d******g
发帖数: 34 | 1 Consider a European call option with strike of $100 and maturity of one year. The current 1-year forward price is $120. The call option is quoted at $18. The dollar interest rate is fixed at 5%.
1. Is there an arbitrage opportunity? If there is, how do you do the
arbitrage trading to profit from it?
2. If the call is quoted at $121, is there an arbitrage opportunity? If there is, how do you do the arbitrage trading to profit from it? | a********e
发帖数: 508 | 2 1. yes. long a call, short 1 stock, long bond with face value 100
2. yes. short a call, long 1 stock
year. The current 1-year forward price is $120. The call option is quoted at
$18. The dollar interest rate is fixed at 5%.
there is, how do you do the arbitrage trading to profit from it?
【在 d******g 的大作中提到】
: Consider a European call option with strike of $100 and maturity of one year. The current 1-year forward price is $120. The call option is quoted at $18. The dollar interest rate is fixed at 5%.
: 1. Is there an arbitrage opportunity? If there is, how do you do the
: arbitrage trading to profit from it?
: 2. If the call is quoted at $121, is there an arbitrage opportunity? If there is, how do you do the arbitrage trading to profit from it?
| c**********e
发帖数: 2007 | 3
Agree.
I think the call price should be $21, not $121. Then the answer is no.
【在 a********e 的大作中提到】
: 1. yes. long a call, short 1 stock, long bond with face value 100
: 2. yes. short a call, long 1 stock
:
: year. The current 1-year forward price is $120. The call option is quoted at
: $18. The dollar interest rate is fixed at 5%.
: there is, how do you do the arbitrage trading to profit from it?
| W*******d
发帖数: 63 | 4 not short stock, should sell foward
borrow $18 to buy call and sell forward at $120
at
【在 a********e 的大作中提到】
: 1. yes. long a call, short 1 stock, long bond with face value 100
: 2. yes. short a call, long 1 stock
:
: year. The current 1-year forward price is $120. The call option is quoted at
: $18. The dollar interest rate is fixed at 5%.
: there is, how do you do the arbitrage trading to profit from it?
| f********k
发帖数: 136 | 5 1. Borrow $118 at 5%, long a call (cost $18), instantly invest the remaining
$100 with 5%, short a forward.
At maturity, excercise the call with $100 to buy a share, delivery the share
to forward contract to receive $120, the cash flow from long call+ short
forward is $20, the net exposure to debt is $18*1.05=$18.9, thus the net
profit =20-18.9=1.1
2. If the call is priced 121, the call is overpriced. Short call, invest the
proceeds of $121 with 5% to generate $127.05 in one year, long a forward.
At maturity, use $120 to buy a share from forward contract, deliver the
share to call if it's exercised, receive $100.The net profit is $127.05-$120
+$100=$107.05. If the call is out of money at maturity, we still can sell the share at spot market to receive some money less than $100 (since it's out of money), then the profit will be less than $107.05 but still positive。The worst case is the stock is deep out-of-money, to say $0, we just loss $120 from forward contract, but still have a profit of $7.05.
If the call is priced at 21, it's fairly priced. Since the initial cost of
the call is 21, the potential cash flow from a long call and short forward
is $20 ($120-$100), so the cost is actually equal to the risk free rate 5% (
$20*1.05=$21). No arbitrage chance.
The key of this problem is just to compare the call price with the
potential cash flow with the oppsite positions in one call and one forward
contract, and compare the potential return with the risk-free rate. | m****r
发帖数: 141 | 6 Would you please recommend some books about these kind of questions ?
thanks
at
【在 a********e 的大作中提到】
: 1. yes. long a call, short 1 stock, long bond with face value 100
: 2. yes. short a call, long 1 stock
:
: year. The current 1-year forward price is $120. The call option is quoted at
: $18. The dollar interest rate is fixed at 5%.
: there is, how do you do the arbitrage trading to profit from it?
| f********k
发帖数: 136 | 7 No infomation is given for the spot market, so the arbitrage should not rely
on the spot market. Instead, the forward contract should be used.
at
【在 a********e 的大作中提到】
: 1. yes. long a call, short 1 stock, long bond with face value 100
: 2. yes. short a call, long 1 stock
:
: year. The current 1-year forward price is $120. The call option is quoted at
: $18. The dollar interest rate is fixed at 5%.
: there is, how do you do the arbitrage trading to profit from it?
| g******e
发帖数: 352 | 8 For 2,
Think about the case that the underlying ends below 97.95 (=120 - 21*1.05).
If you sell call at $21 and long forward, you will get burned in such
situation. The premium you get from selling the call can't cover your loss
on the forward.
remaining
share
the
120
【在 f********k 的大作中提到】
: 1. Borrow $118 at 5%, long a call (cost $18), instantly invest the remaining
: $100 with 5%, short a forward.
