Two-State Option Pricing Model
T-bills currently yield 5.5 percent. Stock in Nina manufacturing is currently selling for $70 per share. There is no possibility that the stock will be worth less than $65 per share in one year.
a. What is the value of a call option with a $60 exercise price? What is the intrinsic value?
b. What is the value of a call option with a $50 exercise price? What is the intrinsic value?
c. What is the value of a put option with a $60 exercise price? What is the intrinsic value?
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a. To calculate the value of a call option with a $60 exercise price in the Two-State Option Pricing Model, we need to consider the two possible states of the stock: a high state and a low state.
First, let's find the value of the call option in the high state. In this state, if the stock price is greater than the exercise price, the intrinsic value of the call option is the difference between the stock price and the exercise price. Therefore, if the stock price in the high state is $70, the intrinsic value of the call option is $10 ($70 - $60).
Next, we need to calculate the probability of the high state occurring. Without specific information about the probabilities, we can assume equal probabilities for simplicity. Therefore, the probability of the high state occurring is 0.5.
To find the value of the call option in the high state, we calculate the present value of the intrinsic value using the risk-free rate, which is the yield of T-bills. The present value is calculated by dividing the intrinsic value by 1 + the risk-free rate. In this case, the present value of the intrinsic value is $10 / (1 + 0.055) = $9.52.
Next, let's find the value of the call option in the low state. In this state, if the stock price is less than or equal to the exercise price, the intrinsic value of the call option is zero since the option would not be profitable to exercise. Therefore, in the low state, the intrinsic value of the call option is $0.
Again, assuming equal probabilities for the high and low states, the probability of the low state occurring is also 0.5.
To find the value of the call option in the low state, we calculate the present value of the intrinsic value, which is $0, using the risk-free rate. The present value of the intrinsic value in the low state is $0 / (1 + 0.055) = $0.
Finally, we calculate the overall value of the call option by taking the weighted average of the values in the high and low states. The overall value of the call option is (0.5 * $9.52) + (0.5 * $0) = $4.76.
Therefore, the value of the call option with a $60 exercise price is $4.76. The intrinsic value is $10, which represents the difference between the stock price and the exercise price.
b. To calculate the value of a call option with a $50 exercise price, we follow the same steps as in part a.
In the high state, if the stock price is greater than the exercise price, the intrinsic value of the call option is the difference between the stock price and the exercise price. Therefore, if the stock price in the high state is $70, the intrinsic value of the call option is $20 ($70 - $50).
The probability of the high state occurring is still 0.5.
The present value of the intrinsic value in the high state is $20 / (1 + 0.055) = $19.05.
In the low state, if the stock price is less than or equal to the exercise price, the intrinsic value of the call option is zero. Therefore, in the low state, the intrinsic value of the call option is $0.
The probability of the low state occurring is still 0.5.
The present value of the intrinsic value in the low state is $0 / (1 + 0.055) = $0.
The overall value of the call option is (0.5 * $19.05) + (0.5 * $0) = $9.53.
Therefore, the value of the call option with a $50 exercise price is $9.53. The intrinsic value is $20.
c. To calculate the value of a put option with a $60 exercise price, we need to consider the same two states as in parts a and b.
In the high state, if the stock price is less than the exercise price, the intrinsic value of the put option is the difference between the exercise price and the stock price. Therefore, if the stock price in the high state is $70, the intrinsic value of the put option is $0 since the stock price is greater than the exercise price.
The probability of the high state occurring is still 0.5.
The present value of the intrinsic value in the high state is $0 / (1 + 0.055) = $0.
In the low state, if the stock price is less than the exercise price, the intrinsic value of the put option is the difference between the exercise price and the stock price. Therefore, if the stock price in the low state is $65, the intrinsic value of the put option is $5 ($60 - $65).
The probability of the low state occurring is still 0.5.
The present value of the intrinsic value in the low state is $5 / (1 + 0.055) = $4.76.
The overall value of the put option is (0.5 * $0) + (0.5 * $4.76) = $2.38.
Therefore, the value of the put option with a $60 exercise price is $2.38. The intrinsic value is $5.