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Contingent Claim Payoffs and Arbitrage Pricing at Maturity

Article Quant Q&A · Author: KMR
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Summary

The question concerns a result in Björk's treatment of arbitrage theory. It describes a European call as a contingent claim whose payoff at time one depends on the stock's move, represented by a random variable with up and down outcomes. The author asks why an arbitrage could arise if the claim's price is not equal to a quantity denoted by X.

The accepted response points to a distinction that resolves the confusion: X is the claim's value at maturity, while the notation for the portfolio value at time one must be interpreted in relation to that payoff. This directs the reader to compare the terminal claim payoff with the portfolio replication or valuation expression. The excerpt gives only a hint, not a worked proof, model assumptions, or a full arbitrage construction, so readers need the surrounding text to complete the argument.

Key ideas

  • The option payoff at maturity is contingent on the stock's outcome.
  • The question turns on interpreting the claim payoff X at time one.
  • A comparison with the portfolio value at time one is central to resolving the apparent arbitrage.
  • The answer is only a hint and does not provide a complete proof.

Tags

OptionsArbitrageDerivatives pricing
Full text
# Confusion about how price of a contingent claim at time 1 could give arbitrage


# Confusion about how price of a contingent claim at time 1 could give arbitrage












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 $X$.

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.

## Answer by msantama (score 0, accepted)

https://quant.stackexchange.com/a/79172

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

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.