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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 $X$.

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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.

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  • $\begingroup$ You mean the final question “(why?)” in the text? $\endgroup$
    – Bob Jansen
    Commented Apr 27 at 7:16
  • $\begingroup$ Yes, I am sorry if it wasn’t clear. $\endgroup$
    – KMR
    Commented Apr 27 at 7:22

1 Answer 1

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Recall what $X$ is -- it is the value of the contingent claim at time $t = 1$. Now think how this relates to $\Pi(1; X)$.

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  • $\begingroup$ I am going to assume that I can breakdown $t=1$ into two further time instants which I will confusingly call, $t=0$ and $t=1$, where $t=0$ is the time just before maturity. Assume that $\Pi(1:X)<X$, then at $t=0$, with stock price $S$ I will short $\Pi(1;X)/S$ number of stocks and buy an option at $\Pi(1;X)$. The initial value of this portfolio is zero. Now at maturity $t=1$, the option will be exercised so it is now worth $X$ and the portfolio is now $X-\Pi(1;X)>0$. So we have an arbitrage opportunity. $\endgroup$
    – KMR
    Commented Apr 29 at 23:23
  • $\begingroup$ I think it’s simpler than you’re imagining. At $t=1$ the payout is fully determined, so the price of the claim must equal that payout. A concrete example might help. If a call option expires \$10 in the money, the price of that contract at expiration must be \$10. Otherwise there is an obvious arbitrage, ie. buy the contract if the price is less than \$10 and sell if it is more than \$10 $\endgroup$
    – msantama
    Commented Apr 30 at 0:45
  • $\begingroup$ I might have convoluted it but would you say buying the contract is the same as the situation I mentioned above? $\endgroup$
    – KMR
    Commented Apr 30 at 0:52
  • $\begingroup$ I don't think you have it quite right. This is a one-period discrete time model, so there is no "time just before maturity". Also, you say "the option will be exercised" but this is not always true. Let me try again. At $t = 1$ we can directly observe the stock price so we know if $Z = u$ or $Z = d$. If $Z = u$ then owning the contingent claim will result in a risk-less positive cash inflow of $su - K$. If it costs less than $su - K$ to purchase the contingent claim, there is an obvious arbitrage opportunity (purchase the contingent claim for an immediate profit). I leave $Z = d$ for you. $\endgroup$
    – msantama
    Commented Apr 30 at 1:49
  • $\begingroup$ By owning do you meaning buying the contingent claim at $t=1$? Also, if we don't exercise the option how would we have the cash inflow of $su-K$? But looking back, I really did not mention that $S=us$ so there would be a possibly of the option not being exercised, I think? $\endgroup$
    – KMR
    Commented Apr 30 at 2:06

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