The Engine Room: Harrison and Pliska Build the Machinery of Continuous Trading

Two years after Harrison and Kreps showed that no-arbitrage and risk-neutral probabilities are the same idea, Michael Harrison teamed up with Stanley Pliska to do the harder job: build the same theory for a world where trading happens continuously, in real time, with prices that never stop moving.

This is the paper that made mathematical finance a real branch of mathematics. It is dense, it is technical, and almost nobody outside academia reads it. But every risk system on every trading floor rests on the results in it. If Harrison-Kreps supplied the idea, Harrison-Pliska supplied the engine room.

The problem: the discrete argument does not survive contact with continuous time

In a binomial tree, everything is finite. Finitely many nodes, finitely many outcomes, finitely many trades. You can define a portfolio strategy by listing what you hold at each node, and you can add up your gains by summing over steps. Nothing can go wrong.

Now let time flow continuously. The stock price wiggles at every instant. Your hedge must be adjusted at every instant. Your trading profit is no longer a sum, it is an integral, and not an ordinary one: you are integrating against a path that is nowhere smooth and has infinite length. Ordinary calculus simply does not apply. Worse, once you allow arbitrary continuous-time strategies, genuinely pathological things become possible. The most famous is the doubling strategy, the martingale betting system: keep doubling your stake after every loss, and you are guaranteed to end up ahead, as long as you are allowed to lose an unbounded amount along the way. In continuous time you can squeeze infinitely many doublings into a finite window. If that counts as a legal trading strategy, then arbitrage exists in every market, and the whole theory collapses before it starts.

So the questions were sharp. What is a legitimate trading strategy? What does "self-financing" mean when you rebalance continuously? And does the beautiful discrete-time result survive?

The key idea via analogy: writing the rulebook before playing the game

Think of it like drafting the rules of a sport. Everyone agrees roughly what football is, but you cannot referee a match until someone has written down, precisely, what counts as a legal move and what does not. Without that, players will find loopholes, and the loopholes will be exploited.

Harrison and Pliska wrote the rulebook, and it has three parts.

First, what a trading strategy is. They modelled prices as semimartingales, a broad class of random processes that includes essentially every price model anyone uses, and defined a portfolio's gains using the stochastic integral, the Ito integral. This is the correct continuous-time analogue of "add up your profits step by step" for paths too jagged for ordinary calculus. A strategy is self-financing if, after the initial investment, every purchase is funded by a sale: no money comes in, no money goes out, you are just moving value between the stock and the cash account.

Second, which strategies are legal. To kill the doubling loophole, you must forbid strategies that require unlimited borrowing on the way to their guaranteed win. Harrison and Pliska imposed admissibility conditions, essentially, your losses must be bounded, you cannot dig an infinitely deep hole. This is not a technicality invented to save the maths. It is a statement about reality: no counterparty will let you keep doubling forever, and no bank has an infinite balance sheet. The rule that saves the theorem is also the rule that describes the world.

Third, the payoff. With the rulebook in place, they proved the continuous-time version of the fundamental theorem, and then went further, to what is arguably the paper's signature result.

The signature result: completeness is a property of the maths, not of your cleverness

Here is the question every derivatives desk implicitly asks: can I hedge this thing? Can I manufacture the payoff of this option by trading the stock and cash, so that whatever happens, I break even?

Harrison and Pliska answered it in the most elegant way imaginable. A market is complete, meaning every payoff can be replicated, if and only if the risk-neutral probability measure is unique. And that, in turn, holds if and only if the price process has a certain martingale representation property: every fair-game process driven by the same source of randomness can be written as an integral against the price itself.

Read that again, because it is genuinely deep. "Can I build any payoff out of stock and cash?" and "Is there only one consistent set of fair odds?" are the same question. They are not analogous, they are not related, they are identical. Hedgeability, a practical, sweaty, trading-desk concern, turns out to be a purely mathematical property of the randomness driving the market.

The intuition is a counting argument. If the stock is driven by one source of randomness, one Brownian motion, then trading the stock gives you one dial to turn, and one dial is enough to neutralise one source of risk. That is why Black-Scholes works, and why the option price is unique. But if the stock is driven by two sources, say its own noise plus a separate, independently wiggling volatility, then you have two risks and only one dial. You cannot hedge them both. And precisely then, the risk-neutral measure stops being unique, and the option no longer has a single arbitrage-enforced price. Every incomplete-market headache in modern finance, stochastic volatility, jumps, credit, is this counting argument biting.

Why it mattered

• It is the licence to use probability theory on trading floors. Every "take the expectation under the risk-neutral measure and discount" calculation performed by every pricing library in the world is legitimate because of the theorems in this paper.
• It explains why some things are hedgeable and some are not, in advance. You do not have to try and fail. Count your sources of risk, count your tradable instruments. If risks outnumber instruments, no amount of ingenuity will produce a perfect hedge, and the price of your exotic is a range, not a number.
• It disarmed the doubling paradox. By taking seriously the fact that you cannot lose unboundedly, it showed that continuous-time finance is coherent. Without this, the field would have been built on sand.
• It set the template for everything after. Girsanov's theorem for changing measures, martingale representation for constructing hedges, the whole standard toolkit of the quant: this paper is where those tools were assembled into a working kit for finance.

The honest limitations

• The market it describes is frictionless and idealised. No transaction costs, no bid-ask spread, no limits on short selling, unlimited liquidity at the quoted price, and continuous rebalancing. Real hedging is discrete, costly and sometimes impossible when you need it most.
• It tells you when a hedge exists, not how much it costs. Existence proofs are not implementations. The replicating strategy the theory promises may require trading enormous quantities at exactly the moments the market is least willing to trade with you.
• Completeness is the exception, not the rule. Almost every realistic model, anything with jumps or genuinely random volatility, is incomplete. So the beautiful "unique price" conclusion applies mainly to the textbook case. For the models people actually use, the theory says: here is a range, choose your own risk premium. That choice, not the maths, is where much of the real money and real risk sits.
• It is a hard read. The paper's practical influence is enormous but almost entirely indirect, filtered through textbooks. Very few of the people who depend on it every day have read a line of it.

The one-line takeaway

Harrison and Pliska built the mathematical machinery that lets derivatives pricing work in continuous time, and proved the field's most quietly profound fact: being able to hedge every payoff and having exactly one set of fair-odds probabilities are the same condition, so whenever your model has more sources of risk than you have instruments to trade, perfect hedging is not hard, it is impossible.