Questions: You work on a proprietary trading desk of a large investment bank, and you have been asked for a quote on the sale of a call option with a strike price of 49 and one year until expiration. The call option would be written on a stock that does not pay a dividend. From your analysis, you expect that the stock will either increase to 72 or decrease to 34 over the next year. The current price of the underlying stock is 49, and the risk-free interest rate is 4% per annum. What is the fair market value for the call option under these conditions? Do not round intermediate calculations. Round your answer to the nearest cent.

Solution Steps

To find the fair market value of the call option, we can use the Binomial Option Pricing Model. Here are the high-level steps:

1. Calculate the up and down factors (u and d) based on the expected stock prices.
2. Determine the risk-neutral probabilities for the up and down movements.
3. Calculate the possible payoffs of the call option at expiration.
4. Discount the expected payoff back to the present value using the risk-free rate.

The up factor \( u \) and down factor \( d \) are calculated based on the expected stock prices: \[ u = \frac{72}{49} \approx 1.4694, \quad d = \frac{34}{49} \approx 0.6939 \]

The risk-neutral probability \( p \) is given by: \[ p = \frac{e^{rT} - d}{u - d} \approx 0.4474 \]

The payoffs of the call option at expiration are: \[ \text{Payoff}_{\text{up}} = \max(0, u \cdot S_0 - K) = \max(0, 72 - 49) = 23 \] \[ \text{Payoff}_{\text{down}} = \max(0, d \cdot S_0 - K) = \max(0, 34 - 49) = 0 \]

The expected payoff of the call option is: \[ \text{Expected Payoff} = p \cdot \text{Payoff}_{\text{up}} + (1 - p) \cdot \text{Payoff}_{\text{down}} \approx 0.4474 \cdot 23 + (1 - 0.4474) \cdot 0 \approx 10.2893 \]

The present value of the expected payoff is calculated as: \[ \text{Call Option Price} = \frac{\text{Expected Payoff}}{e^{rT}} \approx \frac{10.2893}{e^{0.04}} \approx 9.89 \]

Final Answer