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What is the Black-Scholes Equation?
The Black-Scholes equation is a mathematical formula used to price European-style options, which are options that can only be exercised at expiration. It was developed by economists Fischer Black and Myron Scholes in 1973 and has become a cornerstone of option pricing theory. The equation is derived under certain assumptions, including efficient markets, no transaction costs, and constant volatility. It has practical applications in derivatives trading, risk management, and portfolio optimization.
The Black-Scholes equation is as follows:
C = S * N(d1) - X * e^(-r * T) * N(d2)
Where:
C represents the theoretical price of the call option.
S is the current price of the underlying asset.
N(d1) and N(d2) are the cumulative distribution functions of standard normal distribution, evaluated at d1 and d2, respectively.
X is the strike price of the option.
e is the base of the natural logarithm (approximately 2.71828).
r is the risk-free interest rate.
T is the time to expiration of the option, expressed in years.
The d1 and d2 terms are calculated as follows:
d1 = (ln(S / X) + (r + (σ^2) / 2) * T) / (σ * sqrt(T))
d2 = d1 - σ * sqrt(T)
Where:
ln is the natural logarithm.
σ is the volatility of the underlying asset.
The equation provides a way to estimate the fair price of a European call option based on the underlying asset's price, the strike price, the time to expiration, the risk-free interest rate, and the asset's volatility. It's important to note that the Black-Scholes equation assumes a range of assumptions, including constant volatility and no dividends. Various extensions and modifications, such as the Black-Scholes-Merton model, have been developed to account for additional factors or relax some assumptions.
The Black-Scholes model is covered in more detail in module 3 of the CQF program.