Numeraires and forward measures

Definition Work on a filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{0 \le t \le T}, \mathbb{Q})$ in which $\mathcal{F}_0$ is trivial up to $\mathbb{Q}$-null sets, so that every time-$0$ value is a known number. Let $B_t = \exp\bigl(\int_0^t r_s \, ds\bigr)$ be the money-market account, a strictly positive process, and let $\mathbb{Q}$ be a pricing measure for the numeraire $B$. A claim that pays $H$ at $T$, where $H$ is $\mathcal{F}_T$-measurable with $\mathbb{E}^{\mathbb{Q}}[\lvert H \rvert / B_T] < \infty$, then has the time-$t$ value

and the discounted price of every traded asset without payouts is a $\mathbb{Q}$-martingale. A numeraire on $[0, T]$ is an adapted process $N$ such that

1. it is the value of a traded asset or of a self-financing portfolio that makes no payouts on $[0, T]$, any income being reinvested in the position,
2. it is strictly positive, $N_t > 0$ for every $t \in [0, T]$ almost surely, including at $T$,
3. the ratio $N/B$ is a true $\mathbb{Q}$-martingale on $[0, T]$, not only a local martingale.

The account $B$ is a numeraire, with $N/B = 1$. A price measured in units of $N$ is the ratio $V_t/N_t$, the number of units of $N$ that $V_t$ buys. A positive quantity that is not the value of a position anyone can hold, such as the short rate or a statistical index, is not a numeraire, however convenient it would be as a unit.

Definition For a numeraire $N$ on $[0, T]$, the density process with respect to $\mathbb{Q}$ is

for $0 \le t \le T$. It is a strictly positive true $\mathbb{Q}$-martingale with $Z_0^N = 1$, so $\mathbb{E}^{\mathbb{Q}}[Z_T^N] = 1$. The measure

defined for $A \in \mathcal{F}_T$, is a probability on $\mathcal{F}_T$, equivalent to $\mathbb{Q}$ because $Z_T^N > 0$. For $A \in \mathcal{F}_t$ the tower property gives $\mathbb{Q}^N(A) = \mathbb{E}^{\mathbb{Q}}[Z_t^N \mathbf{1}_A]$, which is the statement

The conditional Bayes formula follows. For every $\mathcal{F}_T$-measurable $X$ with $\mathbb{E}^{\mathbb{Q}^N}[\lvert X \rvert] < \infty$,

Each hypothesis on $N$ has a role here. A positive local martingale is a supermartingale, so if $N/B$ were only a local martingale, the normalized terminal value could have $\mathbb{E}^{\mathbb{Q}}[Z_T^N] < 1$ and $\mathbb{Q}^N$ would have total mass below one. If $N$ could reach zero, then $Z_T^N$ would vanish on an event of positive $\mathbb{Q}$-probability, $\mathbb{Q}^N$ would only be absolutely continuous with respect to $\mathbb{Q}$, not equivalent to it, and a payoff divided by $N_T$ would be undefined there.

Definition Change of numeraire theorem. Let $\mathbb{Q}$ be a pricing measure for $B$, let $N$ be a numeraire on $[0, T]$ with density process $Z^N$ and measure $\mathbb{Q}^N$, and let $H$ be $\mathcal{F}_T$-measurable with $\mathbb{E}^{\mathbb{Q}}[\lvert H \rvert / B_T] < \infty$. Then $\mathbb{E}^{\mathbb{Q}^N}[\lvert H \rvert / N_T] = (B_0/N_0)\,\mathbb{E}^{\mathbb{Q}}[\lvert H \rvert / B_T] < \infty$, and for every $t \in [0, T]$

The proof is the conditional Bayes formula with $X = H/N_T$. The terminal density contains $N_T$, which cancels the $N_T$ in the payoff, so $Z_T^N H / N_T = (B_0/N_0)\, H/B_T$. The current density is $Z_t^N = (B_0/N_0)\, N_t/B_t$. Hence

which is $V_t$. Every cancellation pairs factors with the same date, a terminal $N_T$ with a terminal $N_T$ and a current $N_t$ with a current $N_t$. The theorem also shows that $V/N$ is a $\mathbb{Q}^N$-martingale, and the same argument makes $S/N$ a $\mathbb{Q}^N$-martingale for every traded price $S$ without payouts. The unit and the measure change together, so mixing them, for example averaging in units of $N$ under $\mathbb{Q}$, gives wrong prices in general.

