In a one-period binomial model,assume that the current stock price is $100,and that it will rise to $110 or fall to $90 after one month.If the risk-free rate is 0.1668% per month in simple terms,which of the following choices best describes the replicating portfolio for one hundred 99-strike one-month call options?
A)Long 50 shares of stock and long 50 units of $1 bonds.
B)Long 55 shares of stock and long 49 units of $1 bonds.
C)Long 55 shares of stock and short 49 units of $1 bonds.
D)Long 55 shares of stock and short 50 units of $1 bonds.

Question

Quizplus · Accepted Answer

To replicate the payoff of the call options, we calculate the change in the option's value in both the up and down states. If the stock rises to $110, the call option with a strike price of $99 will be worth $11 (since $110 - $99 = $11). If the stock falls to $90, the option will be worth $0 (since it's out of the money). The change in the stock price is $20 ($110 - $90), and the change in the option's value is $11. The number of shares to hold, Δ, is the change in option value divided by the change in stock price, which is $11/$20 = 0.55. For 100 options, this is 55 shares.To find the amount to borrow or lend, we use the formula for the value of the replicating portfolio: V = ΔS - B, where V is the value of the call, S is the stock price, and B is the bond position. At the start, the call option's value can be found by considering its value in the up and down states discounted back at the risk-free rate. However, for simplicity in this explanation, we focus on constructing the portfolio with Δ and B.Given Δ = 55, we need to ensure the portfolio's value matches the call option's initial value. Since we don't have the initial value of the call option directly, we focus on the mechanics of Δ and B. The bond position compensates for the difference in investment in the stock and the total value needed to replicate the option's payoff. Since we're dealing with a simplified explanation, the exact initial value calculation of the call option is not shown, but the principle is to adjust B so that the portfolio matches the option's payoff in both up and down states.Given the positive Δ (55 shares), if we're constructing a portfolio that increases in value like the call option, we would need to finance the purchase of these shares. This implies borrowing, which corresponds to a short position in bonds. The exact amount to borrow (or the number of bonds to short) depends on the initial cost of setting up the replicating portfolio, which balances out to ensure the portfolio's value moves in tandem with the call options' value. Without the specific calculations of the initial option value or the total cost of the shares, the key takeaway is that the portfolio involves borrowing, hence a short position in bonds, to finance the purchase of more shares than the initial capital would allow, aiming to replicate the leveraged payoff structure of the call options.Given the information provided and focusing on the structure of the replicating portfolio rather than the precise numerical calculation of the bond position, the correct choice involves being long on the stock (to gain exposure to its movements) and short on bonds (to finance this position, reflecting the borrowing needed to set up the replicating portfolio).

In a One-Period Binomial Model,assume That the Current Stock Price