So the question asks: Consider 4 following European call and put options with the same maturity time:
Call option with strike price $100$ sell for $45$
Call option with strike price $110$ sell for $40$
Put option with strike price $100$ sell for $36$
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?
You have quoted the book but missed the point of put-call parity. If you buy a call and sell a put at the same exercise price, you will buy one share at that price at expiration, regardless of the stock price today or then. You have two of these pairs available. If you buy one and sell the other...