L21 - Black-Scholes-Merton model

Question 1

What are the assumptions of the BSM model?

Answer 1

1. The risk free rate is:
   
   1. constant
   2. continuously compounded
2. the stock price is:
   
   1. log normal –> means that returns are normal, and stock price cannot be negative if log-normal
   2. has no jumps
   3. constant volatility
3. The stock ays a constant dividend yield
4. The stock and risk-free security can be traded all the time at no cost

Question 2

What is the BSM for a European call option?

Answer 2

• Normal distribution functions –> c.d.f
  
  • give a high value If you put a large value into them, and a small value if a small number is put into it
    
    • so if d1 is large and positive –> N(d1) close to 1
    • if d1 is large and negative –> N(d1) is close to 0
  • delta function is the dividend yield -. minus cause you down receive it when holding the call option so discount it away
  • Ln(S₀/X) –> moneyness
    
    • if the stock > strike –> deep in the money therefore this number becomes very large and in turn, d1 becomes large
    • stock < strike –> deep out the money the ratio is gonna be close to 0 something, a log of anything less than 1 is negative so d1 will be negative and N(d1) will be close to 0

Question 3

Further intuition of the BSM model?

Answer 3