A Black-scholes Options Pricing Model (BSOPM)

  • Soumyaa Rawat
  • Aug 23, 2021
  • Financial Analytics
A Black-scholes Options Pricing Model (BSOPM) title banner

Options Pricing Model 

 

An option is defined as a contract or an agreement between two parties that provide either of the parties the right to buy or sell an asset at a predetermined value before or at the expiration day as mentioned in the contract. 

 

There are two types of options - call option and put option. While the call option, or simply call, is a contract that gives the right to buy an asset, the put option in the money  provides the power to sell an underlying asset. 

 

An options pricing model estimates the value (premium) of an asset according to the options contract and evaluates the fair value of the asset.  

 

(Related blog: Option trading strategies)

 

 

What is the Black-Scholes-Merton model

 

Defined as an options pricing model, the Black-Scholes-Merton (BSM) model is used to evaluate a fair value of an underlying asset for either of the two options - put or call with the help of 6 variables - volatility, type, stock price, strike price, time, and the risk-free rate.  

 

It was conceptualized by Fischer Black, Myron Scholes, and Robert Merton. The Black-Scholes Option Pricing model was developed in 1973 by this trio. In modern financial theory, the Black-Scholes-Merton model is one of the most significant models for options pricing. The first widely used model. 

 

The BSM model takes into consideration a number of factors before calculating options pricing for complex financial instruments. 

 

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It is used to calculate the premium value of a call or put option with the help of current stock prices, expected dividends, option’s strike price (set price/value of an asset), anticipated interest rates, expiry date, and volatility (fluctuation of the asset value in the market). 

 

A popular tool for option pricing, the Black-Scholes Option Pricing Model (BSOPM) is used in different fields for research and studies by analysts, students, and researchers. This alone describes the importance of black-scholes model. 

 

As the model is used to calculate a fair price of options, the main significance of this model is that it helps an investor to hedge the financial instrument while ensuring minimum risk.   

 

The mathematical formulation of the BSM model is as follows -


Black-Scholes-Merton Model, Source


Here,

C = price of a call option

P = price of a put option

S = price of the underlying asset

X = strike price of the option

r = rate of interest

t = time to expiration

s = volatility of the underlying

N represents a standard normal distribution with mean = 0 and standard deviation = 1

 

Apart from other connotations that exist in the model, the BSM model follows a set of simplified assumptions that have led to the formulation of this model. 

 

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Although these options are theoretically acceptable, they slightly deviate from real-life circumstances and are perhaps a bit unrealistic. Here are the major Black-Scholes Option Pricing model assumptions- 

 

  1. The option pricing in this model is done for a European option, as the American options pricing models are slightly different.

 

  1. The volatility of an asset’s value and risk-free rate is known and constant. Perhaps the market is efficient. 

 

  1. The price of financial instruments (for eg- stock shares) is log-normally distributed. This means that the returns, when represented graphically, form a right-skewed curve, leading to the use of log-normal distribution. 

 

It is different from the normal distribution (a bell-shaped curve). On the other hand, the returns of the option are normally distributed, leading to a uniform distribution. 

 

  1. The model is also based on the assumption that an asset can only be exercised (bought/sold) at the time of expiration. While this is highly prevalent in the European stock markets, the American stock markets deviate from this assumption. 

 

  1. Another assumption that is a part of the BSM model is that there are no transaction costs applied in the call option. This also means that no brokerage or middlemen commission is involved in the process. 

 

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Limitations of the model 

 

  1. Volatility of market

 

The BSM model assumes that the stock market is in a state of constant volatility. This means that there is an unrealistic perception that the market faces constant fluctuation in the value of financial instruments like stock shares and that the rate of fluctuation is constant. 

 

However, this perception does not stand true for real-life stock markets as there is no certainty in the rate of fluctuation. 

 

Moreover, no options pricing model or theory can assume that the volatility of the market will remain constant throughout a definite time span.  

 

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  1. Limited to European Market

 

A lot of the features of the BSM model only comply with the reality of the European market. That said, one of the major limitations of the Black-Scholes model is that it is only limited to the European market and does not quite fit well in other markets. 

 

For instance, the way the American market functions is distinct from the European market, and perhaps the model does not work well for options in the American market. 

 

One of the instances that can make this limitation more understandable is that the exercise terms of the two market scenarios differ. In the European market, options can be bought or sold only at the time of expiration. 

 

But, the American market allows investors to trade options before the date of expiration or till the time it happens.  

 

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  1. No Returns on options

 

The third prominent drawback of this model is that it assumes that there are no returns on options. That said, the model calculates options price according to the assumption that the buyer or seller does not gain any returns in the process of trading options. 

 

However, this is far from what actually happens in the real world. In reality, stock shares are always subjected to returns or losses, depending on the volatility level of the market. 

 

Perhaps the assumption that there are no returns on options is fairly a reflection of how limited the model’s concept is. 

 

(Related reading: Types of Pricing Strategies)

 

 

Other Option Pricing Models

 

While the Black-Scholes-Merton option pricing model remains one of the most popular models to calculate the value of options, there are other options pricing models too. Here is a brief introduction to some of the other option pricing models.

 

  1. Binomial Option Pricing model

 

A risk-neutral option pricing model, the Binomial Option Pricing Model is another model used for options pricing. This model allows one to guess different stock prices over the time of expiration of an asset wherein the time of expiration is divided into equal parts. 

 

By looking at the current market situation, one presumes the possible stock prices and whether there will be any losses or gains. Thereafter, the prices are calculated and a binomial distribution is formed that helps to identify which time period will be the most beneficial. 

 

When it comes to the binomial option pricing model vs black scholes, the binomial model is better off in one aspect. Unlike the BSM model, this model allows options to be traded at any particular time, thus scrapping the need to trade an asset at the time of expiration only. 

 

There is no arbitrage in this concept, which implies that no risk is involved in this model. While pricing options, the model follows a binomial pattern wherein two possible outcomes are assumed - more value than the current price and less value than the current price. 

 

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  1. MONTE CARLO SIMULATION


A probability theory concept, Monte Carlo simulations in option pricing models are used to calculate the probability of different outcomes that cannot be easily calculated due to the occurrence of random variables. 

 

Used to anticipate the risk involved in different outcomes, this model assumes that the market is perfectly efficient in nature. Unlike the BSM model, this model assumes that the distribution of the different outcomes is a normal distribution (a bell-shaped curve). 

 

The model must involve a large sample size in order to get more accurate estimates. What’s more, this model is claimed to provide the true value of an option, even though the Black-Scholes model is considered to be more popular. 

 

“The basis of a Monte Carlo simulation involves assigning multiple values to an uncertain variable to achieve multiple results and then averaging the results to obtain an estimate.”Monte Carlo Simulations in Options Pricing 

 

 

Summing Up

 

To sum up, the Black-Scholes model is an option pricing model that takes into consideration a number of factors for estimating a fair price of an asset.

 

Even though the assumptions of this model are quite unrealistic in nature, it is very popular among all other option pricing models and is highly used by students, researchers, and stock market analysts. 

 

Simply put, this option pricing model is a way to estimate the fair value of an asset that is granted to an investor to buy or sell within the time of its expiration.

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