Call – Put Parity

August 19, 2021 by Akshit GUPTA

Call-Put Parity

This article written by Akshit GUPTA (ESSEC Business School, Master in Management, 2019-2022) presents the subject of call-put parity.

Introduction

The call-put parity (also written the put-call parity) is a concept introduced in the 1960s by the economist Hans R. Stoll in a paper named “The Relationship Between Put and Call Option Prices”. The call-put parity shows the relationship between the prices of a put option, a call option, and the underlying asset. The call option and the put option are written on the same underlying asset and have the same expiration date and strike price. The call-put parity is applicable only on European options with a fixed time to expiration (it is not applicable to American options).

Call-put parity relation

The call-put parity relation is given by the equality:

Formula for the call put parity

Where t is the evaluation date (any date between the issuance date and the maturity date of the option), C_t the price of the call option, P_t the price of the put option, S_t the price of the underlying asset, K the strike price of the two options (same strike price for the call and put options), T the maturity date of the two options (same maturity date for the call and put options) and r the risk-free rate.

The call-put parity relation is sometimes written in different ways:

Formula for the call put parity styles

Demonstration

Let us try to find the call-put parity relation for a put option and a call option, which are European options with the same strike price K and the same maturity date T.

Let us consider a portfolio composed a long position in the underlying asset, a long position in the put option, a short position in the call option and a short position of a zero-coupon bond maturing at time T and of final value K.

Let us compute the value of this position at time T. The underlying asset is worth S_T. The zero-coupon bond is worth K. Regarding the call and put options, we can distinguish two cases: S_T > K and S_T < K.

In the first case, the put option finishes out of the money and the call finishes in the money and is worth S_T – K. The value of the position is then equal to: S_T + 0 – (S_T – K) – K, which is equal to zero.

In the second case, the call option finishes out of the money and the put finishes in the money and is worth K – S_T. The value of the position is then equal to: S_T + (K – S_T) – 0 – K, which is equal to zero.

If the value of the position at time T is also equal to 0, then the value of the position at time t is also equal to 0. If there is no arbitrage, then the value of the position by detailing its components satisfies:

Formula for the call put parity without arbitrage

which leads to the formula given above.

Application

The call-put parity formula helps the investors to calculate the price of a put option from the price of a call option, or inversely, to calculate the price of a call option from the price of a put option (the call option and the put option are written on the same underlying asset and have the same expiration date T and strike price K).

Implication

If the put-call parity does not hold true, there exists an arbitrage opportunity for investors. An arbitrage opportunity helps the investors earn profits without taking any risks. But the chances of finding an arbitrage opportunity is low given the high liquidity in the markets.

Example of application of the call-put parity

Assuming the stock of APPLE is trading at $25 in the market, the strike price of a 3-month European call option on Apple stock is $24 and the premium is $5. The risk-free rate is 8%.

Now, using the call-put parity,

Formula for the call put parity styles

we can calculate the price of the 3-month European put option on Apple stock with the same strike price, which is as follows:

The price of the call option (C) is $5, the price of the underlying asset (S) is $25, the present value of the strike price (K) is $23.52, and the risk-free rate (r) is 8% (market data).

As per the formula: P = $5 – $25 + $23.52, the price of the put option (P) is approximately equal to $3.52.

Useful resources

Academic research

Hull J.C. (2015) Options, Futures, and Other Derivatives, Ninth Edition, Chapter 11 – Properties of Stock Options, 256-275.

Stoll H.R. (1969) “The Relationship Between Put and Call Option Prices,” The Journal of Finance, 24(5): 801-824.

About the author

Article written in August 2021 by Akshit GUPTA (ESSEC Business School, Master in Management, 2019-2022).

Option Greeks – Theta

February 4, 2024August 19, 2021 by Akshit GUPTA

Option Greeks – Theta

This article written by Akshit GUPTA (ESSEC Business School, Master in Management, 2019-2022) presents the technical subject of theta, an option Greek used in option pricing and hedging to deal with he passing of time.

Introduction

Theta is a type of option Greek which is used to compute the sensitivity or rate of change of the value of an option contract with respect to its time to maturity. The theta is denoted using the symbol (θ). Essentially, the theta is the first partial derivative of the price of the option contract with respect to the time to maturity of the option contract.

It is shown as:

Formula for the theta

Where V is the value of the option contract and T the time to maturity for the option contract.

Theoretically, as the option contract approaches maturity, the theta of on option contract increases and moves towards zero as the time value or the time value of the option decreases. This is referred to as “theta decay”.

For example, an option contract is trading at a premium of $10 and has a theta of -0.8. Thus, with theta decay, the option price will decrease to $9.2 after one day and further to $6 after five days.

The figure below represent the theta of a call option as a function of the time to maturity:

Figure 1. Theta of a call option as a function of time to maturity.

Source: computation by the author (Model: Black-Scholes-Merton).

