Derivatives - Options - Binomial Model - Arbitrage

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CFA.Rhythm

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reading 60, page 233, questions 2B, 2C, 3B and 3C
First we have to compute the price of the call or the put.
Then, we have to execute an arbitrage transaction that either
1) will earn more than the risk free rate
2) that replicates a loan that will earn less than the risk free rate
AND THIS IS WHERE IT GETS TRICKY, AND I GET LOST
For questions 2B, 2C, 3B and 3C
we have to compute ” at expiration, the value of the combination will be…..”
IS THERE A WRITTEN FORMULA TO COMPUTE THE VALUE OF THE COMBINATION?
SOMETIMES IT IS (SHARES x S+) - (OPTIONS X C+)
THEN FOR #4C IT IS (-SHARES X S+) - (OPTIONS X P+)
~~
QUESTION: IS THERE A SET FORMULA TO COMPUTE THE “VALUE OF THE COMBINATION?”
It seems to be tricky stuff.
Thanks
By looking at questions
 
If the options are priced correctly, you will not earn anything other than riskfree by doing arbitrage. Remember arbitrage means you enter a position where your payoff is independent on market movement.
If option is not priced correctly, you will either get, dependent whether you put or receive money upfront.
- a more than risk free return, if you have to invest your money upfront (e.g., if you long put and long stock in case of underpriced put)
- a less than risk free loan, if you get money upfront and pay back at the end (e.g., short put and short stock).
Whenever the option is overpriced, you sell and if underpriced, you buy.
To maintain risk-free, you need to offset the position of the option with corresponding stock position
- If long call (because call underpriced), you need to short stock since call and stock are positively correlated (i.e., react same way) so you need to offset each other.
- If long put, long stock since call and stock are negatively correlated (i.e., react opposite way)
- Similarly, if you short call –> need to long stock. Short put then short stock.
How you calculate the combination depends on your arbitrage position (short or long, put or call).
Hope it explains the logic
 
elfcfa
thanks for the response. I am really trying to figure out the value of the combination based on the different scenarios.
thanks again. this is tough stuff.
 
Keep also in mind that the payoff is the same whether you calculate the value in case of stock going up (S+) or going down (S-), so as an exercise (or double check in exam), you should try to calculate for both scenarios to see you get the same answer.
 
Is there a trick or a formula to compute the “value of the combination at expiration?”
2B: Show how to execute an arbitrage transaction that will earn more than the risk free rate. Use 100 call options
If St = 110, at expiration the value of this combination will be 80(110) – 100(20) = $6800 ————->>>> (shares x S+) - (calls x C+)
2C: Show how to execute an arbitrage transaction that replicates a loan that will earn less than the risk free rate. Use 100 call options
If St = 110, at expiration the value of this combination will be 100(20) – 80(1100) = $-6800 ———>>> (calls x C+) - (shares x S+)
3B: Suppose a put price is currently $14. Show how to execute an arbitrage transaction that replicates a loan that will earn less than the risk free rate.
If St = 199.5, at expiration the value of this combination will be (-3125 x 199.5) – (10,000 x 0) = $-623,437
—————>>> (-shares x S+) - (#options x P+)
3C: Suppose the put price is currently $11. Show how to execute an arbitrage transaction that will earn more than the risk free rate.
If St = 199.5, at expiration the value of this combination will be (3125 x 199.5) + (10,000 x 0) = $623,437 —————–>>> (shares x S+) + (#options x P+)
~~~~~~
I know I’m skipping a lot of information here.
I just can’t seem to grasp the material or see a pattern in the formulas to compute the value at expiration of the combination. Sometimes they are added together. Sometimes subtracted. Sometimes shares goes first. Sometimes the calls go first.
If anybody can help that would be great. I just seems to be a very important step, with little actual information written about it.
 
Ok. Let’s walk through each case. You need to read this in combination with my previous posting.
2B: Show how to execute an arbitrage transaction that will earn more than the risk free rate. Use 100 call options
No arbitrage call price 16.15 (from 2A), 17.5 –> overpriced call –> short call –> to compensate (i.e., neutralize risk), we need to long stocks –> combination is (long) SHARES - (i.e., short) n* CALLS
2C: Show how to execute an arbitrage transaction that replicates a loan that will earn less than the risk free rate. Use 100 call options
no arbitrage call price 16.15 (2A), 14 –> underpriced call –> need to long call –> to compensate: need to short stocks –> combination is - SHARES +n* CALLS
3B: Suppose a put price is currently $14. Show how to execute an arbitrage transaction that replicates a loan that will earn less than the risk free rate.
no arbitrage put price 12.78 (3A), 14 –> overprice –> short put –> to compensate need short stocks –> combination is - SHARES - n*PUTS
3C: Suppose the put price is currently $11. Show how to execute an arbitrage transaction that will earn more than the risk free rate.
no arbitrage put price 12.78 (3A), 11 –> underprice –> long put –> to compensate need long stocks –> combination is + SHARES + n*PUTS
Let me know if I have answered your question.
 
Also, in case it is not clear yet.
2B: earn more than the risk free rate
Since your position is (long) SHARES - (i.e., short) n* CALLS, you need to put up money upfront. Why: since n<1 and CALLS/SHARES <1, the money you receive from shorting calls is not enough to pay for the purchase of shares –> net net, you need to invest some money upfront.
2C: a loan that will earn less than the risk free
Since your position is - SHARES +n* CALLS, you will receive money upfront –> a loan. See same logic as above.
3B: loan that will earn less than the risk free rate
Since your position is - SHARES - n*PUTS, you will receive money upfront (from both short positions) –> a loan
3C: earn more than the risk free rate.
Since your position is + SHARES + n*PUTS, you need to put up money upfront.
 
Well HEE-HA!
Thank you very much elcfa.
Now I see why I was so confused.
(long shares - calls) is the same thing as (-calls + long shares)
and
(long calls - short shares) is the same thing as (-short shares + long calls)
and
(-shares - -calls) is the same thing as (-calls - -shares)
and
(long stocks + long puts) is the same thing as (long puts + long stocks)
on and on
Now I see it.
thanks again
*** Now I’ll go on and figure out who puts money up front. But I think I can do that.
 
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