Question

must present and explain in words all the steps.

Question A 6-month call option with an exercise price of $50 on a stock that is trading at $52 costs $4.5.

1. Determine whether you should buy the option if the annual risk-free rate is 5% and the annual standard deviation of the stock returns is 12%. Show all the steps and make a conclusion.
2. Using the information in question a) only what should be the cost of a 6-month put option with an exercise price of $50 on the same stock? Show all the steps and explain the rational for your calculations.

Answer 1

Question A 6-month call option with an exercise price of $50 on a stock that is trading at $52 costs $4.5.

1. Determine whether you should buy the option if the annual risk-free rate is 5% and the annual standard deviation of the stock returns is 12%. Show all the steps and make a conclusion.

Following is the formula to calculate the value of call option under the Black-Scholes Model where K is the strike price

C = S*N (d1) - N (d2) *K*e ^ (-r*t)          

Where,

C = call value =?

S = current stock price = $52

N = cumulative standard normal probability distribution

t = days until expiration = 6 months or 0.5 years

Standard deviation, SD = σ = 12%

K = option strike price = $50

r = risk free interest rate = 5% per year

Formula to calculate d1 and d2 are -

d1 = {ln (S/K) +(r+ σ^2 /2)* t}/σ *√t

d2 = d1 – σ *√t Therefore

d1= {ln (52/50) + (0.05 + (0.12^2)/2) * 0.5} / 0.12 * √0.5

= 0.7993

d2 = d1 – σ *√t = 0.7993– 0.12 √0.5 = 0.7144

Now putting the value in the above formula to calculate call price

C = $52 * N (0.7993) – N (0.7144)* $50 * e^ (-0.05*0.5)

=$52 * 0.7879 – 0.7625* $50 * 0.9753

= $3.79

Therefore value of call option is $3.79

We should not buy this call option as its price ($4.5) is more than its actual value ($3.79)

2. Using the information in question a) only what should be the cost of a 6-month put option with an exercise price of $50 on the same stock? Show all the steps and explain the rational for your calculations.

Now price of put option by using following put-call parity equation

P = C –S + K* e^ (-r*t)

Where,

C = price of the call option = $3.79

P= price of the put option =?

S = spot price = $52

Strike price K = $50

The risk-free rate r= 5%

Time period t= 0.5

Now putting all the values in the put-call parity equation

P = $3.79 - $52 + $50 * e^ (-0.05*0.5)

Or P = $3.79 - $52 + $48.77 = $0.55

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