced.However, in reality, options trading is very complex and that is because options pricing models are quite mathematical and complex.
So, how can you evaluate if the option is really worth buying? Let’s find out.
The key requirement in successful options trading strategies involves understanding and implementing options pricing models. In this section, we will get a brief understanding of Greeks in options which will help in creating and understanding the pricing models.
Options Pricing
Options Pricing is based on two types of values
Intrinsic Value of an option
Recall the moneyness concept that we had gone through a few sections ago. When the call option stock price is above the strike price or when put option stock price is below the strike price, the option is said to be “InTheMoney (ITM)”, i.e. it has an intrinsic value. On the other hand, “Out of the money (OTM)” options have no intrinsic value. For OTM call options, the stock price is below the strike price and for OTM put options; stock price is above the strike price. The price of these options consists entirely of time value.
Time Value of an option
If you subtract the amount of intrinsic value from an options price, you’re left with the time value. It is based on the time to expiration. You can enroll for this free online options trading python course on Quantra and understand basic terminologies and concepts that will help you in options trading. We know what is intrinsic and the time value of an option. We even looked at the moneyness of an option. But how do we know that one option is better than the other, and how to measure the changes in option pricing. Well, let’s take the help of the greeks then.
Options Greeks
Greeks are the risk measures associated with various positions in options trading. The common ones are delta, gamma, theta and vega. With the change in prices or volatility of the underlying stock, you need to know how your options pricing would be affected. Greeks in options help us understand how the various factors such as prices, time to expiry, volatility affect the options pricing.
Delta measures the sensitivity of an option’s price to a change in the price of the underlying stock. Simply put, delta is that options greek which tells you how much money a stock option will rise or drop in value with a $1 rise or drop in the underlying stock. Delta is dependent on underlying price, time to expiry and volatility. While the formula for calculating delta is on the basis of the BlackScholes option pricing model, we can write it simply as,
Delta = [Expected change in Premium] / [Change in the price of the underlying stock]
Let’s understand this with an example for a call option:
We will create a table of historical prices to use as sample data. Let’s assume that the option will expire on 5th March and the strike price agreed upon is $140.
Thus, if we had to calculate the delta for the option on 2nd March, it would be $5/$10 = 0.5.
Here, we should add that since an option derives its value from the underlying stock, the delta option value will be between 0 and 1. Usually, the delta options creeps towards 1 as the option moves towards “inthemoney”.
While the delta for a call option increases as the price increases, it is the inverse for a put option. Think about it, as the stock price approaches the strike price, the value of the option would decrease. Thus, the delta put option is always ranging between 0 and 1.
Gamma measures the exposure of the options delta to the movement of the underlying stock price. Just like delta is the rate of change of options price with respect to underlying stock’s price; gamma is the rate of change of delta with respect to underlying stock’s price. Hence, gamma is called the secondorder derivative.
Gamma = [Change in an options delta] / [Unit change in price of underlying asset]
Let’s see an example of how delta changes with respect to Gamma. Consider a call option of stock at a strike price of $300 for a premium of $15.
 Strike price: $300
 Initial Stock price: $150
 Delta: 0.2
 Gamma: 0.005
 Premium: $15
 New stock price: $180
 Change in stock price: $180 – $150 = $30
Thus, Change in Premium = Delta * Change in price of stock = 0.2 * 30 = 6.
Thus, new premium = $15 + $6 = $21
Change in delta = Gamma * Change in stock price = 0.005 * 30 = 0.15
Thus, new delta = 0.2 + 0.15 = 0.35.
Let us take things a step further and assume the stock price increases another 30 points, to $210.
Now,
New stock price: $210
Change in stock price: $210 – $180 = $30
Change in premium = Delta *Change in 0.35*30 = $10.5
Thus, new premium = $21 + $10.5 = $31.5
Change in delta = Gamma * Change in stock price = 0.005 * 30 = 0.15
Thus, new delta = 0.35 + 0.15 = 0.5.
In this way, delta and gamma of an option changes with the change in the stock price. We should note that Gamma is the highest for a stock call option when the delta of an option is at the money. Since a slight change in the underlying stock leads to a dramatic increase in the delta. Similarly, the gamma is low for options which are either out of the money or in the money as the delta of stock changes marginally with changes in the stock option.