Intrinsic and time value of an option contract

Essentially, the price of an option contract consists of two values namely, the intrinsic value and the time value (sometimes called extrinsic value). The intrinsic value in the price of an option contract is the real value or the fundamental value of an option based on the price of the underlying asset at a given point in time.

For example, a call option contract has a strike price of $10 and the underlying asset has a market price of $17. Theoretically, the buyer of a call option can execute the contract and buy the asset at $10 and sell it in the market for $17. He/she can make an immediate profit of $7 if they decide to exercise the option. Thus, the intrinsic value of the option contract is $7.

If the current call option price/premium is $9 in the market and the intrinsic value is $7, then the time value can be calculated as:

Time Value for the theta

Thus, the time value is $9-$7 is equal to $2. The $2 is the time value of an option contract which is determined by the factors other than the price of the underlying asset. As the option approaches maturity, the time value of the option contract declines and tends to zero. The price of an option contract which is at the money or out the money, it consists entirely of the time value as there is no intrinsic value involved.

For example, a call option contract with a strike price of $20, the underlying asset price of $15, and option premium of $3, has a time value equal to the option premium, $3, since the option is out of money.

Calculating Theta for call and put options

The theta for a non-dividend paying stock in a European call and put option is calculated using the following formula from the Black-Scholes Merton model:

Formula for the theta of a call and a put option

Where N’(d₁) represents the first order derivative of the cumulative distribution function of the normal distribution given by:

First_ derivative_Normal_distribution_d1

d₁ is given by:

Formula for d1

d₂ is given by:

Formula for d2

And N(-d₂) is given by:

Formula for -d2

Where S is the price of the underlying asset (at the time of valuation of the option), σ the volatility in the price of the underlying asset, T time to option’s maturity, K the strike price of the option contract and r the risk-free rate of return.

Excel pricer to calculate the theta of an option

You can download below an Excel file for an option pricer (based on the Black-Scholes-Merton or BSM model) which allows you to calculate the theta of a European-style call option.

Example for calculating theta

Let us consider a call option contract with the following characteristics: the underlying asset is an Apple stock, the option strike price (K) is equal to $300 and the time to maturity (T) is of one month (i.e., 0.084 years).

At the time of valuation, the price of the Apple stock (S) is $300, the volatility (σ) of Apple stock is 30% and the risk-free rate (r) is 3% (market data).

The theta of a call option is approximately equal to -0.2636 per trading day.

Using the above example, we can say that after one trading day, the price of the option will decrease by $0.2636 (approximately) due to time decay.

Useful resources

Hull J.C. (2015) Options, Futures, and Other Derivatives, Ninth Edition, Chapter 19 – The Greek Letters, 424–431.

Wilmott P. (2007) Paul Wilmott Introduces Quantitative Finance, Second Edition, Chapter 8 – The Black Scholes Formula and The Greeks, 182-184.

About the author

Article written in August 2021 by Akshit GUPTA (ESSEC Business School, Master in Management, 2019-2022).

Option Greeks – Vega

January 30, 2024August 19, 2021 by Akshit GUPTA

Option Greeks – Vega

This article written by Akshit GUPTA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) explains the technical subject of vega, the option Greek used in option pricing and hedging to take into account the volatility of the underlying asset.

Introduction

Vega is a type of option Greek which is used to compute the sensitivity or rate of change of the value of an option contract with respect to the volatility of the underlying asset. The Vega is denoted using the Greek letter (ν). Essentially, the vega is the first partial derivative of the value of the option contract with respect to the volatility of the underlying asset.

The vega formula for an option is given by

Formula for the gamma

Where V is the value of the option contract and σ is the volatility of the underlying asset.

If the Vega is a very high positive or a negative number, this means that the option price is highly sensitive to the volatility of the underlying asset. The Vega is maximum when the option price is at the money. For example, the strike of an option contract is €100, and the price of the underlying asset is €100. The option is at the money (ATM) and has an intrinsic value of zero. So, the option premium entirely consists of the time value of the option. Thus, the Vega is the highest for at the money option contract since the option value are mostly dependent on the time value (sometimes called the extrinsic value). An increase/decrease in volatility can change the option value significantly for at-the-money options.

Figure 1 below represents the vega of a call option as a function of the price of the underlying asset. The parameters of the call option are a maturity of 3 months and a strike of €100. The market data are a price of the underlying asset between €50 and €150, a volatility of the underlying asset of 40%, a risk-free interest rate of 3% and a dividend yield of 0%.

Figure 1. Vega of a call option as a function of the price of the underlying asset.

Source: computation by the author (Model: Black-Scholes-Merton).

Calculating the vega for call and put options

The vega for a European call or put option is calculated using the following formula:

Formula for the gamma

where

N’(d₁) represents the first order derivative of the cumulative distribution function of the normal distribution given by:

First_ derivative_Normal_distribution_d1

and d₁ is given by:

Formula for d1

where S is the price of the underlying asset (at the time of valuation of the option), σ the volatility in the price of the underlying asset, T time to option’s maturity, K the strike price of the option contract and r the risk-free rate of return.

