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The Black-Scholes Model: The Standard for Option Pricing
Black-Scholes Model: A Revolution in Option Pricing
Published in 1973 by Fischer Black and Myron Scholes, this model provided a mathematical answer to the question: “What is a fair price to pay for a complex right?“
1. The Core Idea: The Risk-Free Portfolio
The foundation of the Black-Scholes model is Replication. The logic is that by mixing options and stocks in the right proportions, one can create a ‘risk-free portfolio’ whose value doesn’t change regardless of how the stock price moves. Therefore, the return on this portfolio must equal the risk-free interest rate ().
2. The Five Key Variables of Option Pricing
In the Black-Scholes formula, the option price ( or ) is determined by the following five variables:
Option Pricing Variables and Their Impact
| Variable | Meaning | Call Option Impact | Put Option Impact |
|---|---|---|---|
| Stock Price (S) | Current price of the underlying asset | Increase (+) | Decrease (-) |
| Strike Price (K) | Pre-determined exercise price | Decrease (-) | Increase (+) |
| Time to Maturity (T) | Time remaining until expiration | Increase (+) | Increase (+) |
| Volatility (σ) | Degree of price fluctuation | Increase (+) | Increase (+) |
| Interest Rate (r) | Risk-free market interest rate | Increase (+) | Decrease (-) |
3. The Logical Flow of Pricing
The Black-Scholes price is derived through the following stochastic process:
Assume that the log-returns of stock prices follow a normal distribution.
Convert the relationship between stock price and strike price into standardized values.
Use standard normal distribution tables to find the probability of exercise.
Bring the pre-determined strike price back to its present value.
💡 Professor’s Tip
‘Volatility ()’ is the only variable in the Black-Scholes formula that cannot be directly observed in the market. Therefore, we often plug the current market option price into the formula to extract the ‘Implied Volatility’. This is the core concept behind the Fear Index (VIX).