C = S*N(d1) - K*e^(-rT)*N(d2)
Where d1 = [ln(S/K) + (r + v²/2)T] / (v*√T), d2 = d1 - v*√T, and N() is the cumulative normal distribution.
An options premium is the price paid by the buyer to the seller (writer) for the right granted by the options contract. It represents the cost of acquiring the option and is determined by several factors including the relationship between the underlying asset price and strike price, time remaining until expiration, implied volatility, risk-free interest rates, and dividends.
The premium consists of two components: intrinsic value and time value (extrinsic value). Intrinsic value is the amount the option is in the money -- for calls, it is the underlying price minus the strike price (if positive). Time value represents the additional premium above intrinsic value, reflecting the probability that the option could become more valuable before expiration.
The Black-Scholes model, developed by Fischer Black, Myron Scholes, and Robert Merton in 1973, is the most widely used theoretical pricing model for European-style options. It assumes that the underlying asset follows a geometric Brownian motion with constant volatility and that markets are frictionless.
The model takes five inputs: underlying price, strike price, time to expiration, risk-free rate, and implied volatility. Higher volatility increases both call and put premiums because greater price uncertainty means more potential for the option to finish in the money. Longer time to expiry also increases premiums as there is more time for favorable price movements.
The Black-Scholes model has known limitations. It assumes constant volatility, but in reality, implied volatility varies across strikes (volatility smile/skew) and time. It also assumes no dividends for the basic model, continuous trading, no transaction costs, and log-normally distributed returns.
Real market prices may deviate from Black-Scholes theoretical values due to these assumptions, supply and demand dynamics, and market microstructure effects. Traders often use the model as a starting point and adjust for known biases. American-style options, which can be exercised early, may also differ from Black-Scholes European-style pricing.