Black-Scholes Model

The potential cost of European-style options is determined using a mathematical model called the Black-Scholes Model. With Robert Merton’s help, economists Fischer Black and Myron Scholes created it in 1973, revolutionizing options pricing and laying the groundwork for contemporary quantitative finance. The concept is widely used in financial markets to value options on stocks, commodities, currencies, and other types of assets.

In this article, we will discuss the formula that helps us calculate the prices of options using the Black-Scholes Model and also see some solved examples for it.

What is the Black Scholes Model?

Black-Scholes model, also known as the Black-Scholes-Merton model, is a mathematical model used for pricing financial derivatives, most commonly options contracts. It was developed by Fischer Black, Myron Scholes, and Robert Merton in the early 1970s and has become a widely used tool in finance.

Black-Scholes formula calculates the theoretical price of European-style options, which can only be exercised at expiration. It provides a means to determine the fair value of an option at a given point in time based on factors such as the current stock price, the option’s strike price, time to expiration, risk-free interest rate, and volatility of the underlying asset.

Assumptions of Black-Scholes Model

The model makes several key assumptions:

• No Dividends: The underlying asset does not pay any dividends during the option’s life.
• Efficient Markets: Market movements are random and follow a geometric Brownian motion with constant drift and volatility.
• Risk-Free Rate: The risk-free interest rate is constant and known.
• Constant Volatility: The volatility of the underlying asset’s returns is constant and known.
• Log-Normally Distributed Returns: The returns on the underlying asset are normally distributed.

Formula for Black Scholes Model

The Black-Scholes formula calculates the theoretical price of a European call or put option. For a call option, the formula is:

[Tex]\bold{C = S_0N(d_1) – Xe^{-rt}N(d_2)}[/Tex]

Where,

• C is the call option price,
• S₀​ is the current price of the underlying asset,
• X is the strike price,
• r is the risk-free interest rate,
• t is the time to expiration, and
• N(d₁​) and N(d₂​) are cumulative distribution functions of the standard normal distribution.

Also,

• [Tex]d_1 = \frac{\ln{\left(\frac{S_0}{X}\right)} + (r + \frac{\sigma^2}{2})t}{\sigma\sqrt{t}} [/Tex]
• [Tex]d_2 = d_1 – \sigma\sqrt{t} [/Tex]

Black Scholes Model Hypotheses

The following presumptions underpin the Black-Scholes Model:

• Efficient Markets: The model makes the assumption that there are no transaction costs and that markets are efficient.
• Constant Volatility: This presupposes that during the option’s lifecycle, volatility will not change.
• Trade Constantly: This presupposes that trade may occur constantly.
• Rate devoid of risk: The model makes the assumption that the rate devoid of risk is always present.
• Log-Normal Distribution: This strategy makes the assumption that the underlying asset’s returns have a log-normal distribution.

Applications of the Black-Scholes Model

Some of the common applications of Black-Scholes Model are:

• Options Pricing: It aids in determining the fair price of options, which is critical for investors and traders making educated purchasing or selling choices.

• Risk Management: By knowing the fair value of options, traders may better control their risk exposure and minimize possible losses.

• Option Strategies: By giving insights into option price dynamics, the model helps to build various option trading strategies such as hedging and speculating.

• Derivatives Valuation: In addition to options, it may be used to evaluate other derivative securities, offering a framework for determining the value of financial instruments having comparable features.

Limitations of Black Scholes Model

The Black-Scholes Model has drawbacks even if it offers insightful information:

• Frictions in the Market: The model does not take into consideration the transaction fees and levies that are present in real-world marketplaces.
• Volatility Assumption: The idea that market volatility is constant is false since market volatility fluctuates.
• Asset Price Movements: The model makes the assumption that asset prices would always fluctuate, which may not always be the case.

Read More,

• What are Derivatives and How it Works?
• What are Options Contracts?
• What are Futures Contracts?
• Difference Between Options and Futures

Solved Problems

Problem 1: Determine the European call option’s cost using the following parameters:

• The current S₀ stock price is $50.
• Price of strike (X): $55
• Expiration date (t): 0.5 years
• Rate of return without risk (r): 5% annually
• Annual volatility (σ): 20%

Solution:

Using the Black-Scholes formula:

[Tex]d_1 = \frac{\ln{\left(\frac{50}{55}\right)} + \left(0.05 + \frac{0.2^2}{2}\right) \times 0.5}{0.2\sqrt{0.5}} [/Tex]

⇒ d₁ ≈ -0.079

and [Tex]d_2 = -0.079 – 0.2\sqrt{0.5} [/Tex]

⇒ d₁ ≈ -0.248

Using standard normal distribution table, we find:

[Tex]N(d_1) \approx 0.469 [/Tex]and [Tex]N(d_2) \approx 0.401[/Tex]

[Tex]C = 50 \times 0.469 – 55 \times e^{-0.05 \times 0.5} \times 0.401[/Tex]

⇒ C ≈ 4.04

So, the price of the call option is approximately $4.04.

