Arbitrage Opportunity

We will determine the six-month forward price of ABC stock, $F$, and then analyze whether an arbitrage opportunity exists given the call and put option prices.

Step 1: Calculate the Six-Month Forward Price

The forward price is given by the standard formula:

F=S0erTF = S_0 e^{rT}F=S0​erT

where:

• S0=50S_0 = 50S0​=50 (current stock price),

• $r=0.10$ (continuously compounded risk-free rate),

• $T=0.5$ years (six months).

F=50e0.10×0.5F = 50 e^{0.10 times 0.5}F=50e0.10×0.5 F=50e0.05F = 50 e^{0.05}F=50e0.05

Approximating e0.05≈1.05127e^{0.05} approx 1.05127e0.05≈1.05127,

$$F\approx 50\times 1.05127=52.56$$

So, the six-month forward price is $52.56$.

Step 2: Check Put-Call Parity

Put-call parity states:

C−P=S0−Ke−rTC – P = S_0 – Ke^{-rT}C−P=S0​−Ke−rT

where:

• $C=8$ (call price),

• $P=7$ (put price),

• $K=F=52.56$,

• S0=50S_0 = 50S0​=50,

• $r=0.10$,

• $T=0.5$.

Calculate the present value of $K$:

Ke−rT=52.56e−0.05K e^{-rT} = 52.56 e^{-0.05}Ke−rT=52.56e−0.05

Approximating e−0.05≈0.95123e^{-0.05} approx 0.95123e−0.05≈0.95123,

Ke−rT≈52.56×0.95123=50K e^{-rT} approx 52.56 times 0.95123 = 50Ke−rT≈52.56×0.95123=50

Now, check the put-call parity equation:

$$8-7=50-50$$ $$1=0$$

This contradiction means that put-call parity is violated, which creates an arbitrage opportunity.

Step 3: Construct an Arbitrage Strategy

Since the left-hand side of the equation ( $C-P=1$ ) is greater than the right-hand side, we exploit this mispricing as follows:

1. Sell the call for $8.

2. Buy the put for $7.

3. Buy the stock for $50.

4. Short the forward contract at $F = 52.56 (which means agreeing to sell the stock in six months at this price).

5. Finance the stock purchase by borrowing $50 at 10% continuous compounding.

Cash Flow at Initiation:

• Buying the stock: $-50$

• Selling the call: $+8$

• Buying the put: $-7$

• No cost to enter the forward contract

• Net cash flow: $-50+8-7=-49$

Cash Flow at Expiry (Six Months Later):

• The stock is delivered at $52.56 under the forward contract.

• The borrowed amount grows to:

  50e0.05≈52.5650 e^{0.05} approx 52.5650e0.05≈52.56

• The put and call will cancel each other since they are at-the-money.

Final Profit Calculation:

• Receive $52.56 from forward contract.

• Repay $52.56 loan.

• Initial arbitrage gain: $1.

Thus, we earn a risk-free arbitrage profit of $1 per share.

Conclusion

Since put-call parity is violated, there exists an arbitrage opportunity. By using a combination of put and call options, stock ownership, and a forward contract, we can lock in a risk-free profit.