Example for calculating vega

At the time of valuation, the price of the Apple stock (S) is $300, the volatility (σ) of Apple stock is 30% and the risk-free rate (r) is 3% (market data).

The vega of the call option is approximately equal to 0.3447963.

Using the above value, we can say that due to a 1% change in the volatility of the underlying asset, the price of the option will change approximately by $0.3447.

Excel pricer to calculate the vega of an option

You can download below an Excel pricer (based on the Black-Scholes-Merton or BSM model) to calculate the vega of an option (call or put).

Useful resources

Hull J.C. (2015) Options, Futures, and Other Derivatives, Ninth Edition, Chapter 19 – The Greek Letters, 424–431.

Wilmott P. (2007) Paul Wilmott Introduces Quantitative Finance, Second Edition, Chapter 8 – The Black Scholes Formula and The Greeks, 182-184.

About the author

Article written in August 2021 by Akshit GUPTA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022).

Option Greeks – Gamma

February 4, 2024August 8, 2021 by Akshit GUPTA

Option Greeks – Gamma

This article written by Akshit GUPTA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) presents the technical subject of gamma, an option Greek used in option hedging.

Introduction

Gamma is a type of option Greek which is used to compute the sensitivity or rate of change of delta (Δ) of an option contract with respect to a change in the price of the underlying in the option contract (S). The gamma of an option is expressed in percentage terms. Denoted by the Greek letter (Γ), the gamma is defined by

Formula for the gamma of an option

Where (Δ) is the delta of the option and S the price of the underlying asset.

Essentially, the gamma is the second partial derivative of the value of the option contract (V) with respect to the price of the underlying asset (S). It measures the convexity of the value of the option contract with respect to the price of the underlying asset. The gamma then corresponds to

Formula for the gamma of an option

Where V is the value of the option and S the price of the underlying asset.

The gamma of an option contract is at its maximum when the price of the underlying asset is equal to the strike price of the option (an at-the-money option). If the price of the underlying moves deeper in the money or out of the money, the value of the gamma approaches zero.

The gamma as a function of the price of the underlying asset for a call option is given below.

Figure 1. Gamma of a call option.

Source: computation by the author (Model: Black-Scholes-Merton).

Also, if the gamma of the option contract is small, it means that the delta of the option moves slowly with the price of the underlying asset.

Calculating gamma for call and put options

The gamma for European call or put options on a non-dividend paying stock is calculated using the following formula from the Black-Scholes-Merton model is:

Formula for the gamma of a call/put option

Where,N’d₁ represents the first order derivative of the cumulative distribution function of the normal distribution given by:

First_ derivative_Normal_distribution_d1

and d₁ is given by:

Formula for d1.png

Excel pricer to calculate the gamma of an option

You can download below an Excel file for an option pricer (based on the Black-Scholes-Merton or BSM model) which allows you to calculate the gamma of a European-style call option.

Delta-gamma hedging

A trader holding a portfolio of option contracts uses gamma hedging to offset the risks associated with the price movement in the underlying asset by buying and selling the option contracts to maintain a constant delta. Generally, the delta is maintained near or at the zero level to attain delta neutrality. The neutrality in the gamma for the option is required to protect the portfolio’s value against sharp price movements in the price of the underlying asset.

Formula for the gamma hedging of a call option

Limitations of gamma hedging

The limitation of gamma hedging includes the following:

Transaction cost – Gamma hedging requires constantly monitoring the markets and buying or selling the option contracts. Due to this practice of buying and selling frequently, the transaction costs are quite high to execute a gamma hedge. Thus, gamma hedging is an expensive strategy to practice.
Loosing delta neutrality – Whenever a trader executes a gamma hedge and trades in option contracts, it is often accompanied with a move in the portfolio’s delta. Thus, to achieve delta neutrality again, the trader must buy or sell additional quantities of the underlying asset, which is time consuming and comes with a transaction cost.

Useful resources

Hull J.C. (2015) Options, Futures, and Other Derivatives, Ninth Edition, Chapter 19 – The Greek Letters, 424–431.

Wilmott P. (2007) Paul Wilmott Introduces Quantitative Finance, Second Edition, Chapter 8 – The Black Scholes Formula and The Greeks, 182-184.

About the author

Article written in August 2021 by Akshit GUPTA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022).

Option Greeks – Delta

February 4, 2024August 5, 2021 by Akshit GUPTA

Option Greeks – Delta

This article written by Akshit GUPTA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) presents the technical subject of delta, an option Greek used in option pricing and hedging.

Introduction

Option Greeks are sophisticated financial metric used by trader to calculate the sensitivity of option contracts to different factors related to the underlying asset including the price of the underlying, its volatility, and time value. The Greeks are used as an effective tool to practice different hedging strategies and eliminate risks in a position. They also help to optimize the options positions at any point in time.

