Numéraire

In a financial market with traded securities, one may use a numéraire to price assets. For instance, let ${\displaystyle M(t)}$ 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 ${\displaystyle S(t)}$ priced in terms of the numéraire (in this case, M), are martingales with respect to a risk-neutral measure, say ${\displaystyle Q}$. That is:

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

Now, suppose that ${\displaystyle N(t)>0}$ 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 ${\displaystyle Q^{N}}$ by the Radon–Nikodym derivative

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

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

${\displaystyle {\begin{aligned}&{}\quad E_{Q^{N}}\left[{\frac {S(T)}{N(T)}}\right]\\&=E_{Q}\left[{\frac {N(T)}{M(T)}}{\frac {S(T)}{N(T)}}\right]/E_{Q}\left[{\frac {N(T)}{M(T)}}\right]\\&={\frac {M(t)}{N(t)}}E_{Q}\left[{\frac {S(T)}{M(T)}}\right]\\&={\frac {M(t)}{N(t)}}{\frac {S(t)}{M(t)}}\\&={\frac {S(t)}{N(t)}}\end{aligned}}}$

This technique has many important applications in LIBOR and swap market models, as well as commodity markets. Jamshidian (1989) first used it in the context of the Vasicek model for interest rates in order to calculate bond options prices. Geman, El Karoui and Rochet (1995) introduced the general formal framework for the change of numéraire technique. See for example Brigo and Mercurio (2001)[1] for a change of numéraire toolkit.

Determining an appropriate numéraire has foundation in several financial pricing models such as options and certain assets. Identifying a risky asset as numéraire has a correlation with the number of underlying assets to model. Underlying shifts are modelled by the following:

${\displaystyle Z_{i}:={\frac {X_{i}}{X_{0}}}}$

${\displaystyle X=(X_{0},X_{1},...,X_{n})\to Z=(1,Z_{1},...,Z_{n})}$

where 1 defines the numéraire.

• Forward measure
• Price index
• Unit of account

• Farshid Jamshidian (1989). "An Exact Bond Option Pricing Formula". The Journal of Finance. 44: 205–209. doi:10.1111/j.1540-6261.1989.tb02413.x.
• Helyette Geman; Nicole El Karoui; J.C. Rochet (1995). "Changes of numéraire, changes of probability measure and option pricing". Journal of Applied Probability. 32 (2): 443–458. doi:10.2307/3215299. JSTOR 3215299. S2CID 124199920.
• Damiano Brigo; Fabio Mercurio (2006) [2001]. Interest Rate Models – Theory and Practice with Smile, Inflation and Credit (2 ed.). Springer Verlag. ISBN 978-3-540-22149-4.
• Allingham M. (2008) Numeraire. In: Palgrave Macmillan (eds) The New Palgrave Dictionary of Economics. Palgrave Macmillan, London. https://doi.org/10.1057/978-1-349-95121-5_1514-2