: At maturity, excercise the call with $100 to buy a share, delivery the share
: to forward contract to receive $120, the cash flow from long call+ short
: forward is $20, the net exposure to debt is $18*1.05=$18.9, thus the net
: profit =20-18.9=1.1
: 2. If the call is priced 121, the call is overpriced. Short call, invest the
: proceeds of $121 with 5% to generate $127.05 in one year, long a forward.
: At maturity, use $120 to buy a share from forward contract, deliver the
: share to call if it's exercised, receive $100.The net profit is $127.05-$120
| f********k
发帖数: 136 | 9 Sorry，I forgot we have to honor the forward contract.
If the call is priced at $121, no matter how spot market going, the cash paid on the forward contract is $120. So, the maximum loss from forward is $120. If the call is exercised finally, we can get 100 back. If the call is out-of-money, the call will not be exercised. But we can get some money back by selling the share at spot market. Even the stock worth $0 finally, we still end with a profit of 7.05 with the very beginning call writing at $121.
I have revised the original post.
In your case, if the call is priced at $21, and spot market ends at 97.95. as I said, no arbitrage chance exists. We borrow $120 at 5%, sell call at $21, and instantly invest $141 with 5%, end up with $148.05 one year later, we can honor forward contract with $120 to get 1 share, sell the share at $97.95 (call expires since out-of-money), so the money in hand 148.05-120+97.95=126, which just enough to pay off the outstanding debt 120*1.05=126. If the stock ends below 97.95, we get net loss.
).
loss
【在 g******e 的大作中提到】
: For 2,
: Think about the case that the underlying ends below 97.95 (=120 - 21*1.05).
: If you sell call at $21 and long forward, you will get burned in such
: situation. The premium you get from selling the call can't cover your loss
: on the forward.
:
: remaining
: share
: the
: 120
| r*****h
发帖数: 505 | 10 stochastic calculus by Shereve
【在 m****r 的大作中提到】
: Would you please recommend some books about these kind of questions ?
: thanks
:
: at
| | | r*****h
发帖数: 505 | 11 shouldn't it be priced 20/1.05?
remaining
share
short
the
forward.
$120
【在 f********k 的大作中提到】
: 1. Borrow $118 at 5%, long a call (cost $18), instantly invest the remaining
: $100 with 5%, short a forward.
: At maturity, excercise the call with $100 to buy a share, delivery the share
: to forward contract to receive $120, the cash flow from long call+ short
: forward is $20, the net exposure to debt is $18*1.05=$18.9, thus the net
: profit =20-18.9=1.1
: 2. If the call is priced 121, the call is overpriced. Short call, invest the
: proceeds of $121 with 5% to generate $127.05 in one year, long a forward.
: At maturity, use $120 to buy a share from forward contract, deliver the
: share to call if it's exercised, receive $100.The net profit is $127.05-$120
| a********e
发帖数: 508 | 12 actually you just need to know the upper/lower bound of option price
nothing fancy about it
【在 m****r 的大作中提到】
: Would you please recommend some books about these kind of questions ?
: thanks
:
: at
| a********e
发帖数: 508 | 13 good point, the logic should be the same though
rely
【在 f********k 的大作中提到】
: No infomation is given for the spot market, so the arbitrage should not rely
: on the spot market. Instead, the forward contract should be used.
:
: at
| m*******r
发帖数: 4468 | 14 do you not pay back the money you borrowed 118x1.05=123.9?
remaining
share
short
net
invest the
forward.
the
$127.05-$120
【在 f********k 的大作中提到】
: 1. Borrow $118 at 5%, long a call (cost $18), instantly invest the remaining
: $100 with 5%, short a forward.
: At maturity, excercise the call with $100 to buy a share, delivery the share
: to forward contract to receive $120, the cash flow from long call+ short
: forward is $20, the net exposure to debt is $18*1.05=$18.9, thus the net
: profit =20-18.9=1.1
: 2. If the call is priced 121, the call is overpriced. Short call, invest the
: proceeds of $121 with 5% to generate $127.05 in one year, long a forward.
: At maturity, use $120 to buy a share from forward contract, deliver the
: share to call if it's exercised, receive $100.The net profit is $127.05-$120
| f********k
发帖数: 136 | 15 I already count it. The only final cash flow from long/short positions
considered is $20, paid $100 and received $120, return $100 to bank,
interests already incured from risk-free investment, so net debt is $18*1.05
, the $100 is already totally off-set itself, since it's borrowed and
invested at the same rate.
If you want more detailed calculations, it's listed below
1.borrowed 118$, paid $18, invested $100 in risk-free. 1-year later, we have debt outstanding $118*1.05=123.9, money in hand from risk-free investment=$100*1.05=$105.
2.Pay $100 to excercise call and receive $120 from forward, so finally, money in hand is 105-100+120=125.
3.After paying off debt, net profit=125-123.9=1.1
【在 m*******r 的大作中提到】
: do you not pay back the money you borrowed 118x1.05=123.9?
:
: remaining
: share
: short
: net
: invest the
: forward.
: the
: $127.05-$120
|
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