The current price of a non-dividend paying stock is $40.A European call option with three months maturity and strike $39 is priced at $2.The risk free rate of interest for three months is 2%.Which of the following statements is correct?
A)The price of the call obeys no-arbitrage restrictions.
B)It is possible to construct a risk-less arbitrage strategy to yield a gain of at least $0.81.
C)It is possible to onstruct a risk-less arbitrage strategy to yield a gain of at least $1.00.
D)It is possible to construct a risk-less arbitrage strategy to yield a gain of at least $1.19.

Question

Quizplus · Accepted Answer

BTo check if the price of the call option obeys no-arbitrage restrictions, we can use the put-call parity for European options, which simplifies to $C + Ke^{-rt} = P + S$ for non-dividend paying stocks, where $C$ is the call price, $K$ is the strike price, $e^{-rt}$ is the discount factor, $P$ is the put price (not directly relevant here as we're not given or asked about it), and $S$ is the current stock price. However, without the price of a put option, we directly look at the possibility of arbitrage given the information.Given:- Current stock price ($S$) = $40- Strike price ($K$) = $39- Call option price ($C$) = $2- Risk-free rate ($r$) = 2% for three monthsIf you buy the call option for $2, your total investment to potentially control the stock is $2. If the stock price is above the strike price at expiration, you can exercise the option, buy the stock at $39, and sell it at the market price. The break-even point at expiration is a stock price of $41 ($39 strike + $2 option premium).To check for arbitrage, consider creating a synthetic position by buying the call and lending/borrowing the present value of the strike price ($Ke^{-rt}$).The present value of the strike price is $39e^{-0.02 \times 0.25} \approx 38.81$. If you buy the call for $2 and borrow $38.81 (the present value of the strike price), your total outlay is $2 + $38.81 = $40.81.If at expiration, the stock is above $39, you exercise the call, buy the stock at $39, repay the borrowed amount (which has grown to $39 due to interest), and sell the stock at its market price. The minimum profit occurs if the stock is exactly at $39, in which case, after repaying the borrowed amount, you break even, but any price above $39 yields a profit.However, since the initial outlay ($40.81) is more than the current stock price ($40), this suggests there's no immediate risk-less arbitrage opportunity based on these prices alone. The mistake in my explanation is assuming an arbitrage exists without correctly calculating the arbitrage profit, which would require a specific strategy showing how the profit is locked in regardless of the stock's future price movement. The correct approach to identify arbitrage involves comparing the cost of a replicating portfolio to the cost of the option, but with the given information, we primarily assess the call's pricing in the context of its intrinsic and time value, which seems reasonable given the stock's current price, the strike price, and the cost of the call. Thus, indicating the call price might obey no-arbitrage restrictions without a clear arbitrage strategy presented. The possibility of constructing a risk-less arbitrage strategy (B) is incorrectly justified without a detailed strategy that exploits price discrepancies. The correct analysis should involve comparing the option's price to the cost of a replicating portfolio or using other arbitrage relationships, which was not accurately done.

The Current Price of a Non-Dividend Paying Stock Is $40