Delta is a type of option Greek which is used to compute the sensitivity or rate of change in price of the option contract with respect to the change in price of the underlying asset. It is denoted by the Greek letter (Δ). The formula for calculating the delta of an option contract is:

Formula for the delta of an option

Where V is the value of the option and S the price of the underlying asset.

For example, if an option on Apple stock has a delta of 0.3, it essentially means that a $1 change in the price of the underlying asset i.e., Apple stock, will lead to a change of $0.3 in the price of the option contract.

When a trader takes a position based on the delta sensitivity of any option contract, it is called delta hedging. The goal is to achieve a delta-neutral portfolio and eliminate the risks associated with movement in the prices of the underlying. Due to the complexity of the tool, delta hedging is generally practiced by professional traders in large financial institutions. In options, the delta of any call option is always positive whereas the delta of a put option is always negative.

Delta formula

Call option

According the Black-Scholes-Merton model, the formula for calculating the delta for a European-style call option on a non-dividend paying stock is given by:

Formula for the delta of a call option

Where N represents the cumulative distribution function of the normal distribution and d₁ is given by:

Formula for d1

Where S is the price of the underlying asset (at the time of valuation of the option), σ the volatility in the price of the underlying asset, T time to maturity of the option, K the strike price of the option, and r the risk-free rate of return.

Put option

According the Black-Scholes-Merton model, the formula for calculating the delta for a European-style put option on a non-dividend paying stock is given by:

Formula for the delta of a put option

Delta as a function of the price of the underlying asset

Call option

The delta as a function of the price of the underlying asset for a European-style call option is represented in Figure 1.

Figure 1. Delta of a call option.

Source: computation by the author (Model: Black-Scholes-Merton).

For a call option, the delta increases from 0 (out-of-the-money option) to 1 (in-the-money option).

Put option

The delta as a function of the price of the underlying asset for a European-style put option is represented in Figure 2.

Figure 2. Delta of a put option.

Source: computation by the author (Model: Black-Scholes-Merton).

For a put option, the delta increases from -1 (in-the-money option) to 0 (out-of-the-money option).

Excel pricer to calculate the delta of an option

You can download below an Excel file for an option pricer (based on the Black-Scholes-Merton or BSM model) which allows you to calculate the delta of a European-style call option.

Delta Hedging

A trader holding an option contract uses delta hedging to offset the risks associated with the price movement in the underlying asset by continuously buying and selling the underlying asset to achieve delta neutrality. This is used by option traders in financial institutions to manage their option book (the delta is computed at the option level and aggregated at the book level) and generate the margin the bank of the option writing activity.

The delta of an option contract keeps on changing as the prices of the underlying and the option contract changes. So, to maintain the delta neutrality the trader must constantly monitor the markets and execute trades to achieve neutrality. The process of continuously buying or selling the underlying asset is called dynamic hedging in options.

At the first order, the change of the value of a delta-hedged call option over the period from t to t+ δt would be equal to the risk-free rate (r) over the period:

Formula for the delta hedging of a call option

Limitations of delta hedging

Although delta hedging is a useful tool to offset the risks associated to the movement in the price of an underlying, it comes with some limitations which are:

Transaction cost

Since delta hedging requires constantly buying or selling the underlying asset, it comes with a high transaction cost. This makes delta hedging an expensive tool to optimize the portfolio against price risk. In practice, traders would adjust their option position from time top time.

Illiquid Markets

When the market for an asset is illiquid, it is difficult to practice delta hedging as the trader will not be able to constantly buy or sell the underlying asset to neutralize the price impact.

Example for calculating delta

At the time of valuation, the price of the Apple stock (S) is $300, the volatility (σ) of Apple stock is 30% and the risk-free rate (r) is 3% (market data).

The delta of a call option is approximately equal to 0.50238.

Using the above value, we can say that due to a $1 change in the price of the underlying asset, the price of the option will change by $0.50238.

Useful resources

Research articles

Black F. and M. Scholes (1973) The Pricing of Options and Corporate Liabilities The Journal of Political Economy, 81, 637-654.

Merton R.C. (1973) Theory of Rational Option Pricing Bell Journal of Economics, 4(1): 141–183.

Books

Hull J.C. (2015) Options, Futures, and Other Derivatives, Ninth Edition, Chapter 19 – The Greek Letters, 424 – 431.

Wilmott P. (2007) Paul Wilmott Introduces Quantitative Finance, Second Edition, Chapter 8 – The Black Scholes Formula and The Greeks, 182-184.

About the author

Article written in August 2021 by Akshit GUPTA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022).

Understanding financial derivatives: options

August 2, 2021 by Alexandre VERLET

In this article, Alexandre VERLET (ESSEC Business School, Master in Management, 2017-2021) explains why financial markets invented options and how they function.

