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Introduction to the Black-Scholes Option Pricing Model

Arriving at the price of a derivative contract, especially options, is challenging in the financial market. The price movement of these contracts relies on various parameters and requires complex mathematical formulas that consider multiple scenarios during the computation process. This is where the Black-Scholes or Black-Scholes-Merton (BSM) model comes into play. In this article, we will cover the black scholes option pricing model in detail.

What is Black Scholes Model?

The Black Scholes Model was formulated in 1973 by economists Fischer Black, Myron Scholes, and Robert Merton. This model computes the theoretical value of option contracts. The model considers stock prices, expected dividends, strike prices, rate of interest, time to expiration, and volatility to provide precise results.

The model presumes that the underlying asset’s price tracks a geometric Brownian motion, which means that the asset’s returns are normally distributed and the price path is continuous.

This assumption is crucial because it allows differential equations to model the option’s price dynamics.

Black-Scholes model assumptions

The Black-Scholes option pricing model categorises assumptions into market behaviour, statistical, and operational classes. Here are some further insights into them.

Market behaviour assumptions

• The model assumes that options are European in nature. That means they can only be exercised at expiration, not before.
• It presumes that the underlying asset does not pay dividends during the option’s life.
• The markets are considered efficient. This implies that prices reflect all available information at any given time.

Statistical assumptions

• Stock prices are presumed to follow a lognormal distribution. That means they can never become negative and are skewed to the right.
• The underlying asset’s price follows a random walk, suggesting that future price movements are unpredictable and independent of past movements.
• The underlying asset’s returns are normally distributed, implying a symmetric distribution around the mean.

Operational assumptions

• The model assumes perfect liquidity in the market, allowing for the prompt buying and selling of assets without impacting their price.
• An arbitrage opportunity, which allows for a risk-free profit, is assumed to be nonexistent in the Black-Scholes model.
• There are no restrictions on short selling. The model assumes that it can be done without any cost.

How does the Black Scholes Model Work?

Here is the Black Scholes model formula:

C = S × N(d1) – X × e^(-r × T) × N(d2)

P = X × e^(-r × T) × N(-d2) – S × N(-d1)

Where:

• ( C ) is the price of the call option
• (P) represents put option price
• ( S_0 ) is the current price of the stock
• ( K ) is the option’s strike price
• ( r ) represents the risk-free interest rate
• ( T ) represents the time to expiration
• ( N ) represents the standard normal distribution’s cumulative distribution function
• ( d_1 ) and ( d_2 ) are computed as follows:

d1 = (ln(S/X) + (r + σ^2/2) × T) / (σ × sqrt(T))

d2 = d1 – σ × sqrt(T)

Black Scholes Model example

Let’s consider a practical example to illustrate how the Black-Scholes Model works. Suppose we have a stock currently priced at Rs. 100 and want to price a call option contract with a strike price of Rs. 105 that expires in a year. Assume the risk-free interest rate is 5% and the stock’s volatility is 20% per annum.

Using the Black-Scholes formula, we first calculate ( d_1 ) and ( d_2 ):

( d_1 ) =  (1 ÷ 0.2√1) [ln(100 ÷ 105) + (0.05 + 0.22 /2​)]

( d_2 ) = d1​ − 0.2

After computing ( d_1 ) and ( d_2 ), we find the values of ( N(d_1) ) and ( N(d_2) ) using the standard normal cumulative distribution function. Finally, we plug these values into the Black-Scholes formula to get the price of the call option.

Importance of the Black-Scholes Model

The Black-Scholes model is a valuable tool in various areas of finance. Here are a few.

Risk management

You can use the model to gauge the sensitivity of an option’s price to various factors, known as the ‘Greeks’. These include delta, which measures the rate of change of the option price to the price of the underlying asset; theta, which measures the rate of time decay; and vega, which measures sensitivity to volatility.

Strategic investment

The model’s insights into option pricing have also led to the development of complex trading strategies. You can use the Black-Scholes Model to identify mispriced options and construct combinations of options and other financial instruments, such as straddles, strangles, and spreads, to take advantage of market conditions.

Economic research

This model helps economists understand market behaviour and predict trends. By analysing how option prices move, researchers can gain insights into investor behaviour and market dynamics.

Financial regulation

By using the model, regulators can assess how these products are priced in the market and ensure they are not used to manipulate prices or mislead investors. The model’s calculations also aid in evaluating the risk level of financial institutions that trade in options, contributing to the overall stability of the financial system.

Corporate finance

This model is handy for valuing employee stock options in corporate finance. Companies can use the model to make informed decisions on how much stock to offer employees and at what price.

Conclusion

The Black-Scholes Model is a complex framework that has revolutionised the field of financial derivatives. Incorporating key market variables into a coherent mathematical model allows for the valuation of options in a way that reflects their accurate theoretical price. While it may not be perfect, its importance in the financial industry cannot be overstated, and it continues to be a benchmark for option pricing theory and practice. To know more about such concepts, subscribe to StockGro.

FAQs

What is the Black-Scholes Model?

The Black-Scholes Model is a mathematical framework for pricing European-style options. It calculates the theoretical price of options based on factors like the underlying asset’s price, strike price, risk-free rate, volatility, and time to expiration.

How do changes in input variables affect the Black-Scholes Model?

In the Black-Scholes Model, the option’s value usually increases if the stock price or volatility increases. If there’s more time until the option expires, that can also increase its value. However, if interest rates rise, the value of a call option might decrease. It is all about balancing these factors.

What is the difference between American and European options?

The fundamental difference between American and European options is when they can be exercised. The advantage of American option contracts is their flexibility, allowing investors to exercise them at any point before they expire. On the other hand, European options can only be exercised at the exact expiration date, limiting when you can use them.

What are the key assumptions of the Black-Scholes Model?

The Black-Scholes Model presumes that stock prices adhere to a certain pattern that can’t go negative, there are no dividends, and you can only use options at their expiration. It also assumes markets move randomly, there are no trading costs, interest rates don’t change, and there is no chance for a risk-free profit.

What if there are dividends during the option’s life?

If dividends declaration occurs during an option’s life, the Black-Scholes Model can be altered to account for them. The model originally didn’t include dividends, but it can be modified by reducing the stock price by the present value of expected dividends. This helps estimate the option’s fair value more accurately.