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I am trying to wrap my head around the 'no arbitrage' (NA) and 'efficient market' (EM) assumptions, in particular their difference.

Please correct me if I am wrong, but NA just claims that there is no trading strategy that generates profit but has zero risk, while by contrast EM states that the (historical) price data of an asset already contains all that there is to know about its future development.

Does one of them imply the other?

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NA, in my opinion, is the lack of a trading strategy to generate excess returns, that means returns greater than those of assets in the same risk category. Note that it doesn't mean no risk.

A market is efficient, again in my own understanding, when all the contents of an information set are reflected in the market's assets prices. There are 3 forms of market efficiency weak, semi-strong and strong.

  • Weak form efficiency is given when the information set contains historical data about the assets, for example past prices, returns, past news about the companies, etc. An example of weak form efficiency would be when past growth data of a company is incorporated into its equity prices.

  • Semi-strong form efficiency occurs when the market is weak form efficient and the information set also includes all currently available public data about the assets in the market. For example press releases, macroeconomic indicators, etc. An example of semi-strong efficiency would be when due to an increase in the interest rate, the cost of capital of a company increases and the equity prices fall.

  • Strong form efficiency takes place when a market is semi-strong form efficient and the information set also includes all available private information about assets in the market. For example, plans yet not publicly available about a new product release or private events in the lives of the members of the management team. An example of strong form efficiency would be if, due to the CEO of the company going through a divorce, the company's shares prices fall.

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A useful way to think about the NA assumption is to conceptualize it as a very strong constraint on equations of a given model. It has been shown that NA unifies not only many theories of financial economics (like the Modigliani-Miller theorem, cash flow valuation, options and asset pricing, among others), but also game theory and decision theory. For references, see O. Varela, Arbitrage in General Equilibrium, (2012) 3 Modern Economy 396 and R.F. Nau, K.F. McCardle, Arbitrage, Rationality, and Equilibrium, (1991) 31 Theory and Decision 199.

Those constraints imply, among others, the existence of general equilibrium and, perhaps more importantly, of a positive linear map that prices all assets, including those that are not traded (!). The proof of this last implication was given in by Dybvig and Ross in (1987) New Palgrave: A Dictionary of Economics, volume 1, J. Eatwell, M. Milgate, P. Newman (eds).

This map, I think, helps to intuitively explain how NA is connected with EMH. If the market is able to price any and all assets, even those that are going to be traded only in the future, it must be strongly efficient in the sense of Fama. Stated more precisely, with perfect information, arbitrage-free prices maximize the amount of information in the economy. Traditional general equilibrium analysis deals with markets that are efficient in this sense. For a precise exposition, see N. Kuksin, General Equilibrium: Arbitrage and Information, Centre for Economic Reform and Transformation Discussion Paper 2007/01.

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In the presence of transaction costs No Arbitrage doesn't have to imply Efficient Markets. For example, prices can be forecast-able but there is no way to profit from that forecast so prices remain wrong but are consistent with each other.

For example, it is possible to have three assets and two states of the world, all priced consistently with each other (in a no arbitrage sense), but to have the asset prices returns violate the efficient market. Consider states High and Low. Asset one has high and low payoffs \$2/\$0 and costs \$1. Asset two has payoffs \$0/\$3 and costs \$1. Asset three pays of \$4/$6 and costs \$4 (which is the same payoff as a portfolio of two of asset one and two of asset two and has the same cost as that portfolio). There is no arbitrage in this economy. However, if there are risk neutral investors that are not borrowing constrained, and using all public information we know that the probability of the high event is 75% and everyone agrees that this is true, then these are are not the efficient prices.

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