Question

Assume that you have 100 shares of XYZ stock which has a volatility of 30% and a current stock price of £60 per share. XYZ pays no dividends. The risk-free interest rate is 3%.

4.1 Use the Black-Scholes option pricing model to value a six-month, at-themoney European put option on XYZ stock with the information as specified above. (10 %)

4.2 Given the information above, what action should you take to hedge using call options and put options, respectively? What would be the total value of these two transactions? (8 %)

4.3 Give and briefly discuss three main problems when using the BlackScholes option pricing model. (9 %)

4.4 Which of the following statements is incorrect? Explain briefly.

A. Three benefits of constructing global investment portfolios are more investment choices, often higher rates of return, and low correlations between markets.

B. An option’s theoretical value depends on a synthetic risk-free portfolio. Both put and call options magnify the possible positive and negative returns of investing in the underlying security.

C. In a collar agreement, like the forward contract hedge, there is no initial out-of-pocket expense associated with this derivative combination. Instead, the manager effectively pays for her desired portfolio insurance by surrendering an equivalent amount of the portfolio’s future upside potential.

D. An option combination known as Strangle, which involves the simultaneous purchase or sale of a call and a put on the same underlying security with the same expiration date, offers a greater riskreward structure than the straddle.

Answer 1

4.1 )Given

no. of shares of XYZ stock = 100

standard deviation = 30% = 0.30

current stock price,S = £60

risk free rate,s = 3%

since the option is at the money option, stock price = strike price

strike price, K =  £60

T = time to expiration of option = 6 months = 0.5 years

d1 = (ln (S/K) + ((r +( s²/2))*T))/s*((T)^(1/2))

= (ln(60/60) + ((0.03 + (((0.30)²/2)*0.5))/0.30*(0.5)^(1/2)

= (0 + (0.03 + 0.045*0.5))/0.212132 = 0.247487 or 0.25 ( rounding off to 2 decimal places)

d2 = d1 - s*(T)^(1/2)

= 0.247487 - 0.30*(0.5)^(1/2) = 0.0353549 or 0.04 ( rounding off to two decimal places )

using d1 and d2 , we will calculate N(d1) and N(d2) which are the probabilities that a random draw from a standard normal distribution will be less than d1 and d2 respectively

these probailities are calculated from cumulative normal distribution table

N(d1) = N(0.25) = (N(0.26) + N(0.24))/2 = (0.5948 + 0.6026)/2 = 0.5987

N(d2) = N(0.04) = 0.5160

now to calculate the value of put option , first we will calculate the value of call otpion for the same values as the put option

Value of call option = S*N(d1) - Ke^-r*T*N(d2) = 60*0.5987 - 60*e^-0.03*0.5*0.5160 = 35.922 - 30.499 = £5.4229

value of put option = Value of call option + Ke^-r*T - S = 5.4229 + 59.1067 - 60 =  £4.5296