Highest Gamma for Atthemoney (ATM) option
Among the three instruments, atthemoney (ATM), outofthemoney (OTM) and inthemoney (ITM); at the money (ATM) has the highest gamma. You can watch this video to understand it in more detail.
Theta measures the exposure of the options price to the passage of time. It measures the rate at which options price, especially in terms of the time value, changes or decreases as the time to expiry is approached.
Vega measures the exposure of the option price to changes in the volatility of the underlying. Generally, options are more expensive for higher volatility. So, if the volatility goes up, the price of the option might go up to and viceversa.
Vega increases or decreases with respect to the time to expiry?
What do you think? You can confirm your answer by watching this video.
One of the popular options pricing model is Black Scholes, which helps us to understand the options greeks of an option.
BlackScholes options pricing model
The formula for the BlackScholes options pricing model is given as:
C is the price of the call option
P represents the price of a put option.
S0 is the underlying price,
X is the strike price,
σ represents volatility,
r is the continuously compounded riskfree interest rate,
t is the time to expiration, and
q is the continuously compounded dividend yield.
N(x) is the standard normal cumulative distribution function.
The formulas for d1 and d2 are given as:
To calculate the Greeks in options we use the BlackScholes options pricing model.
Delta and Gamma are calculated as:
In the example below, we have used the determinants of the BS model to compute the Greeks in options.
At an underlying price of 1615.45, the price of a call option is 21.6332.
If we were to increase the price of the underlying by Rs. 1, the change in the price of the call, put and values of the Greeks in the option is as given below.
As can be observed, the Delta of the call option in the first table was 0.5579. Hence, given the definition of the delta, we can expect the price of the call option to increase approximately by this value when the price of the underlying increases by Rs.1. The new price of the call option is 22.1954 which is
22.1954
22.1911
Let’s move to Gamma, another Greek in option.
If you observe the value of Gamma in both the tables, it is the same for both call and put options contracts since it has the same formula for both options types. If you are going long on the options, then you would prefer having a higher gamma and if you are short, then you would be looking for a low gamma. Thus, if an options trader is having a netlong options position then he will aim to maximize the gamma, whereas in case of a netshort position he will try to minimize the gamma value.
The third Greek, Theta has different formulas for both call and put options. These are given below:
In the first table on the LHS, there are 30 days remaining for the options contract to expire. We have a negative theta value of 0.4975 for a long call option position which means that the options trader is running against time.
He has to be sure about his analysis in order to profit from trade as time decay will affect this position. This impact of time decay is evident in the table on the RHS where the time left to expiry is now 21 days with other factors remaining the same. As a result, the value of the call option has fallen from 21.6332 to 16.9 behaviour 319. If an options trader wants to profit from the time decay property, he can sell options instead of going long which will result in a positive theta.
We have just discussed how some of the individual Greeks in options impact option pricing. However, it is very essential to understand the combined behaviour of Greeks in an options position to truly profit from your options position. If you want to work on options greeks in Excel, you can refer to this blog.
Let us now look at a Python package which is used to implement the Black Scholes Model.
Python Library – Mibian
What is Mibian?
Mibian is an options pricing Python library implementing the BlackScholes along with a couple other models for European options on currencies and stocks. In the context of this article, we are going to look at the BlackScholes part of this library. Mibian is compatible with python 2.7 and 3.x. This library requires scipy to work properly.
How to use Mibian for BS Model?
The function which builds the BlackScholes model in this library is the BS() function. The syntax for this function is as follows:
The first input is a list containing the underlying price, strike price, interest rate and days to expiration. This list has to be specified each time the function is being called. Next, we input the volatility, if we are interested in computing the price of options and the option greeks. The BS function will only contain two arguments.