A historical perspective on options

The history of options is surrounded by legends.. This story is linked to human’s desire to control the unpredictable, sometimes to protect himself from it, often to profit from it. This story is also that of a flower: the tulip. At the beginning of the seventeenth century, in the Netherlands, the tulip was at the origin of the first known speculative bubble. Furthermore, this was historically the first time that options contracts were used on such a large scale. The possibility of profiting from the rise in the price of tulips by paying only a small part of the price aroused great interest on the part of speculators, thus increasing the price of the precious flower tenfold. Soon the price of the tulip reached levels completely unrelated to its market value. Then, suddenly, demand dried up, causing the price to fall even faster than the previous rise. The crisis that followed had serious consequences and confirmed Amsterdam’s loss of world leadership in finance to the benefit of London, which had already taken over the Dutch capital as the world’s center for international trade. Educated by the Dutch experience, the British became increasingly sceptical about options, so much so that they eventually banned them for over a century. The ban was finally lifted towards the end of the 19th century. It was also at this time that options were introduced in the United States.

The American options market entered a new dimension at the end of the 20th century. Indeed, 1973 was a pivotal year in the history of options in more ways than one. In March 1973, a floating exchange rate regime was adopted as the standard for converting international currencies, creating unprecedented instability in the currency market. This was also the year of the “first oil shock”. Also in 1973, the Chicago Board Options Exchange (CBOE), the first exchange entirely dedicated to options, opened its doors. The same year saw the birth of the Options Clearing Corporation (OCC), the first clearing house dedicated to options. Finally, 1973 saw the publication of the work of Fischer Black and Myron Scholes. This work was completed by Robert Merton, leading to the Black-Scholes-Merton model. This model is of capital importance for the evaluation of the price of options.

What’s an option?

There are two types of option contracts: calls and puts. Since these contracts can be both bought and sold, there are four basic transactions. Thus, in options trading, it is possible to either go long (buy a call contract, buy a put contract), or to be short (sell a call contract, sell a put contract). An option contract can therefore be defined as a contract that gives the counterparty buying the contract (the long) the right, but not the obligation, to buy or sell an asset (the underlying) at a predetermined price (the strike price), date (the maturity date) and amount (the nominal value). It is useful to note that the counterparty selling the contracts (the short) is in a completely different situation. This counterparty must sell or buy the underlying asset if the transaction is unfavorable to it. However, if the transaction is favorable, this counterparty will not receive any capital gain, because the counterparty buying the contract (the long) will not have exercised its call option. To compensate for the asymmetry of this transaction, the counterparty selling the option contracts (the short) will receive a premium at the time the contract is initiated. The selling counterparty therefore has a role similar to that of an insurance company, as it is certain to receive the premium, but has no control over the time of payment or the amount to be paid. This is why it is important to assess the amount of the premium.

The characteristic of an option contract

Options contracts can have as underlying assets financial assets (interest rates, currencies, stocks, etc.), physical assets (agricultural products, metals, energy sources, etc.), stock or weather indices, and even other derivatives (futures or forwards). The other important feature of an option contract is its expiration date. Options contracts generally have standardized expiry dates. Expiry dates can be monthly, quarterly or semi-annually. In most cases, the expiration date coincides with the third Friday of the expiration month. In addition, options whose only possible exercise date is the maturity date are called European options. However, when the option can be exercised at any time between signing and expiration, it is called an American option. Ultimately, what will drive the holder of an option contract to exercise his right is the difference between the underlying price and the strike price. The strike price is the purchase or sale price of the underlying asset. This price is chosen at the time the option contract is signed. The strike price will remain the same until the end of the option contract, unlike the price of the underlying asset, which will vary according to supply and demand. In organised markets, brokers usually offer the possibility to choose between several strike prices. The strike price can be identical to the price of the underlying asset. The option is then said to be “at-the-money” (or “at par”).

In the case of a call, if the proposed strike price is higher than the price of the underlying, the call is said to be “out of the money”.

Are you “in the money”?

Let’s take an example: a share is quoted at 10 euros. You are offered a call with a price of 11 euros. If we disregard the premium, we can see that a resale of the call, immediately after buying it, will result in a loss of one euro. For this reason, the call is said to be “out of the money”. On the other hand, when the strike price offered for a call is lower than the price of the underlying asset, the call is said to be “in the money”. Another example: the stock is still trading at 10 euros. This time you are offered a call with a strike price of 9 euros. If you disregard the premium, you can see that you earn one euro if you sell the call immediately after buying it. This is why this call is called “in the money”. Note that our potential gain of one euro is also called the “intrinsic value” of the call. Of course, the intrinsic value is only valid for “in the money” options. For puts, it is the opposite. A put is said to be “out of the money” if its strike price is lower than the price of the underlying asset.

Finally, a put is said to be “in the money” if its strike price is higher than the price of the underlying asset. If you are one of those people who think that you can make money with options by simply buying and selling calls or puts “in the money”, I have bad news for you! In reality, the premiums of the different contracts are calculated in such a way as to cancel out the advantage that “in the money” contracts offer over other contracts.

