Question

Question 1

The following information is given about options on the stock of a certain company.

Current stock price = $23

Exercise price = $20

Continuously compounded risk-free rate = 9.0 percent

Time to expiration = 6 months (T = 0.5)

The variance of the continuously compounded return on the stock = 15.0 percent

The volatility = Sqrt(.15)

d₁ = [ln(S₀/X) + (r_c + σ²/2)T]/σSqrt(T) = [ln(23/20) + (.09 + .15/2)(.5)]/(.3873 x sqrt(.5)) = [.1398 + (.165)(.5)]/.2739 = .2223/.2739 = .8116

d₂ = d₁ - σSqrt(T) = .8116 – (sqrt(.15))(sqrt(.5)) = .8116 – .2739 = .5377

No dividends are expected.

Find the values of N(d₁) and N(d₂) using the excel function ‘=normsdist( ).

What value does the Black-Scholes-Merton model predict for the call?

What is the delta of the call?

If the stock price goes up by $1 to $24, what would be the call price based on the delta?

Suppose that June 20 call has the gamma of .0456. If the stock price goes up by $1 to $24, what would be the delta of the call based on the gamma?

Suppose that the call rho (ρ) is 6.74. If the continuously compounded risk-free rate goes up by 1 percent to 10 percent, what would be the expected call price based on the rho?

Suppose that the call vega (ν) is 4.67. If the volatility of the continuously return on the stock increase by 1 percent, what would be the expected call price based on the vega?

Suppose that the theta (θ) is 3.02. If the time to expiration decreases so that the call expires in 3 months (T = .25), what would be the expected call price based on the theta?