Introduction to the Black-Scholes formula | Finance & Capital Markets | Khan Academy

Introduction

This tutorial provides an introduction to the Black-Scholes formula, a fundamental concept in finance and capital markets. Understanding this formula is crucial for pricing options and derivatives, making it a valuable tool for anyone interested in financial markets and investment strategies.

Step 1: Understand the Basics of the Black-Scholes Formula

• The Black-Scholes formula is used to calculate the theoretical price of European-style options.

It considers various factors such as

• The current stock price
• The strike price of the option
• The time to expiration
• The risk-free interest rate
• The volatility of the stock price

Step 2: Learn the Formula

The Black-Scholes formula can be expressed mathematically. Here’s the formula for a call option:

C = S0 * N(d1) - X * e^(-rt) * N(d2)

Where:

• C = Call option price
• S0 = Current stock price
• X = Strike price of the option
• r = Risk-free interest rate
• t = Time to expiration (in years)
• N(d1) and N(d2) = Cumulative standard normal distributions
• d1 and d2 are calculated as follows:

d1 = [ln(S0/X) + (r + (σ^2)/2)t] / [σ * sqrt(t)]
d2 = d1 - σ * sqrt(t)

• σ = Volatility of the stock price

Step 3: Apply the Formula

To use the Black-Scholes formula

1. Gather necessary inputs: current stock price, strike price, risk-free rate, time to expiration, and volatility.
2. Calculate d1 and d2 using the formulas above.
3. Use the cumulative standard normal distribution to find N(d1) and N(d2).
4. Plug these values into the Black-Scholes formula to find the call option price.

Step 4: Explore Real-World Applications

The Black-Scholes model is widely used by traders and financial analysts for

• Pricing options to make informed trading decisions
• Hedging risks in portfolios
• Valuing employee stock options

Common Pitfalls to Avoid

• Ensure accurate data input: Using wrong figures for volatility or interest rates can lead to incorrect option pricing.
• Understand the limitations: The Black-Scholes model assumes constant volatility and interest rates, which may not always hold true in real markets.

Conclusion

The Black-Scholes formula is an essential tool for pricing options in financial markets. By understanding its components and how to apply it, you can make better investment decisions. For further learning, consider exploring related topics such as implied volatility and other derivatives.