# Black Scholes Model

The Black-Scholes Model, also known as the Black-Scholes-Merton Model, is a mathematical model used to price European-style options. It was developed by economists Fischer Black, Myron Scholes, and Robert Merton in the early 1970s. This model provides a theoretical estimate of the price of European call and put options, and it is based on several assumptions about market behavior.

### Key Assumptions

1. No Dividends: The stock does not pay dividends during the option’s life.
2. Efficient Markets: Markets are efficient, meaning securities are fairly priced, and arbitrage opportunities do not exist.
3. No Transaction Costs: There are no transaction costs or taxes.
4. Constant Risk-Free Rate: The risk-free interest rate is constant and known.
5. Log-Normally Distributed Returns: Stock prices are assumed to follow a log-normal distribution, and their volatility is constant.
6. European Option: The option can only be exercised at expiration, not before.

### The Black-Scholes Formula

For a European call option, the Black-Scholes formula is:C=S0N(d1)−Xe−rTN(d2)C = S_0 N(d_1) – X e^{-rT} N(d_2)C=S0​N(d1​)−Xe−rTN(d2​)

For a European put option, the formula is:P=Xe−rTN(−d2)−S0N(−d1)P = X e^{-rT} N(-d_2) – S_0 N(-d_1)P=Xe−rTN(−d2​)−S0​N(−d1​)

where:d1=ln⁡(S0/X)+(r+σ2/2)TσTd_1 = \frac{\ln(S_0 / X) + (r + \sigma^2 / 2)T}{\sigma \sqrt{T}}d1​=σT​ln(S0​/X)+(r+σ2/2)T​d2=d1−σTd_2 = d_1 – \sigma \sqrt{T}d2​=d1​−σT​

and:

• CCC = price of the call option
• PPP = price of the put option
• S0S_0S0​ = current stock price
• XXX = strike price of the option
• rrr = risk-free interest rate
• TTT = time to expiration (in years)
• σ\sigmaσ = volatility of the stock’s returns
• N(⋅)N(\cdot)N(⋅) = cumulative distribution function of the standard normal distribution

### Interpretation

• d1d_1d1​ and d2d_2d2​ are intermediate variables that incorporate the inputs of the model.
• N(d1)N(d_1)N(d1​) and N(d2)N(d_2)N(d2​) represent the probabilities that the option will be in-the-money under the risk-neutral measure.
• The model outputs the fair price of a call or put option based on the current stock price, strike price, time to expiration, risk-free rate, and volatility.

### Applications

1. Option Pricing: The primary use of the Black-Scholes model is to price European call and put options.
2. Risk Management: Helps in determining the risk associated with holding or writing options.
3. Hedging Strategies: Assists in creating delta-neutral portfolios to hedge against market movements.

### Limitations

1. Assumption of Constant Volatility: In reality, volatility changes over time.
2. No Dividends: The basic model does not account for dividend payments, though extensions exist for this.
3. European-Style Options: The model applies only to options that can be exercised at expiration.
4. Market Assumptions: Assumes no transaction costs, taxes, and perfectly efficient markets, which are unrealistic in practice.

### Extensions and Variations

• Black-Scholes with Dividends: Adjusts the model to account for dividend payments.
• Stochastic Volatility Models: Such as the Heston model, which consider changing volatility over time.
• American Options: For options that can be exercised before expiration, models like the binomial tree model or Monte Carlo simulation are often used.

The Black-Scholes model revolutionized the field of financial economics by providing a systematic way to price options and contributed to the growth of derivatives markets. Despite its limitations, it remains a foundational tool in financial theory and practice.

4o

example of black sholes modelChatGPT

Let’s go through an example to illustrate how to use the Black-Scholes model to price a European call option.

### Given Data

• Current stock price (S0S_0S0​): $100 • Strike price (XXX):$105
• Time to expiration (TTT): 1 year
• Risk-free interest rate (rrr): 5% (0.05)
• Volatility (σ\sigmaσ): 20% (0.20)

### Step-by-Step Calculation

1. Calculate d1d_1d1​ and d2d_2d2​

d1=ln⁡(S0/X)+(r+σ2/2)TσTd_1 = \frac{\ln(S_0 / X) + (r + \sigma^2 / 2)T}{\sigma \sqrt{T}}d1​=σT​ln(S0​/X)+(r+σ2/2)T​d2=d1−σTd_2 = d_1 – \sigma \sqrt{T}d2​=d1​−σT​

Plugging in the given values:d1=ln⁡(100/105)+(0.05+0.202/2)×10.201d_1 = \frac{\ln(100 / 105) + (0.05 + 0.20^2 / 2) \times 1}{0.20 \sqrt{1}}d1​=0.201​ln(100/105)+(0.05+0.202/2)×1​d1=ln⁡(0.9524)+(0.05+0.02)0.20d_1 = \frac{\ln(0.9524) + (0.05 + 0.02)}{0.20}d1​=0.20ln(0.9524)+(0.05+0.02)​d1=−0.0496+0.070.20d_1 = \frac{-0.0496 + 0.07}{0.20}d1​=0.20−0.0496+0.07​d1=0.02040.20=0.102d_1 = \frac{0.0204}{0.20} = 0.102d1​=0.200.0204​=0.102d2=0.102−0.20=−0.098d_2 = 0.102 – 0.20 = -0.098d2​=0.102−0.20=−0.098

1. Calculate N(d1)N(d_1)N(d1​) and N(d2)N(d_2)N(d2​)

N(d1)N(d_1)N(d1​) and N(d2)N(d_2)N(d2​) represent the cumulative distribution function (CDF) of the standard normal distribution at d1d_1d1​ and d2d_2d2​. These values can be found using standard normal distribution tables or a computational tool.

Using a standard normal distribution table or calculator:N(d1)≈N(0.102)≈0.540N(d_1) \approx N(0.102) \approx 0.540N(d1​)≈N(0.102)≈0.540N(d2)≈N(−0.098)≈0.461N(d_2) \approx N(-0.098) \approx 0.461N(d2​)≈N(−0.098)≈0.461

1. Calculate the call option price (CCC)

C=S0N(d1)−Xe−rTN(d2)C = S_0 N(d_1) – X e^{-rT} N(d_2)C=S0​N(d1​)−Xe−rTN(d2​)

Plugging in the calculated values:C=100×0.540−105×e−0.05×1×0.461C = 100 \times 0.540 – 105 \times e^{-0.05 \times 1} \times 0.461C=100×0.540−105×e−0.05×1×0.461C=54.00−105×0.9512×0.461C = 54.00 – 105 \times 0.9512 \times 0.461C=54.00−105×0.9512×0.461C=54.00−45.97C = 54.00 – 45.97C=54.00−45.97C≈8.03C \approx 8.03C≈8.03

The price of the European call option, given the specified parameters, is approximately $8.03. This means an investor would be willing to pay around$8.03 today to have the right, but not the obligation, to buy the stock at $105 in one year. ### Summary 1. Input Values: • S0S_0S0​:$100
• XXX: $105 • TTT: 1 year • rrr: 5% (0.05) • σ\sigmaσ: 20% (0.20) 2. Calculated Values: • d1d_1d1​: 0.102 • d2d_2d2​: -0.098 • N(d1)N(d_1)N(d1​): 0.540 • N(d2)N(d_2)N(d2​): 0.461 • Call Option Price (CCC):$8.03

This example demonstrates how to use the Black-Scholes model to determine the fair price of a European call option based on the given inputs.