Useful resources

ISDA

About the author

Article written in July 2021 by Alexandre VERLET (ESSEC Business School, Master in Management, 2017-2021).

Options

February 4, 2024June 20, 2021 by Akshit GUPTA

Options

This article written by Akshit GUPTA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) presents an introduction to Options.

Introduction

Options is a type of derivative which gives the buyer of the option contract the right, but not the obligation, to buy (for a call option) or sell (for a put option) an underlying asset at a price which is pre-determined, and a date set in the future.

Option contracts can be traded between two or more counterparties either over the counter or on an exchange, where the contracts are listed. Exchange based trading of option contracts was introduced to the larger public in April 1973, when Chicago Board Options Exchange (CBOE)) was introduced in the US. The options market has grown ever since with over 50 exchanges that trade option contracts worldwide.

Terminology used for an option contract

The different terms that are used in an option contract are:

Option Spot price

The option spot price is the price at which the option contract is trading at the time of entering the contract.

Underlying spot price

The underlying spot price is the price at which the underlying asset is trading at the time of entering the option contract.

Strike price

Strike price is essentially the price at which the option buyer can exercise his/her right to buy or sell the option contract at or before the expiration date. The strike price is pre-determined at the time of entering the contract.

Expiration date

The expiration date is the date at which the option contracts ends or after which it becomes void. The expiration date of an option contract can be set to be after weeks, months or year.

Lot size

A lot size is the quantity of the underlying asset contained in an option contract. The size is decided and amended by the exchanges from time to time. For example, an Option contract on an APPLE stock trading on an exchange in USA consists of 100 underlying APPLE stocks.

Option class

Option class is the type of option contracts that the trader is trading on. It can be a Call or a Put option.

Position

The position a trader can hold in an option contract can either be Long or Short depending on the strategy. A Long position essentially means Buying the option and a short position means Selling or writing the option contract.

Option Premium

Option premium is the price at which the option contracts trade in the market.

Benefits of using an option contract

Trading in option contracts gives the traders certain benefits which can be categorised as:

Hedging Benefits

Hedging is an essential benefit of the option contract. For an investor or a trader holding an underlying stock, an option contract provides them with the opportunity to offset their risk exposure by buying or selling an option contract as per their market outlook. If an trader holding stocks of APPLE is bearish about the market and expects the market to fall, he/she can buy a PUT option which essentially gives him/her the right to sell the security at a pre-determined price and date. Such a contract protects the trader from significant losses which he/she might incur if the stock price for APPLE goes down significantly.

Cost Benefits

While buying an option contract, the traders benefits from the leverage effect which exchanges across the world provides. Leverage helps the traders to multiply the size of their holdings with lesser capital investment. This also helps them to earn higher profits by taking limited risks.

Choice Benefits

In traditional trading, traders have a limited degree of flexibility as they can only buy or sell assets based on their outlook. Whereas, Option contracts provides a great choice to the traders as they can take different positions in call and put options (Long and short positions) and for different strikes and maturities.
They can also use different strategies and spreads to execute and manage their positions to earn profits.

Types of option contracts

The option contracts can be broadly classified into two categories: call options and put options.

Call options

A call option is a derivative contract which gives the holder of the option the right, but not an obligation, to buy an underlying asset at a pre-determined price on a certain date. An investor buys a call option when he believes that the price of the underlying asset will increase in value in the future. The price at which the options trade in an exchange is called an option premium and the date on which an option contract expires is called the expiration date or the maturity date.

For example, an investor buys a call option on Apple shares which expires in 1 month and the strike price is $90. The current apple share price is $100. If after 1 month,
The share price of Apple is $110, the investor exercises his rights and buys the Apple shares from the call option seller at $90.

But, if the share prices for Apple falls to $80, the investor doesn’t exercise his right and the option expires because the investor can buy the Apple shares from the open market at $80.

Put options

A put option is a derivative contract which gives the holder of the option the right, but not an obligation, to sell an underlying asset at a pre-determined price on a certain date. An investor buys a put option when he believes that the price of the underlying asset will decrease in value in the future.

For example, an investor buys a put option on Apple shares which expires in 1 month and the strike price is $110. The current apple share price is $100. If after 1 month,
The share price of Apple is $90, the investor exercises his rights and sell the Apple shares to the put option seller at $110.
But, if the share prices for Apple rises to $120, the investor doesn’t exercise his right and the option expires because the investor can sell the Apple shares in the open market at $120.

Different styles of option exercise

The option style doesn’t deal with the geographical location of where they are traded. However, the contracts differ in terms of their expiration time when they can be exercised. The option contracts can be categorized as per different styles they come in. Some of the most common styles of option contracts are:

American options

American style options give the option buyer the right to exercise his option any time prior or up to the expiration date of the contract. These options provide greater flexibility to the option buyer but also comes at a high price as compared to the European style options.

