Understanding the Black-Scholes Pricing Model for Options in Microsoft Excel

The Black–Scholes pricing model is a mathematical method to calculate the fair value of European-style options, which are contracts that give the buyer the right, but not the obligation, to buy or sell an underlying asset at a fixed price on a specific date. The model was developed by Fischer Black, Myron Scholes, and Robert Merton in 1973, and it won them the Nobel Prize in Economics in 1997.

The model works by using five input variables: the current price of the underlying asset, the strike price of the option, the time to expiration, the risk-free interest rate, and the volatility of the asset. The model assumes that the asset price follows a random walk with constant drift and volatility, and that there are no transaction costs, taxes, dividends, or arbitrage opportunities. The model also assumes that the option can only be exercised at maturity, which is why it only applies to European-style options.

The model uses a partial differential equation, known as the Black–Scholes equation, to derive the price of a call option, which is the option to buy the asset. The price of a put option, which is the option to sell the asset, can be obtained by using a mathematical relationship called put-call parity. The model also provides a way to calculate the Greeks, which are measures of how sensitive the option price is to changes in the input variables.

The Black–Scholes model is widely used by traders and investors to price and hedge options, as well as to evaluate other financial instruments that are based on options, such as warrants, convertible bonds, and employee stock options. The model is also the basis for many extensions and modifications that aim to address its limitations and assumptions, such as the Black–Scholes–Merton model, the binomial model, and the stochastic volatility model.

Basic Theory:

The Black–Scholes model is designed to calculate the theoretical price of a financial option based on several key factors:

1. Current Stock Price (S): The current market price of the underlying asset.
2. Strike Price (K): The price at which the option holder can buy (call option) or sell (put option) the underlying asset.
3. Time to Maturity (T): The time remaining until the option expires.
4. Risk-Free Interest Rate (r): The interest rate on a risk-free investment, typically represented by a government bond.
5. Volatility (σ): The standard deviation of the stock’s return, representing its price volatility.

Procedures for Implementation in Excel:

To calculate the Black–Scholes option price in Excel, follow these steps:

1. Set Up Excel Spreadsheet:
   • Create a table with columns for each parameter: Current Stock Price (S), Strike Price (K), Time to Maturity (T), Risk-Free Interest Rate (r), and Volatility (σ).
   • Label additional columns for calculations, including d1, d2, and the option price.
2. Implement the Black–Scholes Formulas:
   • Use the provided formulas for d1 and d2.
   • Calculate the call option price (C) and put option price (P) using the Black–Scholes formulas.

Example:

Let’s consider a scenario with the following parameters:

• Current Stock Price (S): $100
• Strike Price (K): $95
• Time to Maturity (T): 1 year
• Risk-Free Interest Rate (r): 0.05
• Volatility (σ): 0.2

Calculation in Excel:

Now, using the Black–Scholes formulas mentioned earlier, we get:

d1 ≈ 0.532

d2 ≈ 0.332

Call Option Price (C) ≈ $11.62

Put Option Price (P) ≈ $4.82

Alternative Approaches:

While the Black–Scholes model is widely used, it’s essential to be aware of its assumptions and limitations. Alternative models, such as the Binomial Option Pricing Model or the Monte Carlo Simulation, may be considered for more complex scenarios or when the assumptions of the Black–Scholes model are not met.