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No-Arbitrage Condition in a One-Period Binomial Market

Article Quant Q&A · Author: Aaron Mitropolous
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Summary

The document poses a no-arbitrage question for a one-period market with a savings account and a stock that can end at either an up or down value. The initial prices are normalized, and the risk-free growth factor lies strictly between the two possible stock outcomes, each of which has positive probability.

Under these conditions, the model has no arbitrage: a risk-neutral probability can be assigned to the two outcomes so that the discounted stock price is a martingale. The response itself does not show this argument or derive the probability; it points to a textbook proposition for a proof. Thus, the market setup and key parameter ordering are available, but the document provides little worked explanation for applying the result or extending it to other models.

Key ideas

  • The model has a savings account and a stock with two possible terminal values.
  • The risk-free growth factor lies between the down and up stock outcomes.
  • This ordering is the standard no-arbitrage condition for a one-period binomial market.
  • The cited response refers readers to a textbook proposition rather than providing a proof.

Tags

Derivatives pricingArbitrageStatistics
Full text
# One-Period Binomial Model


# One-Period Binomial Model












So, I'm required to consider the one-period Binomial market model for a particular question. We're told that the savings account is \$1 at time 0 and \$β at time 1. The stock price is given by S0 = 1 and S1 = ξ where ξ is a random variable taking two possible values u and d, each with positive probability. It is assumed that 0 < d < β < u.

How can I prove or disprove whether this model has arbitrage opportunities?

## Answer by Bob Jansen (score 0)

https://quant.stackexchange.com/a/63967

This is Tomas Björk's Arbitrage Theory in Continuous Time Proposition 2.3. The book also contains a proof.

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.