Probelm 2: A stock is trading at $50, a European call option has a $55 strike price, it will expire in six months (or 0.5 years), the risk-free interest rate is 4% annually, and the stock volatility is 20% annually. Utilizing the Black – Scholes Model, determine the call option’s theoretical price.

Solution:

Given:

• S₀ = 50
• K = 55
• T = 0.5
• r = 0.04
• σ = 0.20

Using the Black – Scholes formula for a call option:

[Tex]d_1 = \frac{\sigma \sqrt{T} \ln\left(\frac{S_0}{K}\right) + (r + \frac{\sigma^2}{2})T}{\sigma \sqrt{T}}[/Tex] and [Tex]d_2 = d_1 – \sigma \sqrt{T}[/Tex]

[Tex]d_1 = \frac{\ln\left(\frac{S_0}{K}\right) + \left(r + \frac{\sigma^2}{2}\right)T}{\sigma \sqrt{T}}\\ d_2 = d_1 – \sigma \sqrt{T} [/Tex]

Calculate d1and d2​:

[Tex]d_1 = \frac{\ln\left(\frac{50}{55}\right) + \left(0.04 + \frac{0.20^2}{2}\right) \times 0.5}{0.20 \times \sqrt{0.5}} \approx -0.1814[/Tex], and

[Tex]d_2 = -0.1814 – 0.20 \times \sqrt{0.5} \approx -0.4814[/Tex]

Using a standard normal distribution table,

N(d₁​)≈0.4286 and N(d₂) ≈ 0.3159

Now, calculate the call option price:

C=50×0.4286−55×e^−0.04×0.5×0.3159 ≈ 2.5804

So, the theoretical price of the call option is approximately $2.58.

Practice Problems

Problem 1: Given the following parameters, calculate the price of a European call option using the Black-Scholes formula.

• Current price of the underlying asset (S₀): $50
• Strike price (K): $55
• Time to expiration (t): 0.5 years
• Risk-free interest rate (r): 5% per annum
• Volatility (σ): 20% per annum

Problem 2: Using the same parameters as in problem 1, calculate the price of a European put option.

Problem 3: Keeping all other parameters constant, investigate how changes in volatility affect the price of a call option. Assume the following:

• S₀ = $50
• K = $55
• t = 0.5 years
• r = 5%
• Volatility ranges from 10% to 30% in increments of 5%.

Problem 4: Explore the impact of time to expiration on the price of a call option. Keep all other parameters constant and vary the time to expiration from 0.25 years to 1 year in increments of 0.25 years. Use the following parameters:

• S₀ = $50
• K = $55
• r=5%
• σ=20%

Problem 5: Examine how changes in the risk-free interest rate affect the price of a call option. Keep all other parameters constant and vary the interest rate from 2% to 8% in increments of 1%. Use the following parameters:

• S₀ = $50
• K = $55
• t = 0.5years
• σ = 20%

FAQs: Black-Scholes Model

What is Option Contract?

An option contract is a legally binding agreement between two parties where the offeror promises to keep an offer open for the offeree to accept within a specified period without revoking it.

What is Future Contract?

A futures contract is a standardized legal agreement to buy or sell a particular asset at a predetermined price on a specific date or during a specific month.

What is the key difference between Americal Style Options and European Style Options?

The key difference between American Style Options and European Style Options lies in when they can be exercised. American options can be exercised at any time before the expiration date, while European options can only be exercised at the expiration date.

Who developed Black-Scholes Model?

The Black-Scholes Model was developed by Fischer Black, Myron Scholes, and Robert C. Merton.

What is the Formula for Black-Scholes Model?

Formula for Black-Scholes model is given as [Tex]\bold{C = S_0N(d_1) – Xe^{-rt}N(d_2)}[/Tex].

Can Black-Scholes Model be used to calculate the Price of Future Contract as Well?

No, options are secondary derivatives because they derive their values from futures contracts themselves.