a) In-the-money options are options that would be profitable if exercised immediately. Out-of-the-money options are options that would not be profitable if exercised immediately. At-the-money options are options that have a strike price equal to the current price of the underlying asset.
- Call on ABL with a strike of $29.1: This option is in-the-money because the strike price is lower than the current price of ABL shares ($30).
- Call on ABL with a strike-price of $32: This option is out-of-the-money because the strike price is higher than the current price of ABL shares ($30).
- Put on HHT with a strike-price of $38.82: This option is at-the-money because the strike price is equal to the current price of HHT shares ($37.55).
b) To calculate the expected return on HHT shares, we can use the probability of each outcome and the associated returns.
- 70% chance of increasing by 10%: Expected return = 70% * 10% = 7%
- 30% chance of decreasing by 17%: Expected return = 30% * -17% = -5.1%
Therefore, the expected return on HHT shares is 7% - 5.1% = 1.9%.
To calculate the expected return on a protective put position, we need to consider the cost of the put option.
- Put option price: $1
- Strike price: $38.82
If the stock decreases below the strike price, the put option will provide protection, limiting the loss. If the stock increases, the put option will not be exercised and the return will be the same as the return on the stock. Therefore, the expected return on the protective put position will be the same as the expected return on HHT shares, which is 1.9%.
c) To create a risk-free portfolio, the delta of the short call option needs to offset the delta of the stock position.
Delta measures the change in the value of an option for a one-point change in the price of the underlying asset. If the delta of the stock position is 1, the delta of the short call option should be -1 to create a risk-free portfolio.
d) The risk-neutral probability of HHT shares increasing 10% can be calculated using the risk-neutral valuation principle. In the risk-neutral world, the expected return is equal to the risk-free rate.
Risk-neutral probability = (Expected return - Risk-free rate) / (Upside return - Downside return)
- Expected return = 10%
- Risk-free rate = 1.24%
- Upside return = 10%
- Downside return = -17%
Risk-neutral probability = (10% - 1.24%) / (10% - (-17%)) = 0.105 / 0.27 ≈ 0.389 (rounded to 3 decimal places)
Therefore, the risk-neutral probability of HHT shares increasing 10% is approximately 0.389.
e) The strategy that has one long stock and one short call with the same strike price is called a covered call strategy. This strategy involves owning the underlying stock while simultaneously selling a call option against it.
f) To find the Black-Scholes price of the call on ABL with a strike price of $29.1, we can use the Black-Scholes formula. The formula is:
C = S * N(d1) - X * e^(-r * T) * N(d2)
Where:
- C = Call option price
- S = Current price of the underlying asset ($30)
- N(d1) = Cumulative standard normal distribution function of d1
- X = Strike price of the call option ($29.1)
- e = Euler's number (approximately 2.71828)
- r = Risk-free rate per year (1.24%)
- T = Time to expiration (6 months or 0.5 years)
- N(d2) = Cumulative standard normal distribution function of d2
To calculate d1 and d2, we use the following formulas:
d1 = (ln(S/X) + (r + (σ^2)/2) * T) / (σ * sqrt(T))
d2 = d1 - σ * sqrt(T)
Where:
- ln = Natural logarithm
- σ = Annual standard deviation of the stock price (20%)
By plugging in the values into the formulas and calculating N(d1) and N(d2), we can find the Black-Scholes price of the call option.