Potential Arbitrage profit or proof problem

Question

So the question asks: Consider 4 following European call and put options with the same maturity time:

1. Call option with strike price $100$ sell for $45$

2. Call option with strike price $110$ sell for $40$

3. Put option with strike price $100$ sell for $36$

4. Put option with strike price $110$ sell for $42$

Given the continuous compounding interest rate r = 0.05 and the maturity time T = 1. Can you choose a portfolio using some of the options from the table and the Bank account to find an arbitrage profit? If yes, be specic about your arbitrage portfolio. If no, prove your argument.

So so far I got:

First consider the put-call parity:

So the market follows

$C^E − P^E = S(0) − X^{e^{−rT}}$

if there is no arbitrary profit.

If $C^E − P^E> S(0) − X^{e^{−rT}}$

In this case an arbitrage strategy can be constructed as follows: At time 0

• buy one share for S(0);

• buy one put option for $P^E$;

• write and sell one call option for $C^E$;

• invest the sum $C^E−P^E−S(0)$ (or borrow, if negative) on the money market at the interest rate r.

The balance of these transactions is 0. Then, at time T

• close out the money market position, collecting (or paying, if negative) the sum $(C^E − P^E − S(0))e^{rT}$ ;

• sell the share for X either by exercising the put if S(T) ≤ X or settling the short position in calls if S(T) > X.

The balance will be $(C^E − P^E − S(0))e^{rT} + X > 0$

If $C^E − P^E < S(0) − X^{e^{−rT}}$

At time 0 • sell short one share for S(0);

• write and sell a put option for $P^E$;

• buy one call option for $C^E$;

• invest the sum $S(0)−C^E+P^E$ (or borrow, if negative) on the money market at the interest rate r.

The balance of these transactions is 0. At time T

• close out the money market position, collecting (or paying, if negative) the sum $(S(0) − C^E + P^E)e^{rT}$ ;

• buy one share for X either by exercising the call if S(T) > X or settling the short position in puts if S(T) ≤ X, and close the short position in stock.

The balance will be $(S(0) − C^E + P^E)e^{rT} − X$ > 0

But how do I suppose to find a specific arbitrary profit without knowing the S(0) which is the current stock price? Or, how to prove if no Arbitrage profit exists without the current stock price?

Update:

At time 0:

• sell one share with strike price 110 for S(0);

• write and sell a put option for PX with strike price 100;

• buy one call option for CE with strike price 100;

• investing the sum $S(0)−C^E+P^X$

• buy one share with strike price 110 for S(0);

• buy one put option for $P^E$ with strike price 100;

• write and sell one call option for $C^X$ with strike price 110;

• invest the sum $C^X−P^E−S(0)$ on the money market at the interest rate r.

• investing the balance $( C^E−P^X ) –( C^X-P^E )$ on the money market at the interest rate r.

The balance of these transactions is 0. At time T:

If exercised at time t ≤ T,

• borrow a share and sell it for X to settle the obligation as writer of the call option

• investing X to buy at option E.

• Investing E at rate r

• At time T, use the call to buy a share for E and close your short position in stock. The arbitrage profit will be ($(C^E−P^X) – (C^X-P^E))e^{rT} +(X-E)e^{r(T−t)} –(X –E)>0$.

If not exercised at all

• close the short position in stock,

• End up with the option and an arbitrage profit of $[(C^E−P^X)−(C^X-P^E)] e^{rT} > 0$.

Answer 1

There are two strike prices in this problem $E=100$ and $X=110$. The PCP conditions are $C^E − P^E = S(0) − E^{e^{−rT}}$ and $C^X − P^X = S(0) − X^{e^{−rT}}$. Subtracting the second equation from the first we get $C^E − P^E -C^X + P^X= (X-E)^{e^{−rT}}$. This gets rid of the unknown S(0) term.

Let's evaluate the right hand side numerically:

$C^E − P^E -C^X + P^X = 10 e^{-0.05}= 9.5$

So in theory this portfolio of options (long the 100 call and the 110 put, short the 100 put and the 110 call) should be worth 9.5

In the market place though it trades at 45-36-40+42 = 11. So it is overpriced and needs to be sold for an arbitrage profit of 1.5