The Black–Scholes model is used to price European options. For a call option:
\[ C = S \cdot N(d_1) - K \cdot e^{-rT} \cdot N(d_2) \] with \[ d_1 = \frac{\ln(S/K) + (r + \frac{\sigma^2}{2})T}{\sigma\sqrt{T}}, \quad d_2 = d_1 - \sigma\sqrt{T} \]
Here, \(S\) is the current stock price, \(K\) is the strike price, \(T\) is the time to expiration (in years), \(r\) is the risk‑free interest rate, and \(\sigma\) is the volatility.
The implied volatility (IV) is the value of \(\sigma\) that, when substituted into the model, gives the observed market option price.
The Greeks measure the sensitivity of an option's price to various factors:
The volatility smile is a pattern in which options with strike prices far from the current price tend to have higher IV. This phenomenon suggests that the market expects larger moves in the underlying asset for these options.
High IV may indicate market uncertainty or expensive options, while low IV might suggest lower expected movement or undervalued premiums.
Below are some common strategies along with their payoff diagrams. The diagrams now display each individual leg’s net payoff (which may be negative) as light gray dashed lines, and the net payoff (sum of all legs) as a solid colored line. All diagrams share the same x‑axis ([50,150]) and y‑axis ([–20, 60]) scales.