What volatility smile shows — implied volatility varies with strike price, contradicting Black-Scholes

What implied volatility tells you — the market's expectation of future price movement

Implied volatility indicates the market's anticipated future price fluctuation of an asset

What the Black-Scholes assumptions are — constant volatility, no dividends, log-normal prices, no transaction costs

Black-Scholes assumes constant volatility, no dividends, log-normal price distribution, and no transaction costs

What the Black-Scholes formula prices — European call and put options

Black-Scholes prices European call and put options using volatility, interest rates, and time to expiration

How does implied volatility decay affect the pricing of exotic options, particularly as the expiration date approaches?

Implied volatility decay increases the value of long-dated exotic options as expiration nears

How do implied volatility, beta, and alpha influence the pricing and risk management of equity options?

Implied volatility, beta, and alpha affect option pricing and risk management by indicating market sentiment, systemic risk, and stock performance respectively

How does the Black-Scholes formula for pricing European call and put options incorporate the concepts of stochastic volatility and risk-neutral valuation? support: The Black-Scholes formula incorporates stochastic volatility by assuming that the volatility of the underlying

is a random process, and risk-neutral valuation through the use of a risk-free interest rate and a discount factor