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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.

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closed as off-topic by skoestlmeier, Helin, Lliane, lehalle, Attack68 Mar 2 at 19:08

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    $\begingroup$ Could you please rewrite the question more "cleanly". Could you tell what the current YMH Price is ? If I interpreted it correctly both the Call and Put look mispriced $\endgroup$ – user23564 Feb 1 at 6:51
  • $\begingroup$ Well I assumed the YMH_0 at 50, calculated Risk neutral Pu and Pd =1/2, Priced call/put 5*1/2*exp(-0.05/4)=2.47. The call is over priced and Put is under priced? This is what I concluded. $\endgroup$ – user23564 Feb 1 at 8:42
  • $\begingroup$ why do you have $p_{u}=p{d}=1/2$, if the statement of the problem says $p_{u}=0.6$ and $p_{u}=0.4$ $\endgroup$ – ZRH Feb 1 at 8:53
  • $\begingroup$ Well I assumed they're physical probability. My calculations are wrong too, I guess better use the probabilities given $\endgroup$ – user23564 Feb 1 at 22:51
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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.

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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.

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  • $\begingroup$ you are getting the exact same result as I do, I would say all answers are incorrect $\endgroup$ – ZRH Feb 5 at 6:28
  • $\begingroup$ @ZRH indeed :) the official answers is very misleading, there’s no way one would call a cheap straddle an « arbitrage » $\endgroup$ – Ezy Feb 5 at 9:03
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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.

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  • $\begingroup$ Thanks for providing the expected answer. Compared to the 2 other anwers we gave you the solution assumes that the forward contract is not available for trading which does not make much sense. Also it assumes that the binomial model accurately reflects the terminal distribution of the spot which also does not make much sense. So all in all what really happens here is that according to the binomial model the straddle looks « cheap ». There is no way in real life you would call this an « arbitrage opportunity » so for that reason i call this question idiotic :) $\endgroup$ – Ezy Feb 5 at 8:31

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