Question

In: Finance

An investor is said to take a position in a “collar” if she buys the asset,...

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

Solutions

Expert Solution

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


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