Options in Trading

Balamurugan

7 min readJan 22, 2021

The following article about options is broken down into three sections,

1. Theory about options

2. Implementation of theoretical concepts in Python

3. Implementation used in practice

Theory

What are options?

Options are financial instruments that derive its value from underlying securities such as stocks, commodities, interest rates etc. Hence options are called as derivatives and classified as a derivative product. An option contract gives the buyer the opportunity to buy or sell an underlying asset (depending on the type of contract they hold) at a predefined price and date. The option buyer is not required to buy or sell the asset if they decide not to.

Types of options

There are two types of options-

Call option: The ‘Call Option’ gives the buyer of the option the right to buy a particular asset at a stated price on or before the maturity date in return for a premium paid upfront to the seller. Call options becomes more valuable as the value of the underlying asset increases.

Put option: The Put Option gives the buyer the right to sell a particular asset at a stated price anytime on or before the maturity date in return for a premium paid up front. Put options become more valuable when the price of the underlying stock falls.

Terms used in option trading:

Premium: Amount of money that the buyer of an option pays to the seller for the right, but not the obligation, to exercise the option.

The option premium is broken down into two components:

1. Intrinsic Value: The market price of an option minus the strike price — representing the profit the holder of the option would book if he or she exercised the option, took delivery of the underlying asset and sold it in the current marketplace.

2. Time Value: The time value is calculated by subtracting the intrinsic value of the option from the option premium.

Lot Size: Lot size refers to a fixed number of units of the underlying asset that form part of a single options contract. The standard lot size is different for each stock and is decided by the exchange on which the stock is traded.

Strike Price: The predefined price on an option at which the contract can be exercised is known as the strike price.

Expiry Date: Each option contract will have a specific expiry date after which the contract expires. For NSE, it is the last Thursday of every month for monthly options. There are also weekly options available for certain instruments.

Moneyness:

1. In the Money (ITM): An in-the-money option is an option that has intrinsic value. With regard to a call option, it is an option with a strike price below the current market price and vice-versa for a put option.

2. At the Money (ATM): At-the-money option is a scenario where the current market price is equal to the strike price. In that case, both the call and put options would be at the money, and the intrinsic value of both would be zero. However, that doesn’t mean that the options have no value because they may still have time value.

3. Out of the Money (OTM): An out-of-the-money option is an option that has only time value and no intrinsic value. An out-of-the-money call option is an option with a strike price that is higher than the current market price and vice-versa for a put option.

Example:

Breakdown of all the above terminologies for a Call option of Infosys listed in National Stock Exchange expiring on 29th August 2019.

Participants in option trading:

Buyer of Calls (Long Call): Buyers of call options lose money which is the equivalent of the premium value, if the underlying asset price (ST) is lower than the strike price (SP). If the underlying asset price (ST) is more than the strike price (SP) and continually increasing, the holders’ loss is decreasing until the underlying asset price reaches the breakeven point (premium paid), and since then the call options buyer profit from their long call positions.

Profit = max(ST — SP, 0) — premium

The payoff diagram is shown below for buyer of an Infosys call option with premium = 10 and Strike Price (SP) = 800.

Seller of Calls (Short Call): A short call position is the opposite of a long call option position (the other side of the trade). Seller of a call option receives cash in the beginning and make money if the option expires worthless or with intrinsic value lower than the premium.

Profit = min(SP– ST, 0) + premium

The payoff diagram is shown below for seller of an Infosys call option with premium = 10 and Strike Price (SP) = 800.

Buyer of Puts (Long Put): Buyers of put options lose money which is the equivalent of the premium value, if the underlying asset price (ST) is higher than the strike price (SP). If the underlying asset price (ST) is less than the strike price (SP) and continually decreasing, the holders’ loss is decreasing until the underlying asset price reaches the breakeven point (premium paid), and since then the put options buyer profit from their long put positions.

Profit = max(SP — ST, 0) — premium

The payoff diagram is shown below for buyer of an Infosys put option with premium = 10 and Strike Price (SP) = 800.

Seller of Puts (Short Put): A short put position is the opposite of a long put option position (the other side of the trade). Seller of a put option receives cash in the beginning and make money if the option expires worthless or with intrinsic value lower than the premium.

Profit = min(ST — SP, 0) + premium

The payoff diagram is shown below for seller of an Infosys put option with premium = 10 and Strike Price (SP) = 800.

Pricing an option

The Black-Scholes model is the most commonly used option pricing model. After its introduction in 1973 and refinements thereafter, the model has become the de-facto standard for pricing European-style stock options. This model has been adopted by many of the world’s foremost investment banks and hedge funds.

Assumptions

The model is based on the core assumption that the market consists of at least one risky asset (such as a stock) and one (essentially) risk-free asset, such as a government bond and bank deposits.

Assumptions about the assets in the market are:

1. The continuously compounded returns on the stock are normally distributed

2. The volatility of continuously compounded returns is known and constant

3. The risky asset does not pay a dividend

Assumptions about the market itself are:

1. The risk-free rate is known and constant

2. It is possible to buy and sell any amount of the stock

3. There are no transaction costs in the market

Variables

The main variables used in the Black-Scholes model include:

1. Price of underlying asset (S) is a current market price of the asset

2. Strike price (K) is a price at which an option can be exercised

3. Volatility (σ) is a measure of how much the security prices will move in the subsequent periods. Volatility is the trickiest input in the option pricing model as the historical volatility is not the most reliable input for this model as it is a forward looking estimate

4. Time until expiration (T) is a time between calculation and option’s exercise date

5. Interest rate (r) is a risk-free interest rate

Pricing formula

The pricing formula for European call and put options as per Black-Scholes model is given below,

Implementation in Python

Payoff Diagram for Long Call

The payoff diagram for other options can be implemented in a similar manner.

Black-Scholes model Implementation for European call option

The Black Scholes model implementation for a Put option can be defined similarly by inverting the signs of a few parameters as explained in the previous section.

Practical Use

The above function can be used to value options using the Black Scholes model. As an example, let’s consider the below Call option of Infosys listed in National Stock Exchange at INR 10.05, expiring on 29th August 2019 with strike price INR 790.