A company issues an option grant with an outperformance feature,against the S&P 500.Assume S&P 500 = 1100,S = 46,k = 45,σ = 0.30,r = 0.04,and 10 years until expiration.The S&P 500 has a dividend yield of 2.5%,standard deviation of 20.0% and a 0.45 correlation coefficient with the stock.What is the value of the outperformance feature?
A) $2.25
B) $3.29
C) $4.11
D) $4.78

Question

Quizplus · Accepted Answer

To calculate the value of the outperformance feature, we use a two-index model. The option will pay out if the stock outperforms the S&P 500. Therefore, we need to calculate the expected values of the stock and the S&P 500 and then determine the probability that the stock outperforms the S&P 500.

The expected value of the S&P 500 is:

E(S&P 500) = 1100 x (1 + 0.025) = 1127.5

The expected value of the stock is:

E(S) = 46 x (1 + 0.025) = 47.15

We can use these expected values and the correlation coefficient to calculate the expected value of the stock in a scenario where the S&P 500 outperforms the stock:

E(S|S&P 500>S) = E(S) + σ(S) x ρ x (E(S&P 500)/S&P 500 - 1)

E(S|S&P 500>S) = 47.15 + 0.3 x 0.45 x (1127.5/1100 - 1) = 47.43

The probability that the S&P 500 outperforms the stock can be calculated using a standardized normal distribution:

P(S&P 500>S) = N(d2)

d2 = (ln(S/E(S&P 500)) + (r - 0.025 + σ(S&P 500)^2/2) x T) / (σ(S&P 500) x √T)

d2 = (ln(47.15/1127.5) + (0.04 - 0.025 + 0.2^2/2) x 10) / (0.2 x √10) = -3.20

P(S&P 500>S) = N(-3.20) = 0.0007

The value of the outperformance feature can be calculated:

Value = P(S&P 500>S) x (E(S|S&P 500>S) - k)

Value = 0.0007 x (47.43 - 45) = $0.00245

Therefore, the value of the outperformance feature is $0.00245, or $2.45. The closest option is B, $3.29.

A Company Issues an Option Grant with an Outperformance Feature,against