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The article "What is ... a Free Lunch?" nicely explains why market models with arbitrage opportunity are unlikely to describe financial markets of the real world.

Are there other properties of market models that are unlikely to be found in the real world?

For simplicity, let us stick to time-continuous models driven by Brownian motions.

For example, does the existence of two stocks $$ dX_1=r_1X_1 dt+ \sigma dW_1 $$

$$ dX_2=r_2X_2 dt+\sigma dW_2 $$ with $r_1\gg r_2$, $\sigma>0$ and independent Brownian motions $W_1,W_2$ contradict any common market axiom? There is no arbitrage-opportunity in this case, but my intuition still tells me that these two stocks cannot coexist in a realistic financial market, since I cannot imagine anyone that would buy $X_2$ if they could instead buy $X_1$. Is my intuition flawed here? If yes, are there any other signs of non-realistic market models?

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Generally the properties market prices must satisfy are of 2 kinds: no arbitrage properties and no-disequilibrium properties.

"No arbitrage" is stronger in that it creates a sure-profit incentive for its elimination. Any economic agent who sees a \$50 USD bill on the floor has an incentive to pick it up. On the other hand, the example you give is a disequilibrium: the stocks have the same risk $\sigma$ but one has a better expected return. There is an economic incentive to eliminate the disequilibrium by investing in 1 rather than 2, but it does not represent a riskless profit opportunity. Only economic agents who had already decided to purchase stock 2 have an incentive to switch to stock 1 when they discover the disequilibrium situation.

In general arbitrage conditions are rare to find (they require that 2 securities are perfect substitutes), and so most loosely related securities (for example stocks of two different companies, say Toyota and Honda) are priced by equilibrium considerations. Only in the field of Financial Derivatives do no-arbitrage conditions solve all our problems. Most questions in Economics (eg. why are oranges more expensive than apples) involve equilibrium considerations, supply (oranges are more difficult to grow) and demand (oranges are more nutritious, can be squeezed to make juice, etc.). "No arbitrage" does not help in relative pricing of oranges and apples (or Toyota and Honda) because they are goods with different properties.

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Yes, many! For example, stock prices aren't actually continuous or smooth. They are discrete and jump...

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Well under the risk-neutral measure, r1=r2=risk-free rate. In practice you'd add the cost of financing each stock and the dividend yields in the mix, so that you may have very different values for two stocks with the same volatility.

Now in the real, physical measure, you would expect the so-called market price of risk to be similar for all assets. That would imply that r1 ~ r2 for similar volatility. However there are very valid reasons why this isn't the case.

One is that the model is too simplistic, and there many more sources of risk that just a Brownian motion, so even if you observe a window of time when the two stocks behave like two constant-parameter Brownian motions (which won't be the case anyway), with similar volatilities, the fact that the drifts are vastly different could simply indicate that part of the process (e.g. a large, low-probability jump process on one of the two stocks) is not apparent over that window (it hasn't jumped in the period).

Second there are supply and demand dynamics at play whereby some stocks may simply be constantly bid up for example by index trackers.

Those are only two of many.

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