0
$\begingroup$

I have been reading the book Tomas Bjork's Arbitrage Theory in Continuous Time and could not understand how there could be arbitrage if the price of a contingent claim is not $X$.

enter image description here

To give some context, $X$ represents a European call option and $Z$ is a random variable representing how to stock will move at time 1. So if the stock price is $S$ at $t=0$, the stock price at $t=1$ can be written as $sZ$ where $Z=u$ would be the stock moving up and $Z=d$ would be the stock moving down.

I would like to figure it out myself so maybe just a hint on how this could be would be very helpful.

$\endgroup$

1 Answer 1

Reset to default
1
$\begingroup$

At the option expiration we imagine that one can buy and sell the stock at the market price and buy and sell the options at the option price. This enables a round trip. You can buy an option, convert it to a share, and sell the share. If this results in a profit, you have free money at no risk. If this results in a loss, just go the other way around the cycle. Buy a share and sell an option. Based on the loss in the other direction, you can count on the option being exercised and you have a profit.

$\endgroup$
4
  • $\begingroup$ I would like to check if I understood the answer correctly if you don't mind. Suppose the stock went up and now the price is $su$ and the strike is $K$ but the price of the option $p$ is less than $su-K$. Then I can buy the option at price $p$, exercise the option and sell the stock. With that, I will end up with $su-K-p$ profit without risk. Similarly if $p$ is greater than $su-K$, then I can sell the option at $p$ buy a share at $su$ and sell it at strike price $K$. $-p<K-su$ so I made a profit of $p-(su-K)$. $\endgroup$
    – KMR
    Commented Apr 27 at 4:05
  • 1
    $\begingroup$ Exactly right. This shows that if $p \neq su-K$ you can make a profit one way or the other, so we must have $p=su-K$ $\endgroup$
    – Ross Millikan
    Commented Apr 27 at 4:08
  • $\begingroup$ Thank you for your time! $\endgroup$
    – KMR
    Commented Apr 27 at 4:09
  • 1
    $\begingroup$ The claims of arbitrage that I have seen all work this way. You identify a set of trades that result in owning the same securities (usually none) at the end but having some extra cash, so the more you do it the richer you get. If you are asked to look for an arbitrage opportunity you should look for ways to make trades that cancel out the securities like we have here. $\endgroup$
    – Ross Millikan
    Commented Apr 27 at 4:17

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .