No Free Lunch: Harrison and Kreps Find the Grammar of Arbitrage

By the late 1970s, quantitative finance had a formula it trusted and an explanation it could not quite articulate. Black and Scholes had priced options. Cox, Ross and Rubinstein had shown the same answer falls out of a simple tree. Both derivations kept producing the same strange artefact: a set of made-up probabilities, the risk-neutral probabilities, that were not the real odds of anything, yet under which every price was just a discounted expected value.

Why did that keep happening? Was it a coincidence of the particular models people had chosen? Or was something deeper going on?

Michael Harrison and David Kreps answered the question in 1979, and the answer was as clean as it was surprising. The risk-neutral probability is not an artefact of any model. It is what "no free lunch" means, translated into the language of probability. The two statements are the same statement.

The problem: pricing without a theory of value

Traditional economics prices an asset by asking what people want. You model investors' preferences, their tolerance for risk, their beliefs, and you grind out an equilibrium. It works, but it depends on a mountain of assumptions about human beings that nobody can verify.

Option pricing seemed to sidestep all that. Black, Scholes, Merton and the binomial crowd never asked what investors preferred. They asked only: what price would allow someone to make money out of nothing? Rule that price out, and only one price remains. That is a far more robust foundation, because "you cannot make money out of nothing" is something almost everyone will agree to.

But this raised a question nobody had properly answered. Exactly how far does that argument go? When does no-arbitrage pin down a unique price, and when does it merely narrow the range? What is the general structure here, underneath all the specific models?

The key idea via analogy: the bookmaker's odds

Think about a bookmaker taking bets on a horse race. The bookmaker quotes odds on every horse. Now, those odds are not really the bookmaker's honest opinion of who will win. They are set so that the book balances: whatever the outcome, the bookmaker is not exposed. A punter cannot construct a set of bets that guarantees a profit regardless of which horse comes first.

Here is the mathematical fact underneath that intuition. A set of odds is free of guaranteed-profit combinations if and only if you can convert those odds into a proper set of probabilities that add up to one. If the implied probabilities summed to less than one, you could bet on every horse and profit no matter what. The absence of a free lunch is equivalent to the existence of a consistent probability assignment.

Harrison and Kreps showed that a securities market works exactly like the bookmaker's board. If no combination of trades can produce a riskless profit out of nothing, then there must exist a probability measure, an alternative set of odds on how the world unfolds, under which every traded asset's price, once you strip out the risk-free interest rate, is a martingale. A martingale is simply a process whose best forecast of tomorrow is today's value: a fair game, no drift, no edge. And conversely, if such odds exist, no arbitrage is possible.

That equivalence is the whole paper in one sentence. It is now called the fundamental theorem of asset pricing, and it has two halves that are worth separating:

• No arbitrage means fair-odds probabilities exist. There is at least one way of reweighting the future under which nobody has an edge.
• If those fair-odds probabilities are unique, every derivative has exactly one price. Uniqueness corresponds to what is called a complete market: every possible payoff can be manufactured by trading the assets you already have. When you can build a copy of an option out of stock and bonds, the option's price is forced. When you cannot, no-arbitrage alone gives you only a range of admissible prices, and something outside pure arbitrage logic must break the tie.

That second bullet is the part practitioners underrate. It explains precisely why Black-Scholes gets a unique answer (you can replicate the option) and why models with jumps or stochastic volatility do not (you cannot). It is not a quirk of those models. It is a structural fact about how much information trading in the underlying gives you.

Why it mattered

• It turned derivatives pricing into a theorem, not a trick. Before Harrison and Kreps, risk-neutral probabilities looked like a computational convenience that happened to work. After, they were understood as the necessary shadow cast by the absence of arbitrage. Any model, present or future, must have them.
• It handed the field to the probabilists. Once you know pricing is "find the martingale measure and take an expectation," the whole apparatus of modern probability theory, martingales, changes of measure, stochastic integration, becomes directly applicable. The result was an explosion of tractable models. Almost every derivatives paper written since is, in effect, an exercise in the Harrison-Kreps program.
• It clarified what completeness buys you. The distinction between a market where you can hedge perfectly and one where you cannot is the difference between a price and a price range. That distinction shapes how banks actually think about the risk of exotic books.
• It is model-free. The theorem does not care whether the stock follows a random walk, a jump process or something nobody has invented yet. It is about the logical structure of prices, so it survives every change of fashion in modelling.

The honest limitations

• It says a fair-odds measure exists, not how to find it. The theorem is an existence result. Turning it into an actual number requires you to specify a model of how prices move, and there the arguments start again.
• Real markets are incomplete, and that is where the money and the danger live. Perfect replication requires continuous, costless trading in a world with no jumps. None of that is true. So in practice you almost never get a single arbitrage-enforced price; you get a range, and traders pick a point inside it based on judgment, supply and demand, and their own risk appetite. The theorem tells you the range exists. It does not tell you where to sit in it.
• "No arbitrage" is an idealisation too. The theorem's frictionless world has no bid-ask spread, no collateral, no funding costs, no limits on short selling. Add those and even the definition of arbitrage gets slippery. A great deal of post-2008 research is about repairing this.
• It is a statement about consistency, not correctness. A market can be perfectly free of arbitrage and still be pricing things absurdly relative to reality. Internal consistency is not the same as being right.

The one-line takeaway

Harrison and Kreps proved that "there is no free lunch" and "there exists a set of pretend probabilities that makes every asset a fair game" are two ways of saying exactly the same thing, which is why risk-neutral pricing keeps appearing in every derivatives model ever written, and why perfect hedging is precisely the condition under which an option has one price rather than a range.