Question

Question 3

a.   Explain the assumptions in the Black-Scholes-Merton model?

b.   What is the price of a European call option on a non‐dividend‐paying stock with the stock price is £73, with a strike price is £73, volatility is 40% pa. risk‐free interest rate is 10% pa, and the time to maturity is 6 months?

c.   Without applying the Black‐Scholes model, what is the price of a 6 month European put on the same stock in b) with strike price of £70

If possible, please provide a detailed step by step as I would like to fully understand and not just copy answers. Thank you :)

Answer 1

a. The assumptions in the Black-Scholes Model are-

1. Lognormal Distribution of stock Prices- It is assumed that the stock prices are distributed lognormally in the Black-Scholes Model to adhere to the fact that stock prices cannot go negative. The stock returns are assumed to be normally distributed.

2. European Options- The BS model assumes and calculates option prices for European options and hence assumes that the options can be exercised only on the expiry day.

3. Random Walk- As no one can predict the movement of the underlying i.e. the stock, we assume a random walk to mimic the movement of the stock prices.

4. Zero Dividends- Although we assume zero dividends, the formula can be changed to adjust for dividends also.

5. No arbitrage, no transaction costs, etc.

b. The Black-Scholes formula is given as-

Putting all the values, we get the call price as = $9.91.

c. We use the Put-call parity formula for this.

Call + Xexp(-rt) = Put + Share

9.91 + 70 x exp(-0.1 x 0.5) = Put + 73

Put = $3.49