Risk-Neutral Measures and No-Arbitrage in a One-Period Market
Summary
The document examines why a no-arbitrage condition in a one-period market does not generally identify a unique probability measure. With a risk-free asset and a risky asset that can end at either a lower or higher value, the risk-free return lying between those outcomes prevents a sure dominance. That inequality alone allows many probability assignments for the two outcomes.
The answers clarify the distinction between physical probabilities and risk-neutral probabilities. In a market with a traded stock and a forward, arbitrage-free pricing requires the forward price to align with the stock’s discounted expectation under a risk-neutral measure; an arbitrary probability choice need not satisfy that condition. The text also notes that a mispriced derivative can fail to have discounted martingale dynamics and may produce gains or losses under delta hedging. It is a brief conceptual exchange rather than a full proof, and it does not develop the general conditions for existence or uniqueness of equivalent martingale measures.
Key ideas
- A one-period no-arbitrage inequality can hold under multiple probability measures.
- Risk-neutral probabilities must price traded assets consistently with the risk-free return.
- A traded forward can expose an inconsistent stock expectation through arbitrage.
- A derivative price inconsistent with martingale dynamics can create systematic delta-hedging gains or losses.
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Full text
# Different risk neutral measure # Different risk neutral measure I don't understand in the following example how there can be a single risk neutral measure. The risk free asset price $B$ at time $t = 1$ is $1+R$. An other asset $S$ at time $t=1$ can take two values: $d$ with probability $q_d$ and $u$ with probablity $q_u$. We assume that $d \leq u$. In order to not have arbitrage we must have that the asset can't dominate the risk-free asset and that the risk-free asset can't dominate the asset so: $$d \leq 1+R \leq u$$ Yet I don't see how this simple inequality forces a single risk-neutral measure $Q$. For example taking $d =1, u =4, R = 1$, then the inequality is respected and we can take whatever probability like for example $q_u = 0.5, q_d = 0.5$ or $q_u = 0.1, q_d = 0.9$. In my book there are saying this inequality means that the expectation of the asset is equal to $1+R$. But this is not correct right? since I can choose whatever probability measure (as long as both event don't have probability $0$) and my market is arbitrage free. Even in the more general case (multiple asssets, continuous time...) I don't see how for a market to be arbitrage free a portfolio expectation should always be equal to the risk-free asset. For a market to be arbitrage free we just need that: nothing can completely dominate the risk-free asset (in the sense we can build a portfolio such that with probability $1$ the portfolio value in the future is bigger than the risk-free asset) or be dominated by the risk-free asset, but with this condition there are a lot of different risk-neutral probability measure that we can find and the expectation will not be equal to the risk free asset. ## Answer by dm63 (score 3) https://quant.stackexchange.com/a/80080 Another example: if the probabilities of u and d are both 0.5, then you will calculate the forward price of the stock as $(u+d)/2$. If this is not equal to $1+R$, you will have an arbitrage between the spot and forward price of the stock. ## Answer by user121416 (score 0) https://quant.stackexchange.com/a/80079 If the discounted asset (call option) is not a martingale then you can hedge it with a martingale (the stock) and earn more/less than the risk free rate, guaranteed. For example, value an option with the wrong vol (so that it is not a martingale) and you will make the delta hedge consistently gain/lose money.
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