If we are interested in computing the implied volatility, we will not input the volatility but instead will input either the call price or the put price. In case we are interested in computing the putcall parity, we will enter both the put price and call price after the list. The value returned would be:
(call price + price of the bond worth the strike price at maturity) – (put price + underlying asset price)
The syntax for returning the various desired outputs are mentioned below along with the usage of the BS function. The syntax for BS function with the input as volatility along with the list storing underlying price, strike price, interest rate and days to expiration:
Attributes of the returned value from the abovementioned BS function:
The syntax for BS function with the input as callPrice along with the list storing underlying price, strike price, interest rate and days to expiration:
Attributes of the returned value from the abovementioned BS function:
The syntax for BS function with the input as putPrice along with the list storing underlying price, strike price, interest rate and days to expiration:
Attributes of the returned value from the abovementioned BS function:
The syntax for BS function with the inputs as callPrice and putPrice along with the list storing underlying price, strike price, interest rate and days to expiration:
Attributes of the returned value from the abovementioned BS function:
While BlackScholes is a relatively robust model, one of its shortcomings is its inability to predict the volatility smile. We will learn more about this as we move to the next pricing model.
Derman Kani Model
The Derman Kani model was developed to overcome the longstanding issue with the Black Scholes model, which is the volatility smile. One of the underlying assumptions of Black Scholes model is that the underlying follows a random walk with constant volatility. However, on calculating the implied volatility for different strikes, it is seen that the volatility curve is not a constant straight line as we would expect, but instead has the shape of a smile. This curve of implied volatility against the strike price is known as the volatility smile.
If the Black Scholes model is correct, it would mean that the underlying follows a lognormal distribution and the implied volatility curve would have been flat, but a volatility smile indicates that traders are implicitly attributing a unique nonlognormal distribution to the underlying. This nonlognormal distribution can be attributed to the underlying following a modified random walk, in the sense that the volatility is not constant and changes with both stock price and time. In order to correctly value the options, we would need to know the exact form of the modified random walk.
The Derman Kani model shows how to take the implied volatilities as inputs to deduce the form of the underlying’s random walk. More specifically a unique binomial tree is extracted from the smile corresponding to the random walk of the underlying, this tree is called the implied tree. This tree can be used to value other derivatives whose prices are not readily available from the market – for example, it can be used in standard but illiquid European options, American options, and exotic options.
What is the Heston Model?
Steven Heston provided a closedform solution for the price of a European call option on an asset with stochastic volatility. This model was also developed to take into consideration the volatility smile, which could not be explained using the Black Scholes model.
The basic assumption of the Heston model is that volatility is a random variable. Therefore there are two random variables, one for the underlying and one for the volatility. Generally, when the variance of the underlying has been made stochastic, closedform solutions will no longer exist.
But this is a major advantage of the Heston model, that closedform solutions do exist for European plain vanilla options. This feature also makes calibration of the model feasible. If you are interested in learning about these models in more detail, you may go through the following research papers,
 Derman Kani Model – “The Volatility Smile and Its Implied Tree” by Emanuel Derman and Iraj Kani.
 Heston Model – “A ClosedForm Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options”
So far, you have understood options trading and how to analyse an option as well as the pricing models used. Now, to apply this knowledge, you will need access to the markets, and this is where the role of a broker comes in.
Opening an options trading account
How to choose a broker for Options Trading?
Before we open an options trading account with a broker, let’s go over a few points to take into consideration when we choose a broker.
 Understand your aim when you tread the options trading waters, whether it is a way of hedging risk, as a speculative instrument, for income generation etc.
 Does the broker provide option evaluation tools of their own? It is always beneficial to have access to a plethora of tools when you are selecting the right option.
 Enquire the transaction costs or the commission charged by the broker as this will eat into your investment gains.
 Some brokers give access to research materials in various areas of the stock market. You can always check with the broker about access to research as well as subscriptions etc.
 Check the payment options provided by the broker to make sure it is compatible with your convenience.
Searching for the right broker
Once the required background research is done, you can choose the right broker as per your need and convenience. In the global market, a list of the top brokers is provided below:
List of Top International Brokers (Options Trading)
The list of top international options brokers is given below:
 Etrade ($0.65 per options contract)
 Ally Invest ($0.5 per contract traded)
 TD Ameritrade ($0.65 fee per contract)
 Interactive Brokers (starts at $0.25 per options contract)
 Schwab Brokerage ($0.65 per options contract)
List of Top Indian Brokers (Options Trading)
The list of top Indian options brokers is given below:
 Zerodha
 ICICI Direct
 HDFC Securites
 ShareKhan
 Kotak Securities
 Angel Broking
 Axis Direct
Great! Now we look at some options trading strategies which can be used in the real world.
