Arbitrage Opportunity - My Assignment Hero : Get Expert's Help

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, $F$ , 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:

$F = S_0 e^{rT}$

where:

$S_0 = 50$ (current stock price),
$r = 0.10$ (continuously compounded risk-free rate),
$T = 0.5$ years (six months).

$e^{0.10 times 0.5}$ $F = 50 e^{0.05}$

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

$F \approx 50 \times 1.05127 = 52.56$

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

Step 2: Check Put-Call Parity

Put-call parity states:

$C – P = S_0 – Ke^{-rT}$

where:

$C = 8$ (call price),
$P = 7$ (put price),
$K = F = 52.56$ ,
$S_0 = 50$ ,
$r = 0.10$ ,
$T = 0.5$ .

Calculate the present value of $K$ :

$K e^{-rT} = 52.56 e^{-0.05}$

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

$e^{-rT} approx 52.56 times 0.95123 = 50$

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:

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: $- 50$
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:

$e^{0.05} approx 52.56$
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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Arbitrage Opportunity