9) Analytic solutions to the Black-Scholes equation

Question 1

What are the Black-Scholes formulas for European call and put option prices

Answer 1

Question 2

What are the Delta values for European call and put options, and how do they behave

Answer 2

Question 3

What are the main Greeks in the Black-Scholes model and what do they represent

Answer 3

Question 4

What is implied volatility and what does the volatility smile indicate about the Black-Scholes model

Answer 4

• Implied volatility is the value of σ that, when plugged into the Black-Scholes formula, gives the observed market price of an option
• It is usually found using an iterative method, such as Newton-Raphson

However:

• Implied volatility often varies depending on the strike price and time to maturity, contradicting the Black-Scholes assumption of constant volatility
• This variation forms patterns like the volatility smile, frown, or wry smile, which reflect real market behaviour and show the model’s limitations