Sunday, June 24, 2012

Derivatives: Arbitrage opportunities for Option trading


Arbitrage opportunity implies 2 things:
-NO INVESTMENT
-NO RISK






Notations that we will use:
C:Value of an american call
P:Value of an American Put
r: Nominal risk Free rate
S0: Price of the share at t=0
St: Price of the share on the expiration date
K: Strike price of the option

CALL option :Upper Bounds:

Call Option gives the right to buy a share at a certain price specified in the contract but no matter what happens  the option will never be worth more than the stock’s price, therefore we can say that the upper bound of the option’s price is the Stock Price.
Therefore                           C≤ S0       
If the C>S0 we would have an arbitrage opportunity:                                                                        C>S0 ==> C-S0>0

So we could make a riskless profit by:
1)Short call : C
2)Buy Stock: S0
This would give us the following table:                                            t=T
t=0
StK
St>K
Short Call: C
0
K-St
Buy Stock: -S0
St
St
We get C-S00                                                 St                                                                         K
It proves that whatever the price is we have no loss, so there is arbitrage, as there is no risk and no investment.


CALL options: Lower Bounds:

A lower boud for a call option would never be lower than S0 –Ke-RT,
Therefore we have this as a lower bound: C≥S0-Ke-RT
So if we have S0-Ke-RT-C ≥0 we would have an arbitrage opportunity

So we could make a riskless profit by:
-Short Stock: -S0 
-Lend money :Buy bond: -Ke-rt
-Long call: -C                                                                                   t=T
t=0
St≤K
St>K
Buy stock S0
-St
-St
Buy Bond (lend money) -Ke-rt
K
K
Long Call -C
0
St-K
S0-Ke-rt-C ≥ 0                                                    K-St≥0                                                0

The total is more than 0 in both case รจ Arbitrage opportunity

PUT options: Upper bounds:

PUT Option is the right to sell a share at a certain price specified in the contract
The upper bound for a put option will never be higher than Ke-rt≥ P
Therefore we have an arbitrage opportunity if
P≥Ke-RT so     P-Ke-rt≥ 0
So we could make a riskless profit if :
-Short put : P
-buy bond (lend money): -Ke-rt
                                                                                                          t=T
t=0
St≤K
St>K
Short put : P
-(K-St)
-(0)
Buy Bond –Ke-RT
K
K
P- Ke-RT ≥ 0                                                            St                                                               K

The total is more than 0  ==>Arbitrage opportunity


Put Options: Lower Bounds:

The lower bound for a put option will never be lower than P≥Ke-RT-S0
Therefore an arbitrage opportunity arise if:
P<Ke-RT-S0  so   Ke-RT- S0 –P> 0

We could make a riskless profit if:
-Sell bond(borrow money) Ke-RT
-Buy Stock: -S0
-Long Put: -P

t=0
St≤K
St>K
Sell bond Ke-RT
-K
-K
Buy stock –S0
St
St
Long Put –P
K-St
0
Ke-RT-S0-P≥0                                                         0                                                    St-K

The total is more than 0 in both case ==> Arbitrage opportunity
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