Put-Call Parity Arbitrage from a Mispriced Synthetic Forward

# Put Call Parity Arbitrage Question


# Put Call Parity Arbitrage Question












I am incredibly stuck on the following question... Any help would be greatly appreciated.

According to your binomial model, the price of YMH in 3 months will be either USD 55 or USD 45, with probabilities 0.6 and 0.4, respectively. Two European options, a call and a put, on YMH with maturity 3 months and exercise price $50 are available. The price of the call option is USD 2.72 while the price of the put option is USD 2.23. If you can borrow up to USD 10,000 for 3 months at 0.5%, the arbitrage profit you can generate now is

(A) USD 40.65. (B) USD 42.90. (C) USD 47.35. (D) USD 50.50.

## Answer by ZRH (score 0)

https://quant.stackexchange.com/a/43835

I see none of the possible answers as correct. Based on the info on your model and the 0.5% stocklending rate, the current price of YMH is

$(0.6*55+0.4*45)/(1+0.005*0.25)=50.94$

Now if I assume the depo/loan rate is also 0.5%, the cashflows would be as follows:

T=0: Sell 10'000 YMH at 50.94, receive 509'363,Buy 10'000 50C on YMH, pay 27'200, Sell 10'000 50P on YMH, receive 22'300. Receive in total 504'463, deposit at 0.5%

T=1 Buy on the forward (50C+50P) 10'000 YMH, pay 500'000, Receive back from deposit 504'463*(1+0.005*.25)=505'094. Receive in total 5'094.

So the future arbitrage profit is 5'094, thus 50.94 ct/shr. Even if discounting that back to the date of inception, it will not be a materially different number.

## Answer by Ezy (score 0)

https://quant.stackexchange.com/a/43877

Let's assume that your binomial model is correct enough that it reflects the price of the forward $F$ on YMH accurately. Then this means that the forward price of YMH as seen from today is $F=\$51$.

Then assuming that you enter the following positions with the provided costs at inception

- buy 1 call at $K=\$50$. Cost: $C=\$2.72$

- sell 1 put at $K=\$50$. Cost: $C=-\$2.23$

- enter a short position of 1 forward contract expiring at $T$ for an exercise value of $\$51$. Cost $C=\$0$

- money you need to borrow to finance this portfolio with a 3 month term loan: $\$0.49$

At expiry here is what happens

- receive 1 share and pay $\$50$ to settle the call-put position

- provide this share to the person long the forward and receive $\$51$ in proceeds

- payback the loan plus interest of $\$0.4906125$

Total profit from the operation: $P=1-0.4906125=\$0.5093875$

This is a return of more than $103.8\%$ on the amount you borrowed which is completely enormous. So one of the following must happen

1) my reasoning is completely wrong and there's something i am missing

2) some of the inputs of your question are incorrect

3) the various answers provided are all incorrect.

## Answer by Jason (score 0)

https://quant.stackexchange.com/a/43894

Here is the "official" answer I was given.

In the up state, the call and put option payoffs are $5 and $0, respectively. In the down state, the call and put option payoffs are $0 and $5, respectively. Hence, a portfolio of one call and one put option delivers $5 in 3 months regardless of the state. The cost of that portfolio is $2.72+$2.23=$4.95. The implied riskless rate is 5 − 1 = 1.01%. 4.95 Hence, if we can borrow at 0.5%, we should do so to exploit the arbitrage opportunity. If we borrow $10,000 at 0.5% today, we need to return $10, 000 × 1.005 = $10,050 in 3 months. So the arbitrage strategy should involve buying 10,050 = 2, 010 units of the 5 above call/put portfolio. Note that the total payoff of the strategy in 3 months will be 2, 010 × 5 − 10, 050 = 0 dollars. The cash flow associated with the arbitrage strategy now is 10, 000 − 2, 010 × 4.95 = 50.50 dollars. Therefore, the correct answer is D.