Options Pricing Theory:

Explain the Black$-$Scholes$-$Merton $($BSM$)$ model for option pricing, including its assumptions, mathematical formulation, and key components. Discuss the practical applications of the BSM model in option valuation and risk management, and analyze its limitations and extensions. Compare and contrast alternative option pricing models, such as the Binomial model and the Heston model, highlighting their respective advantages and disadvantages.