17 January 2023

Among the important concepts in modern financial theory, the Black Scholes model, developed in 1973, is one of the most prominent. It is also called BSM or the Black-Scholes-Merton model and is a mathematical equation that is used to determine the fair price of an options contract.

This article discusses the Black-Scholes equation, its assumptions, its calculation, and more. Let’s dive in.

The Black Scholes Merton model is a mathematical simulation of the dynamics of a financial market that contains instruments like futures, options, swaps, and forwards. The fair price of financial instruments is determined using this model. It is also called the Black Scholes option pricing model and it indicates that every option has a unique price irrespective of the risk associated with the underlying security and returns expected from it.

Many finance experts believe that the Black Scholes Model has helped to make options and stock markets more efficient. Although it has been designed only for European options, the model still offers a fair understanding of the impact of changes in the prices of underlying stocks on the price of the option and options trading strategies. Investors can use the model to optimise their investment portfolio and manage risk.

The Black Scholes formula estimates the value of a call option by multiplying the current stock prices by a probability factor (D_{1}) and then subtracting from it the product of discounted exercise payment time and a second probability factor (D_{2}).

D_{1} is the cumulative standard normal probability distribution function. However, D_{1} and D_{2} are hard to explain due to the absence of adequate explanations in original and subsequent research papers about the Black Scholes model. D_{2} is defined as the risk-adjusted probability of exercising an option. D_{1} is the difference between the present value of the contingent receipt of stock and the current stock price.

Below is the mathematical notation of the Black Scholes Formula:

**C= SN(D _{1}) – Ke^{-rt}N(D_{2})**

Here,

- d1=lnStK+r+22tt
- d
_{2 }= d_{1}– t - C= call option price
- S= current stock price or price of the underlying security
- K= strike price
- r = risk-free interest rate
- t = time to maturity
- N= normal distribution

You can use a BSM calculator to figure out the value of price of an options contract. While estimating the values of the call and put option using the Black Scholes calculator, investors must input six main variables. These are stock price, strike price, option term, dividend yield, volatility, and risk-free interest rate. The Black Scholes option calculator will thus give the call option price.

The first step is to find the 5 main values required by the Black Scholes formula. They are mentioned below:

- Current stock price (S)
- The exercise price of the call option (C)
- Time to maturity (t)
- Risk-free interest rate
- Stock volatility based on the standard deviation of historical log returns

The following example illustrates the price calculation of a European call option:

- Time to maturity = 6 months
- Exercise price = $50
- Trading price of the stock = $52
- Cost of the call option = $4.5
- Annual risk-free rate = 5%
- Standard deviation of stock returns =12% annually

To determine whether it is a good idea to buy the option, the value of the call option must be derived using the Black Scholes option pricing model. This will be compared with the current option price to assess if it is fairly priced.

First, the values of d_{1} and d_{2} need to be calculated:

**d1=ln 6050+5%+12%22*0.5 12%2*0.5=0.7993**

**d2=ln 6050+5%-12%22*0.5 12%2*0.5=0.7144**

The next step is to calculate the standardised normal distribution probability. The NORMSDIST function in Microsoft Excel will be used for this.

**N(d _{1}) = 0.7879**

**N(d _{2}) = 0.7625**

Using these values in the Black Scholes Formula, C= $3.788

The model has estimated an option value lower than the premium on currently traded call options. The possible reasons behind this can be an overvalued option or a low estimate of volatility.

The Black-Scholes model is based on multiple assumptions. They are mentioned below:

The volatility of a stock measures the price change over time. The Black Scholes model assumes that volatility will remain the same over time. However, volatility can never be constant over a long time. The stochastic- process generated estimates are used in place of Black Scholes constant volatility in advanced option valuation models.

The Black Scholes model also assumes that the underlying stocks do not pay any dividends to the investors during the life of an option. However, in reality, investors can earn dividends from their holdings. Thus, adjustments were later made to the initial Black Scholes model to include dividends. The basic model is commonly adjusted for dividends by subtracting the discounted value of the expected dividend from the current stock price.

Similar to volatility, the Black-Scholes model also assumes the interest rates to be constant. The risk-free rate represents this rate in the model and is constant. In reality, a risk-free return rate doesn’t exist. Thus, due to its credibility, the 30-day rate of U.S Government Treasury bills is often used for calculation. However, when volatility increases, the treasury rates also change.

According to this assumption of the Black Scholes model, the probability of an increase or decrease in the price of the underlying stock is the same at any given time. This means that it is not possible to consistently predict market movement.

The model also assumes that the stock prices are lognormally distributed. The basis of this assumption is the principle that asset prices are bounded by zero and cannot be negative.

The model is also based on the assumption that there are no buying and selling fees or trading barriers on options and stocks.

Another assumption behind the Black Scholes model is that any number of options and stocks can be bought or sold at any time.

The Black Scholes Model is used to determine the present value of a call option, the final value of which depends on the stock price at the maturity date. Since the stock prices are variable, the call option’s value keeps changing too. Investors who wish to trade the option contract must have a way to estimate the expected values involved in the call option. This will help to find the value that they can get by buying the option and what they will have to pay if they exercise it.

While the Black Scholes option pricing model is widely used, it has some limitations too. Below are the limitations of the model:

The model works accurately to determine the prices of European options but is not an accurate measure of the stock options in the U.S (American options). This is because it is based on the assumption that options can be exercised only on the maturity date.

While risk-free interest rates are an assumption behind the BSM model, they do not exist in the real world.

Brokerage fees, commissions, and other costs mostly accompany trades. The Black Scholes model ignores these costs and assumes that there is no friction in the market. This is far from reality. Ignoring these costs may lead to inaccurate valuations.

The model assumes that there are no dividends or interest earnings from stock options. However, the trading market relies primarily on returns.

The Black Scholes model is an important mathematical formula that forms an integral part of modern financial history. It is among the most accurate ways to compute the fair value of a European call option. While it has some limitations and the calculations are complex, options traders can still benefit from it, provided they understand the model in detail.

**Ans: **The Black Scholes Merton model is used to compute the fair prices of options. It makes some assumptions and computes the price of European-style call options using known variables such as strike price, time till expiration, and current price.

**Ans: **The model first multiplies the net present value of the strike price with cumulative standard normal distribution. It then multiplies the cumulative standard normal probability distribution with the stock price and subtracts the result of the previous step from this product.

**Ans: **The Black Scholes equation uses the risk-free interest rate, the current price of the underlying security, volatility, the option’s strike price, and time until maturity as inputs for calculation.

**Ans: **Implied volatility is used in the Black Scholes model. This kind of volatility is used to estimate future variability of the underlying assets that the contracts depict.

**Ans: **No, this model employs various metrics and figures and hence may not be considered an accurate representation of real situations.

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