Risk-Neutral Pricing and Arbitrage in a One-Period Binomial Market

Summary

The document poses a basic discrete-time finance problem: a savings account grows by a fixed factor, while a stock can move to one of two values. The stock’s initial value is normalized, and the two possible outcomes lie on opposite sides of the savings account’s growth factor. The question asks how to find and prove the risk-neutral probability measure and whether the market permits arbitrage.

The setup points to the standard binomial-market approach: choose probabilities that make the discounted stock price a martingale, then use the strict ordering of the down, risk-free, and up factors to assess whether both outcomes can be replicated without arbitrage. However, the document contains only the question and no worked derivation, proof, or explicit conclusion. It is useful as an exercise prompt, but readers will need to supply the calculations themselves. Its scope is limited to a one-period model with two stock outcomes and a deterministic savings account; it does not address option replication in detail, despite the title.

Key ideas

• A one-period binomial model has two possible stock prices at the next time step.
• The risk-neutral measure is found by requiring the discounted stock price to have the martingale property.
• The stated ordering places the risk-free growth factor between the stock’s down and up factors.
• The document asks about arbitrage but provides no solution or proof.

Tags

# Replicating portfolio of an option and to find inital price


# Replicating portfolio of an option and to find inital price












I am very new to financial math so I am not sure how to do with this question. A friend sent me this question to practice but I am unsure how to begin. I read about call option . Can that be used for any option? Any help would be appreciated.

> Consider the following discrete time one-period market model. The savings account is at \$1 at time 0 and \$$\beta$ at time 1. The stock price is given by $S_0 = 1$ and $S_1 = \xi$ where $\xi$ is a random variable taking two possible values $u$ and $d$, each with positive probability. Moreover, assume that $0 < d < \beta < u$. Find, with proof, the risk-neutral measure of this model. Does this model have arbitrage opportunities?

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.