Options Trading Strategies
There are quite a few options trading strategies which can be used in today’s trading landscape. One of the most popular options trading strategies is based on Spreads and Butterflies. Let’s look at them in detail.
Spreads and Butterflies
Spreads or rather spread trading is simultaneously buying and selling the same option class but with different expiration date and strike price. Spread options trading is used to limit the risk but on the other hand, it also limits the reward for the person who indulges in spread trading.
Thus, if we are only interested in buying and selling call options of security, we will call it a call spread, and if it is only puts, then it will be called a put spread.
Depending on the changing factor, spreads can be categorised as:
 Horizontal Spread – Different expiration date, Same Strike price
 Vertical Spread – Same Expiration date, Different Strike price
 Diagonal Spread – Different expiration date, Different Strike price
Remember that an option’s value is based on the underlying security (in this case, stock price). Thus, we can also distinguish an option spread on whether we want the price to go up (Bull spread) or go down (Bear spread).
Bull call spread
In a bull call spread, we buy more than one option to offset the potential loss if the trade does not go our way.
Let’s try to understand this with the help of an example.
The following is a table of the available options for the same underlying stock and same expiry date:
Normally, if we have done the analysis and think that the stock can rise to $200, one way would be to buy a call stock option with a strike price of $180 for a premium of $15. Thus, if we are right and the stock reaches $200 on expiry, we buy it at the strike price of $180 and pocket a profit of ($20 $15) = $5 since we paid the premium of $15.
But if we were not right and the stock price reaches $180 or less, we will not exercise the option resulting in a loss of the premium of $15. One workaround is to buy a call option at $180 and sell a call option for $200 at $10.
Thus, when the stock’s price reaches $200 on expiry, we exercise the call option for a profit of $5 (as seen above) and also pocket a profit of the premium of $10 since it will not be exercised by the owner. Thus, in this way, the total profit is ($5 + $10) = $15.
If the stock price goes above $200 and the put option is exercised by the owner, the increase in the profit from bought call option at $180 will be the same as the loss accumulated from the sold call option at $200 and thus, the profit would always be $15 no matter the increase in the stock price above $200 at expiry date.
Let’s construct a table to understand the various scenarios.
You can go through this informative blog to understand how to implement it in Python.
Bear put spread
The bull call spread was executed when we thought the stock would be increasing, but what if we analyse and find the stock price would decrease. In that case, we use the bear put spread.
Let’s assume that we are looking at the different strike prices of the same stock with the same expiry date.
One way to go about it is to buy the put option for the strike price of 160 at a premium of $15 while selling a put option for the strike price of $140 for the strike price of $10.
Thus, we create a scenario table as follows:
In this way, we can minimize our losses by simultaneously buying and selling options. You can go through this informative blog to understand how to implement it in Python.
Butterfly Spread
A butterfly spread is actually a combination of bull and bear spreads. One example of the Butterfly Options Strategy consists of a Body (the middle double option position) and Wings (2 opposite end positions).
 Its properties are listed as follows:
 It is a threeleg strategy
 Involves buying or selling of Call/Put options (unlike Covered Call Strategy where a stock is bought and an OTM call option is sold)
 Can be constructed using Calls or Puts
 4 options contracts at the same expiry date
 Have the same underlying asset
 3 different Strike Prices are involved (2 have the same strike price)
 Create 2 Trades with these calls
Other Trading Strategies
We will list down a few more options trading strategies below:
Summary
We have covered all the basics of options trading which include the different Option terminologies as well as types. We also went through an options trading example and the option greeks. We understood various options trading strategies and things to consider before opening an options trading account.
If you have always been interested in automated trading and don’t know where to start, we have created a learning track for you at Quantra, which includes the. “Trading using Option Sentiment Indicators” course.
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