Complete And Incomplete Markets

Black-Scholes gives a single, unique price for an option. That should feel surprising: two traders can disagree about how likely the stock is to go up, yet must agree on the option's fair value. The reason is a very specific property of the model's world, and it is a property most real markets only approximate — the market is complete.

A hardware store with every part in stock

Imagine you need to build a custom bracket. If the hardware store stocks every possible size of bolt, washer, and bracket piece, you can assemble exactly what you need from parts, and the price of your custom bracket has to equal the sum of the parts — otherwise you'd just buy the pieces and skip the overpriced pre-made version. Now imagine the store only stocks three bolt sizes. Some brackets you simply cannot build from what's available, no matter how you combine them; for those, there's no "correct" price derivable from the parts, only a matter of what someone is willing to pay. A complete market is the fully-stocked hardware store: there are enough tradable securities (the stock and a risk-free bond, continuously tradable, in the Black-Scholes world) to build — replicate — any payoff exactly. An incomplete market is missing parts: jumps, stochastic volatility, transaction costs, or trading restrictions mean some payoffs simply cannot be assembled from what trades.

The formula

Completeness has a precise mathematical face: the market is complete if and only if the risk-neutral measure is unique.

$$\text{Complete} \iff \exists! \; \mathbb{Q} \text{ such that } \frac{V_0}{B_0} = \mathbb{E}^{\mathbb{Q}}\!\left[\frac{V_T}{B_T}\right]$$

In plain English: $\mathbb{Q}$ is the risk-neutral probability measure — a set of "as-if" probabilities under which every tradable asset's discounted price is a fair game (a martingale). $V_T$ is a payoff at time $T$, and $B_T$ discounts it back using the risk-free bond. If there is exactly one such $\mathbb{Q}$ consistent with the prices of the tradable securities, every payoff has one unique arbitrage-free price — you compute it by taking the expectation under that one $\mathbb{Q}$. If more than one $\mathbb{Q}$ is consistent with observed prices, the market is incomplete: different valid $\mathbb{Q}$'s give different "fair" prices for the same untradeable payoff, and no amount of arbitrage argument picks one over another.

Worked example 1: a complete one-period model

A stock at $100 can go to $120 or $90 in one period; the risk-free rate is 0%. There is exactly one $\mathbb{Q}$ that prices both outcomes consistently with the stock's own price: solve $100 = q(120) + (1-q)(90)$ for $q$, giving $q = 10/30 = 0.333$. With only two possible outcomes and two tradable instruments (the stock and cash), that single $q$ is forced — the market is complete for any payoff defined on these two outcomes. A call struck at $100 pays $20 or $0, and its unique price is $0.333 \times 20 + 0.667 \times 0 = 6.67$, i.e. $6.67. Anyone quoting a different price is offering an arbitrage.

Worked example 2: an incomplete one-period model

Now suppose the stock can move to three outcomes — $120, $100, or $90 — but you still only have the stock and cash to trade with, two instruments for three outcomes. Solving for a risk-neutral $q_1, q_2, q_3$ (summing to 1, matching the $100 stock price) is one equation short: infinitely many probability triples satisfy $100 = 120q_1 + 100q_2 + 90q_3$. A digital option paying $1 only if the stock lands exactly at $100 can be priced anywhere between $0 and roughly $0.77 depending on which valid $q_2$ you pick, and no arbitrage argument narrows it further. You would need a third tradable instrument — say, an option already trading in the market — to pin down a unique price.

What this means in practice

Real markets are incomplete: prices jump, volatility is itself random, and you cannot trade continuously for free. Practitioners patch this in two ways. First, add more tradable hedging instruments — a liquid options market for many strikes turns an otherwise incomplete stock-and-bond market into something close to complete, which is exactly why exotic-options desks hedge with vanilla options, not just the underlying. Second, when replication genuinely can't be made exact, abandon the idea of "the" fair price and instead price via Superhedging And Quantile Hedging or Utility Indifference Pricing, both of which give a range of defensible prices rather than one number.