Numéraire

Key takeaways

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

Reference excerpt

The numéraire (or numeraire) is a basic standard by which value is computed. In mathematical economics it is a tradable economic entity in terms of whose price the relative prices of all other tradables are expressed. In a monetary economy, one of the functions of money is to act as the numéraire, i.e. to serve as a unit of account and therefore provide a common benchmark relative to which the value of various goods and services can be measured against. Using a numeraire, whether monetary or some consumable good, facilitates value comparisons when only the relative prices are relevant, as in general equilibrium theory. When economic analysis refers to a particular good as the numéraire, one says that all other prices are normalized by the price of that good. For example, if a unit of good g has twice the market value of a unit of the numeraire, then the (relative) price of g is 2. Since the value of one unit of the numeraire relative to one unit of itself is 1, the price of the numeraire is always 1.

Change of numéraire In a financial market with traded securities, one may use a numéraire to price assets. For instance, let M ( t ) {\displaystyle M(t)} be the price at time t of $1 that was invested in the money market at time 0. The fundamental theorem of asset pricing says that all assets S ( t ) {\displaystyle S(t)} priced in terms of the numéraire (in this case, M), are martingales with respect to a risk-neutral measure, say Q {\displaystyle Q} . That is:

S ( t ) M ( t ) = E Q [ S ( T ) M ( T ) ] {\displaystyle {\frac {S(t)}{M(t)}}=E_{Q}\left[{\frac {S(T)}{M(T)}}\right]}

Now, suppose that N ( t ) > 0 {\displaystyle N(t)>0} is another strictly positive traded asset (and hence a martingale when priced in terms of the money market). Then we can define a new probability measure Q N {\displaystyle Q^{N}} by the Radon–Nikodym derivative

d Q N d Q = M ( 0 ) M ( T ) N ( T ) N ( 0 ) = N ( T ) M ( T ) {\displaystyle {\frac {dQ^{N}}{dQ}}={\frac {M(0)}{M(T)}}{\frac {N(T)}{N(0)}}={\frac {N(T)}{M(T)}}}

Then it can be shown that S ( t ) {\displaystyle S(t)} is a martingale under Q N {\displaystyle Q^{N}} when priced in terms of the new numéraire N ( t ) {\displaystyle N(t)} :

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Numéraire

Start with the simplest possible case. Write down what Numéraire 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 Numéraire 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 Numéraire 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 Numéraire

In research: Numéraire 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 Numéraire 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: Numéraire is common in secondary-school and first-year university syllabi. It links to neighbouring topics Finance theories, General equilibrium theory, Mathematical finance, so understanding it makes those chapters shorter.
In everyday life: Look for Numéraire 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.

How to study Numéraire in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Numéraire 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 Numéraire out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Numéraire in simple terms?

The numéraire (or numeraire) is a basic standard by which value is computed. In mathematical economics it is a tradable economic entity in terms of whose price the relative prices of all other tradables are expressed.

Why does Numéraire 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 Numéraire?

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 Numéraire.