Suppose VS's stock price is currently $20. In the next six months it will either fall by 50 percent or rise by 50 percent. What is the current value of a call option with an exercise price of $15 and expiration of one year? The six-month risk-free interest rate is 5 percent (periodic rate). Use the two-stage binomial method.
A)$5.00
B)$2.14
C)$7.86
D)$8.23

Question

Quizplus · Accepted Answer

The two-stage binomial model involves calculating the option value at each possible state in the future and then discounting those values back to the present using the risk-free rate. First, calculate the stock price movements: if the stock rises by 50%, it will be $30 ($20 * 1.5), and if it falls by 50%, it will be $10 ($20 * 0.5). For a call option with an exercise price of $15, the option will be worth $15 ($30 - $15) in the up state and $0 in the down state (since the stock price is below the exercise price). In the second stage, we calculate the expected option value at the end of the year, considering both the up and down scenarios. However, the question simplifies to a single period for calculation purposes, so we adjust the values for a six-month period and then use the risk-free rate to discount the expected option value back to the present. Given the risk-free rate of 5% for six months, the present value factor is 1 / (1 + 0.05) = 0.9524. The expected value of the option in six months, considering both up and down scenarios, would typically involve calculating the probabilities of each scenario. However, without specific probabilities, we can't directly apply the risk-neutral valuation here. Instead, we focus on the information given and the method requested, which implies a simplification or an approximation.The correct answer involves recognizing that the binomial model's output must be discounted back to the present value. The provided choices suggest specific values rather than a process to calculate them directly from the given information. The correct answer (D) $8.23, implies a calculation that considers the potential future values of the option, their probabilities, and the discounting back at the risk-free rate. However, without the explicit probabilities or a detailed step-by-step calculation provided in the question, the exact process involves estimating the expected payoff of the option in the future and then discounting that payoff to its present value using the risk-free rate. The value of $8.23 suggests a higher value than the intrinsic value ($5, from $20 - $15) due to the time value and the potential for the stock to increase above the exercise price, discounted back at the risk-free rate.

Suppose VS's Stock Price Is Currently $20