The Arbitrage Implied by the Binomial Options Pricing Model

1
The Arbitrage Implied by the Binomial Options
Pricing Model

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2
Assumptions
7.4 interest, compounded semiannually Strike
price 25.50 Actual Price of Option 1.52
30
27
24
25
22
20
3
Result of Binominal Option Pricing Model Analysis
7.4 interest, compounded semiannually Strike
price 25.50 Actual Price of Option 1.52
Click here if you want to see where these results
come from.
30
The objective of this PowerPoint presentation is
to show you how an arbitrageur would make a
riskless profit.
C2.6028
27
H.90
C1.4497
24
25
C0
H.52056
22
H0
20
The theoretical call premium is 1.45.
4
Arbitrage at time 0
Time 0 Summary You own 520.56 stocks You
owe 10,973.44.
Buy Low, Sell High Theoretical call premium
1.45 Actual call price 1.52
7.4 interest, compounded semiannually Strike
price 25.50 Actual Price of Option 1.52
30
C2.6028
Time 0 cash flows Sell 10 call options
1,520.00 (1000 shares1.52) Buy 520.56
stocks at 24 - 12,493.42 (1000
shares0.52056) Borrow 10,973.42
net cash flow at time 0 0.00
27
H.90
C1.4497
24
25
C0
H.52056
22
H0
Buy stocks (for hedge), Sell call options
20
5
Arbitrage at time 1/2
Entering Time 1/2 You own 520.56 stocks.
With interest you now owe 11,379.44 (10,973.44(
13.7)
End of Time 1/2 Summary You have no stock.
Options will not be exercised. You have no
debit. You have 72.86.
End of Time 0 You owned 520.56 stocks
You owed 10,973.44.
7.4 interest, compounded semiannually Strike
price 25.50 Actual Price of Option 1.52
If the price of the stock is 22 six months from
now, then regardless whether the price goes up to
25 or down to 20 at expiration, the option will
be out of the money. Therefore, the option will
be worthless at expiration regardless, and
therefore the option will be worthless six months
from now if the price is 22.
30
C2.6028
27
H.90
C1.4497
24
25
Time 1/2 cash flows if P_at_1/222 Options are
worthless Sell 520.56 stocks at 22
11,452.30 Pay Back loan plus interest
- 11,379.44 net cash flow at time 0
72.86
C0
H.52056
22
H0
20
FV of 72.86 72.86(13.7)75.56.
6
Arbitrage at time 1/2
Entering Time 1/2 You own 520.56 stocks.
With interest you now owe 11,379.44 (10,973.44(
13.7)
End of Time 1/2 Summary You will have 900
shares of stock. You will owe 21,624.34.
7.4 interest, compounded semiannually Strike
price 25.50 Actual Price of Option 1.52
30
Time 1/2 cash flows if P_at_1/227 buy 379.44
stocks at 27 - 10,244.90 Borrow an
additional 10,244.90 net cash flow
at time 0 0 !
C2.6028
27
H.90
C1.4497
24
25
If the price of the stock is 27 six months from
now, then according to our analysis, the
arbitrageur should increase his/her hedge ratio
from .52056 to .90. To do so, the arbitrageur
will need to buy an additional 379.44 shares. To
fund this purchase, the arbitrageur would need to
borrow.
C0
H.52056
22
H0
20
7
Arbitrage at time 1
Entering Time 1 You own 900 stocks. With
interest you now owe 22,424.44 (21,624.34(13.7
)
End of Time 1/2 Summary You will have 900
shares of stock. You will owe 21,624.34.
7.4 interest, compounded semiannually Strike
price 25.50 Actual Price of Option 1.52
30
Time 1 cash flows if P130 buy 10 options to
reverse - 4,500.00 sell 900 stocks
at 30 27,000.00 Pay back loan
- 22,424.44 net cash flow at
time 0 75.56
C2.6028
27
H.90
C1.4497
24
25
If the price of the stock is 30 at expiration,
then the option will be in the money by 4.50
30-25.5. To reverse out of the contract, you
would have to buy 10 contracts for 4500
4.510100. (Remember each option contract is
for 100 shares.)
C0
H.52056
22
H0
20
8
Arbitrage at time 1
Entering Time 1 You own 900 stocks. With
interest you now owe 22,424.44 (21,624.34(13.7
)
7.4 interest, compounded semiannually Strike
price 25.50 Actual Price of Option 1.52
30
Time 1 cash flows if P125 sell 900 stocks at
25 22,500.00 Pay back loan
- 22,424.44 net cash flow at time
0 75.56
C2.6028
27
H.90
C1.4497
24
25
If the price of the stock is 25 at expiration,
then the option will be out of the money and
expire worthless. The arbitrageur would then be
free to sell all of his/her stock.
C0
H.52056
22
H0
20
9
Conclusion
7.4 interest, compounded semiannually Strike
price 25.50 Actual Price of Option 1.52
30
C2.6028
Regardless what happens, the arbitrageurs profit
will equal 75.56 one year from now.
27
H.90
C1.4497
24
25
C0
H.52056
22
H0
20
10
Appendix Notes on theoretical call price and
hedge ratio calculations
7.4 interest, compounded semiannually Strike
price 25.50 Actual Price of Option 1.52
30
C2.6028
If the price six months from now is 22, then we
know that the option will expire out of the money
regardless what happens thereafter. Hence, the
option would be worthless six months from now.
27
H.90
C1.4497
24
25
C0
H.52056
22
H0
20
Click here to go back to the results slide.
The theoretical call premium is 1.45.