European options

European style options can only be exercised on the expiration or maturity date of the contract. Thus, they offer less flexibility to the option buyer in terms of his rights. However, the European options are cheaper as compared to the American options.

Bermuda options

Bermuda options are a mix of both American and European style options. These options can only be exercised on a specific pre-determined dates up to the expiration date. They are considered to be exotic option contracts and provide limited flexibility to the option buyer to exercise his claim.

Different underlying assets for an option contract

The different underlying assets for an option contract can be:

Individual assets: stocks, bonds

Option traders trading in individual assets can take positions in call or put options for equities and bonds based on the reports provided by the research teams. They can take long or short positions in the option contract. The positions depend on the market trends and individual asset analysis. The option contracts on individual assets are traded in different lot sizes.

Indexes: stock indexes, bond indexes

Options traders can also trade on contracts based on different indexes. These contracts can be traded over the counter or on an exchange. These traders generally follow the macroeconomic trends of different geographies and trade in the indices based on specific markets or sectors. For example, some of the most known exchange traded index options are options written on the CAC 40 index in France, the S&P 500 index and the Dow Jones Industrial Average Index in the US, etc.

Foreign currency options

Different banks and investment firms deal in currency hedges to mitigate the risk associated with cross border transactions. Options traders at these firms trade in foreign currency option contracts, which can be over the counter or exchange traded.

Option Positions

Option traders can take different positions depending on the type of option contract they trade. The positions can include:

Long Call

When a trader has a long position in a call option it essentially means that he has bought the call option which gives the trader the right to buy the underlying asset at a pre-determined price and date. The buyer of the call option pays a price to the option seller to buy the right and the price is called the Option Premium. The maximum loss to a call option buyer is restricted to the amount of the option premium he/she pays.

Long Call

With the following notations:
   C_T = Call option value at maturity T
   S_T = Price of the underlying at maturity T
   K = Strike price of the call option

The graph of the payoff of a long call is depicted below. It gives the value of the long position in a call option at maturity T as a function of the price of the underlying asset at time T.

Payoff of a long position in a call option
Long call

Short Call

When a trader has a short position in a call option it essentially means that he has sold the call option which gives the buyer of the option the right to buy the underlying asset from the seller at a pre-determined price and date. The seller of the call option is also called the option writer and he/she receive a price from the option buyer called the Option Premium. The maximum gain to a call option seller is restricted to the amount of the option premium he/she receives.

Short call

With the following notations:
   C_T = Call option value at maturity T
   S_T = Price of the underlying at maturity T
   K = Strike price of the call option

The graph of the payoff of a short call is depicted below. It gives the value of the short position in a call option at maturity T as a function of the price of the underlying asset at time T.

Payoff of a short position in a call option
Short call

Long Put

When a trader has a long position in a put option it essentially means that he/she has bought the put option which gives the trader the right to sell the underlying asset at a pre-determined price and date. The buyer of the put option pays a price to the option seller to buy the right and the price is called the Option Premium. The maximum loss to a put option buyer is restricted to the amount of the option premium he/she pays.

Long Put

With the following notations:
   P_T = Put option value at maturity T
   S_T = Price of the underlying at maturity T
   K = Strike price of the put option

The graph of the payoff of a long put is depicted below. It gives the value of the long position in a put option at maturity T as a function of the price of the underlying asset at time T.

Payoff of a long position in a put option
Long put

Short Put

When a trader has a short position in a put option it essentially means that he has sold the call option which gives the buyer of the option the right to sell the underlying asset from the seller at a pre-determined price and date. The seller of the put option is also called the option writer and he/she receive a price from the option buyer called the Option Premium. The maximum gain to a put option seller is restricted to the amount of the option premium he/she receives.

Short Put

With the following notations:
   P_T = Put option value at maturity T
   S_T = Price of the underlying at maturity T
   K = Strike price of the put option

The graph of the payoff of a short put is depicted below. It gives the value of the short position in a put option at maturity T as a function of the price of the underlying asset at time T.

Payoff of a short position in a put option
Short put

Useful Resources

Academic research

Hull J.C. (2015) Options, Futures, and Other Derivatives, Ninth Edition, Chapter 10 – Mechanics of options markets, 235-240.

Business analysis

CNBC Live option trading for APPLE stocks

About the author

Article written in June 2021 by Akshit GUPTA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022).

History of Options Markets

February 4, 2024June 20, 2021 by Akshit GUPTA

History of the options markets

This article written by Akshit GUPTA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) presents an introduction to the History of the Options markets.

Introduction

Options are a type of derivative contracts which give the buyer the right, but not the obligation, to buy or sell an underlying security at a pre-determined price and date. These contracts can either be traded over-the-counter (OTC) through dealer or broker network or can be traded over an exchange in a standardized form.

