Question

An investor is said to take a position in a “collar” if she buys the asset, buys an out-of-the-money put option on the asset, and sells an out-of-the-money call option on the asset. The two options should have the same time to expiration. Suppose Marie wishes to purchase a collar on Riggs, Inc., a non-dividend-paying common stock, with six months until expiration. She would like the put to have a strike price of $34 and the call to have a strike price of $63. The current price of the stock is $45 per share. Marie can borrow and lend at the continuously compounded risk-free rate of 5 percent per year and the annual standard deviation of the stock’s return is 50 percent.

  

Use the Black-Scholes model to calculate the total cost of the collar that Marie is interested in buying. (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

	Cost of collar

Answer 1

We use Black-Scholes Model to calculate the value of the call option.

The value of a call option is:

C = (S₀ * N(d₁)) - (Ke^-rt * N(d₂))

where :

S₀ = current spot price

K = strike price

N(x) is the cumulative normal distribution function

r = risk-free interest rate

t is the time to maturity in years

d₁ = (ln(S₀ / K) + (r + σ²/2)*T) / σ√T

d₂ = d₁ - σ√T

σ = standard deviation of underlying stock returns

First, we calculate d₁ and d₂ as below :

• ln(S₀ / K) = ln(45 / 63). We input the same formula into Excel, i.e. = LN (45 / 63)
• (r + σ²/2)*T = (0.05 + (0.50²/2)*0.50
• σ√T = 0.50 * √0.50

d₁ = -0.7042

d₂ = -1.0578

N(d₁), and N(d₂) are calculated in Excel using the NORMSDIST function and inputting the value of d₁ and d₂ into the function.

N(d₁) = 0.2407

N(d₂) = 0.1451

Now, we calculate the values of the call option as below:

C = (S₀ * N(d₁))   - (Ke^-rt * N(d₂)), which is (45 * 0.2407) - (63 * e^{(-0.05 * 0.50)})*(0.1451)    ==> $1.9148

Value of call option is $1.9148

We use Black-Scholes Model to calculate the value of the put option.

The value of a put option is:

P = (K * e^-rt)*N(-d₂) - (S₀)*N(-d₁)

where :

S₀ = current spot price

K = strike price

N(x) is the cumulative normal distribution function

r = risk-free interest rate

t is the time to maturity in years

d₁ = (ln(S₀ / K) + (r + σ²/2)*T) / σ√T

d₂ = d₁ - σ√T

σ = standard deviation of underlying stock returns

First, we calculate d₁ and d₂ as below :

• ln(S₀ / K) = ln(45 / 34). We input the same formula into Excel, i.e. =LN (45 / 34)
• (r + σ²/2)*T = (0.05 + (0.50²/2)*0.50
• σ√T = 0.50 * √0.50

d₁ = 1.0403

d₂ = 0.6867

N(-d₁), and N(-d₂) are calculated in Excel using the NORMSDIST function and inputting the value of d₁ and d₂ into the function.

N(-d₁) = 0.1491

N(-d₂) = 0.2461

Now, we calculate the values of the put option as below:

P = (K * e^-rt)*N(-d₂) - (S₀)*N(-d₁), which is (34 * e^{(-0.05 * 0.50)})*(0.2461) - (45 * (0.1491) ==> $1.4520

Value of put option is $1.4520

cost of collar = cost of put option - cost of call option = $1.4520 - $1.9148 = -$0.4628