mathematics

Fundamental theorem of asset pricing

Fundamental theorem of asset pricing is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Fundamental theorem of asset pricing rather than just read about it. In short: The fundamental theorems of asset pricing (also: of arbitrage, of finance), in both financial economics and mathematical finance, provide necessary and sufficient conditions for a market to be arbitrage-free, and for a market to be complete. An arbitrage opportunity is a way of making money with no initial investment without any possibility of loss.

Key takeaways

Fundamental theorem of asset pricing belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Fundamental theorem of asset pricing to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Fundamental theorem of asset pricing from memory before moving on to harder problems.

Reference excerpt

The fundamental theorems of asset pricing (also: of arbitrage, of finance), in both financial economics and mathematical finance, provide necessary and sufficient conditions for a market to be arbitrage-free, and for a market to be complete. An arbitrage opportunity is a way of making money with no initial investment without any possibility of loss. Though arbitrage opportunities do exist briefly in real life, it has been said that any sensible market model must avoid this type of profit. The first theorem is important in that it ensures a fundamental property of market models. Completeness is a common property of market models (for instance the Black–Scholes model). A complete market is one in which every contingent claim can be replicated. Though this property is common in models, it is not always considered desirable or realistic.

Discrete markets In a discrete (i.e. finite state) market, the following hold:

The First Fundamental Theorem of Asset Pricing: A discrete market on a discrete probability space ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},P)} is arbitrage-free if, and only if, there exists at least one risk neutral probability measure that is equivalent to the original probability measure, P. The Second Fundamental Theorem of Asset Pricing: An arbitrage-free market (S,B) consisting of a collection of stocks S and a risk-free bond B is complete if and only if there exists a unique risk-neutral measure that is equivalent to P and has numeraire B.

In more general markets When stock price returns follow a single Brownian motion, there is a unique risk neutral measure. When the stock price process is assumed to follow a more general sigma-martingale or semimartingale, then the concept of arbitrage is too narrow, and a stronger concept such as no free lunch with vanishing risk (NFLVR) must be used to describe these opportunities in an infinite dimensional setting. In continuous time, a version of the fundamental theorems of asset pricing reads: Let S = ( S t ) t ≥ 0 {\displaystyle S=(S_{t})_{t\geq 0}} be a d-dimensional semimartingale market (a collection of stocks), B {\displaystyle B} the risk-free bond and ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},P)} the underlying probability space. Furthermore, we call a measure Q {\displaystyle Q} an equivalent local martingale measure if Q ≈ P {\displaystyle Q\approx P} and if the processes ( S t i B t ) t {\displaystyle \left({\frac {S_{t}^{i}}{B_{t}}}\right)_{t}} are local martingales under the measure Q {\displaystyle Q} .

The First Fundamental Theorem of Asset Pricing: Assume S {\displaystyle S} is locally bounded. Then the market S {\displaystyle S} satisfies NFLVR if and only if there exists an equivalent local martingale measure. The Second Fundamental Theorem of Asset Pricing: Assume that there exists an equivalent local martingale measure Q {\displaystyle Q} . Then S {\displaystyle S} is a complete market if and only if Q {\displaystyle Q} is the unique local martingale measure.

See also Arbitrage pricing theory Asset pricing Financial economics § Arbitrage-free pricing and equilibrium Rational pricing

References Sources

Worked examples

Example 1 — a first encounter with Fundamental theorem of asset pricing

Start with the simplest possible case. Write down what Fundamental theorem of asset pricing claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Fundamental theorem of asset pricing before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Fundamental theorem of asset pricing ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Fundamental theorem of asset pricing

In research: Fundamental theorem of asset pricing appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Fundamental theorem of asset pricing in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Fundamental theorem of asset pricing is common in secondary-school and first-year university syllabi. It links to neighbouring topics Corporate development, Financial economics, Mathematical finance, so understanding it makes those chapters shorter.
In everyday life: Look for Fundamental theorem of asset pricing outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Fundamental theorem of asset pricing in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Fundamental theorem of asset pricing means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Fundamental theorem of asset pricing out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Fundamental theorem of asset pricing in simple terms?

Why does Fundamental theorem of asset pricing matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Fundamental theorem of asset pricing?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Fundamental theorem of asset pricing.