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Aaron Brown (in the book, The Poker Face of Wall Street, p. 196), discusses four approaches to deriving the same Black-Scholes-Merton option-pricing formula:

Ed Thorp, Myron Scholes, Robert Merton, and Fischer Black all had almost the same formula [for option-pricing], but each had a different reason for believing it was true. Ed showed that it was a way to make money, Scholes that it was required for market efficiency, Merton that it had to be true or there would be arbitrage, and Black that it was required for market equilibrium. Black's insight turned out to be the most important...

What is the difference between "market efficiency", "no arbitrage", and "market equilibrium", and why would equilibrium be considered the most important insight?

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In three bullet points:

  • Efficiency: the obtained prices maximize assumed utilities of different agents. In their paper "The Valuation of Option Contracts and a Test of Market Efficiency", Cohen, Black and Scholes compare the theoretical value of options to their market price. The efficiency is in this sense: can agents obtain more or less in practice than in theory using options?
  • No arbitrage: it is possible to synthesize different prices combining (linearly, in the case of Merton) existing instruments. In his paper "On the pricing of contingent claims and the Modigliani-Miller theorem", Merton nevertheless seems to contradict Aaron Brown since he writes "Although the formula can be derived using the arbitrage technique of Black and Scholes, the alternative approach of continuous-time portfolio strategies is used instead." He writes later in the paper "Market prices need not be general equilibrium prices." Hence he does not use equilibrium argument for sure. But during the proof, he writes "[...] then to rule out arbitrage,[...]. He definitely need and uses a no arbitrage argument.
  • Equilibrium: close to efficiency but every agent in the market has a utility function, and the obtained prices simultaneously solves all of them (it is typically a Nash equilibrium). I did not found a paper by Black dealing simultaneously with derivatives and market equilibrium. Nevertheless Black published papers on equilibria, like "International capital market equilibrium with investment barriers".

To compare the three concepts:

  • an agent does not need to have a utility function to implement arbitrage, it is a very local and objective behaviour: observe prices, buy low and sell high...
  • efficiency is to be optimal for one agent. That for he needs to objective and quantify his utility and maximize it. It means the considered agent is rational an advanced way.
  • last but not least: market equilibrium. It is a global property: everyone is optimal simultaneously, and knows others are optimal (and know how they manage to be).
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