A brief history

The history of the use of options can be dated back to ancient times. In early 4th century BC, a philosopher, and an astronomer, named Thales of Miletus calculated a surplus olive harvest in his region during the period. He predicted an increase in demand for the olive presses due to an increase in the harvest. To benefit from his prediction, he bought the rights to use the olive presses in his region by paying a certain sum. The olive harvest saw a significant surplus that year and the demand for olive presses rose, as predicted by him. He then exercised his option and sold the rights to use the olive presses at a much higher prices than what he actually paid, making a good profit. This is the first documented account of the use of option contracts dating back to 4th century BC.

The use of option contracts was also seen during the Tulip mania of 1636. The tulip producers used to sell call options to the investors when the tulip bulbs were planted. The investors had the right to buy the tulips, when they were ready for harvest, at a price pre-determined while buying the call option. However, since the markets were highly unstandardized, the producers could default on their obligations.
But the event laid a strong foundation for the use of option contracts in the future.

Until 1970s, option contracts were traded over-the-counter (OTC) between investors. However, these contracts were highly unstandardized leading to investor distrust and illiquidity in the market.

In 1973, the Chicago Board Options Exchange (CBOE)) was formed in USA, laying the first standardized foundation in options trading. In 1975, the Options clearing corporation (OCC) was formed to act as a central clearing house for all the option contracts that were traded on the exchange. With the introduction of these 2 important bodies, the option trading became highly standardized and general public gained access to it. However, the Put options were introduced only in 1977 by CBOE. Prior to that, only Call options were traded on the exchange.

With the advent of time, options market grew significantly with more exchanges opening up across the world. The option pricing models, and risk management strategies also became more sophisticated and complex.

Market participants

The participants in the options markets can be broadly classified into following groups:

Market makers: A market maker is a market participant in the financial markets that simultaneously buys and sells quantities of any option contract by posting limit orders. The market maker posts limit orders in the market and profits from the bid-ask spread, which is the difference by which the ask price exceeds the bid price. They play a significant role in the market by providing liquidity.
Margin traders: Margin traders are market participants who make use of the leverage factor to invest in the options markets and increase their position size to earn significant profits. But this trading style is highly speculative and can also lead to high losses due to the leverage effect.
Hedgers: Investors who try to reduce their exposure in the financial markets by using hedging strategies are called hedgers. Hedgers often trades in derivative products to offset their risk exposure in the underlying assets. For example, a hedger who is bearish about the market and has shares of Apple, will buy a Put option on the shares of Apple. Thus, he has the right to sell the shares at a high price if the market price for apple shares goes down.
Speculators: Speculative investors are involved in option trading to take advantage of market movements. They usually speculative on the price of an underlying asset and account for a significant share in option trading.

Types of option contracts

The option contracts can be broadly classified into two main categories, namely:

Call options

But, if the share prices for Apple falls to $80, the investor doesn’t exercise his right and the option expires because the investor can buy the Apple shares from the open market at $80.

Put options

For example, an investor buys a put option on Apple shares which expires in 1 month and the strike price is $110. The current apple share price is $100. If after 1 month,

The share price of Apple is $90, the investor exercises his rights and sell the Apple shares to the put option seller at $110.

But, if the share prices for Apple rises to $120, the investor doesn’t exercise his right and the option expires because the investor can sell the Apple shares in the open market at $120.

Different style of options

American options

European options

Bermuda options

Useful Resources

Chapter 10 – Mechanics of options markets, pg. 235-240, Options, Futures, and Other Derivatives by John C. Hull, Ninth Edition

Wikipedia Options (Finance)

The Street A Brief History of Stock Options

About the author

Article written in June 2021 by Akshit GUPTA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022).

Call-Put Parity

Introduction

Call-put parity relation

Demonstration

Application

Implication

Example of application of the call-put parity

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Option Greeks – Theta

Introduction

Intrinsic and time value of an option contract

Calculating Theta for call and put options

Excel pricer to calculate the theta of an option

Example for calculating theta

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Option Greeks – Vega

Introduction

Calculating the vega for call and put options

Example for calculating vega

Excel pricer to calculate the vega of an option

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Option Greeks – Gamma

Introduction

Calculating gamma for call and put options

Excel pricer to calculate the gamma of an option

Delta-gamma hedging

Limitations of gamma hedging

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Option Greeks – Delta

Introduction

Delta formula

Call option

Put option

Delta as a function of the price of the underlying asset

Call option

Put option

Excel pricer to calculate the delta of an option

Delta Hedging

Limitations of delta hedging

Transaction cost

Illiquid Markets

Example for calculating delta

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Books

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A historical perspective on options

What’s an option?

The characteristic of an option contract

Are you “in the money”?

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Options

Introduction

Terminology used for an option contract

Option Spot price

Underlying spot price

Strike price

Expiration date

Lot size

Option class

Position

Option Premium

Benefits of using an option contract

Hedging Benefits

Cost Benefits

Choice Benefits