Black-Scholes Model for Options Pricing

The Black-Scholes Model is a mathematical model used for calculating the theoretical price of European-style options. It was developed by economists Fischer Black and Myron Scholes in collaboration with Robert Merton in the early 1970s. The model has been widely adopted in finance and is foundational for understanding options pricing. The original model is designed for non-dividend-paying stocks and assumes constant volatility and interest rates. The formula for the Black-Scholes Model is as follows:​

Key concepts in the Black-Scholes Model include:

1. Call Option Price : Represents the price an investor pays for the right to buy a stock at the strike price before expiration.

2. Put Option Price : Represents the price an investor pays for the right to sell a stock at the strike price before expiration.

3. European Style: The model is specifically designed for European options, which can only be exercised at expiration.

4. No Dividends: The original model assumes that the underlying stock does not pay dividends during the option's life.

5. Constant Volatility and Interest Rates: The model assumes that volatility and interest rates remain constant over the option's life.

6. Risk-Neutral Pricing: The model uses a risk-neutral approach, assuming that the expected return on the stock is the risk-free rate.

While the Black-Scholes Model has been influential, it does have limitations. It assumes constant volatility, which may not hold in real markets, and it may not accurately price options on assets that pay dividends. Additionally, the assumption of continuous trading is not always realistic. Various extensions and modifications, such as the Black-Scholes-Merton Model, have been developed to address some of these limitations.