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. The risk-free gross return per time step is 1.001668. If a 98-strike call option is trading at $2, how much arbitrage profit can you make in present value terms?
A) $2.09
B) $4.09
C) $6.09
D) $12.09

Question

Quizplus · Accepted Answer

First, calculate the payoff of the call option in both the up and down states. If the stock price goes up to $110, the call option with a strike price of $98 will have a payoff of $12 ($110 - $98). If the stock price goes down to $90, the call option will expire worthless, so the payoff is $0. Next, calculate the risk-neutral probabilities. Let $p$ be the risk-neutral probability of the stock price going up. The risk-free rate of return is given as 1.001668, which means the risk-free return for one period is 0.1668%. The risk-neutral probability $p$ is calculated using the formula: $$p = \frac{e^{rT} - d}{u - d}$$where $e^{rT}$ is the risk-free gross return, $u$ is the gross return if the stock price goes up, $d$ is the gross return if the stock price goes down, and $T$ is the time to maturity (which is 1 in this case). Given that $u = \frac{110}{100} = 1.1$, $d = \frac{90}{100} = 0.9$, and $e^{rT} = 1.001668$, we find:$$p = \frac{1.001668 - 0.9}{1.1 - 0.9} = \frac{0.101668}{0.2} = 0.50834$$The expected payoff of the call option under the risk-neutral measure is:$$E[\text{Payoff}] = p \times \text{Payoff if up} + (1-p) \times \text{Payoff if down}$$$$E[\text{Payoff}] = 0.50834 \times 12 + (1-0.50834) \times 0 = 6.10008$$To find the present value of this expected payoff, we discount it back at the risk-free rate:$$PV = \frac{6.10008}{1.001668} = 6.089$$The arbitrage profit is the difference between the present value of the expected payoff and the current price of the call option:$$Arbitrage\ Profit = PV - \text{Price of the call option}$$$$Arbitrage\ Profit = 6.089 - 2 = 4.089$$Rounding to two decimal places, the arbitrage profit in present value terms is approximately $4.09.

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