Arbitrage Opportunity

The current price of ABC stock is $50. The term structure of interest rates (continuously compounded) is flat at 10%. What is the six-month forward price of the stock?  Denote this as F. The six-month call price at strike F is equal to $8. The six-month  put price at strike F is equal to $7. Explain why there is arbitrage opportunity given  these prices.

Das, Sanjiv; Rangarajan Sundaram. Derivatives (The Mcgraw-hill/Irwin Series in Finance, Insureance and Real Estate) (p. 214). McGraw-Hill Higher Education. Kindle Edition.

We will determine the six-month forward price of ABC stock, FFF, 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=0.10r = 0.10r=0.10 (continuously compounded risk-free rate),

T=0.5T = 0.5T=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≈50×1.05127=52.56F approx 50 times 1.05127 = 52.56F≈50×1.05127=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=8C = 8C=8 (call price),

P=7P = 7P=7 (put price),

K=F=52.56K = F = 52.56K=F=52.56,

S0=50S_0 = 50S0​=50,

r=0.10r = 0.10r=0.10,

T=0.5T = 0.5T=0.5.

Calculate the present value of KKK:

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=50−508 – 7 = 50 – 508−7=50−50 1=01 = 01=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=1C – P = 1C−P=1 ) is greater than the right-hand side, we exploit this mispricing as follows:

Sell the call for $8.

Buy the put for $7.

Buy the stock for $50.

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

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

Cash Flow at Initiation:

Buying the stock: −50-50−50

Selling the call: +8+8+8

Buying the put: −7-7−7

No cost to enter the forward contract

Net cash flow: −50+8−7=−49-50 + 8 – 7 = -49−50+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.

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