Question

Suppose the stock price is $40 and the effective annual interest rate is 8%.

a. Draw on a single graph payoff and profit diagrams for the following options:

(i) 35-strike call with a premium of $9.12.

(ii) 40-strike call with a premium of $6.22.

(iii) 45-strike call with a premium of $4.08.

b. Consider your payoff diagram with all three options graphed together. Intuitively, why should the option premium decrease with the strike price?

Answer 1

Given that the EAR = 8%

• 35-strike call with a premium of $9.12
  • \( \begin{align*} \text{purchased call payoff} &= \max(0, \text{spot price at expiration} - \text{strike price})\\ &= \max(0, S - 35)\\ &= \begin{cases} 0 & S<35 \\ S-35 & S\geq 35 \end{cases} \\ \text{purchased call profit} &= \max(0, \text{spot price at expiration} - \text{strike price}) - \text{future value of option premium} \\ &= \max(0, S-35) - 9.12(1.08) \\ &= \max(-9.8496, S-35-9.8496)\\ &= \max(-9.8496, S-44.8496)\\ &= \begin{cases} -9.8496 & S<35 \\ S-44.8496 & S\geq 35 \end{cases} \end{align*} \)
• 40-strike call with a premium of $6.22
  • \( \begin{align*} \text{purchased call payoff} &= \max(0, \text{spot price at expiration} - \text{strike price})\\ &= \max(0, S - 40)\\ &= \begin{cases} 0 & S<40 \\ S-40 & S\geq 40 \end{cases} \\ \text{purchased call profit} &= \max(0, \text{spot price at expiration} - \text{strike price}) - \text{future value of option premium} \\ &= \max(0, S-40) - 6.22(1.08) \\ &= \max(-6.7176, S-40-6.7176)\\ &= \max(-6.7176, S-46.7176)\\ &= \begin{cases} -6.7176 & S<40 \\ S-46.7176 & S\geq 40 \end{cases} \end{align*} \)
• 45-strike call with a premium of $4.08
  • \( \begin{align*} \text{purchased call payoff} &= \max(0, \text{spot price at expiration} - \text{strike price})\\ &= \max(0, S - 45)\\ &= \begin{cases} 0 & S<45 \\ S-45 & S\geq 45 \end{cases} \\ \text{purchased call profit} &= \max(0, \text{spot price at expiration} - \text{strike price}) - \text{future value of option premium} \\ &= \max(0, S-45) - 4.08(1.08) \\ &= \max(-4.4064, S-45-4.4064)\\ &= \max(-4.4064, S-49.4064)\\ &= \begin{cases} -4.4064 & S<45 \\ S-49.4064 & S\geq 45 \end{cases} \end{align*} \)
• Option premium decreases with the strike price because: the payoff of a long call is \( \max(0,S-K) \). As \( K \) increases, the payoff gets worse and the option becomes less valuable. Ceteris paribus, the higher strike price, the lower the premium.

The option premium decrease with the strike price because if the payoff of a long call is \( \max(0, S-K) \) then as \( K \) increases, the becomes less valuable.