Definition Assume $\mathbb{E}^{\mathbb{Q}}[1/B_T] < \infty$, and take $B_0 = 1$. The default-free zero-coupon bond that pays $1$ at $T$ has the time-$t$ price

for $0 \le t \le T$. It is strictly positive because $1/B_T > 0$, it satisfies $P(T,T) = 1$, and $P(t,T)/B_t$ is the conditional expectation of the integrable variable $1/B_T$, so it is a true $\mathbb{Q}$-martingale on $[0, T]$. The bond is therefore a numeraire on $[0, T]$ without any further check, and its density process is

The bond matures at $T$, so it serves as a numeraire on $[0, T]$ only. When the rates are deterministic, $B_T$ is a number, $P(0,T) = 1/B_T$ and $Z^{T} \equiv 1$, and the bond price process is the discount factor $D(t,T) = B_t/B_T$. With stochastic rates $P(t,T)$ is a random process, and a deterministic discount function cannot stand in for it.

Definition The $T$-forward measure $\mathbb{Q}^{T}$ is the measure of the bond numeraire, defined on $\mathcal{F}_T$ by

with $B_0 = 1$. Since $P(T,T) = 1$, numeraire invariance gives, for every $\mathcal{F}_T$-measurable $H$ with $\mathbb{E}^{\mathbb{Q}}[\lvert H \rvert / B_T] < \infty$ and every $t \in [0, T]$,

The random discount is absorbed into the measure, not dropped. If moreover $\mathbb{E}^{\mathbb{Q}}[\lvert H \rvert] < \infty$, the two expectations at time $0$ are linked by

because $\mathbb{E}^{\mathbb{Q}}[H/B_T] = P(0,T)\,\mathbb{E}^{\mathbb{Q}}[H] + \operatorname{Cov}^{\mathbb{Q}}(H, 1/B_T)$. So the formula $P(0,T)\,\mathbb{E}^{\mathbb{Q}}[H]$ is right only when $H$ and $1/B_T$ are uncorrelated under $\mathbb{Q}$, for example when rates are deterministic. For a traded asset $S$ without payouts on $[0, T]$, the ratio $S_t/P(t,T)$ is a $\mathbb{Q}^{T}$-martingale that ends at $S_T$, so the forward price for delivery at $T$ is $S_t/P(t,T) = \mathbb{E}^{\mathbb{Q}^{T}}[S_T \mid \mathcal{F}_t]$. Each maturity has its own forward measure, and two forward measures differ in general.

Definition Let $W^{\mathbb{Q}}$ be a $d$-dimensional $\mathbb{Q}$-Brownian motion for $(\mathcal{F}_t)$, and let the bond have the $\mathbb{Q}$-dynamics

on $[0, T]$, where the bond loading $\sigma_P(\cdot, T)$ is predictable with values in $\mathbb{R}^d$ and $\int_0^T \lvert \sigma_P(s,T) \rvert^2 \, ds < \infty$ almost surely. The drift is $r_t$ because $P/B$ is a $\mathbb{Q}$-martingale. Then $dZ^{T}_t = Z^{T}_t \, \sigma_P(t,T)^{\top} dW^{\mathbb{Q}}_t$, so the density is the stochastic exponential of $\int \sigma_P^{\top} dW^{\mathbb{Q}}$, and it is a true martingale because it equals $P(t,T)/(B_t P(0,T))$. Girsanov’s theorem, in the form for a density $\mathcal{E}(\int \theta^{\top} dW^{\mathbb{Q}})$ with $\theta = \sigma_P(\cdot, T)$, makes

a $\mathbb{Q}^{T}$-Brownian motion on $[0, T]$. So $dW^{\mathbb{Q}}_t = dW^{T}_t + \sigma_P(t,T)\,dt$, and an Itô process with $dX_t = \mu_t \, dt + \sigma_{X,t}^{\top} dW^{\mathbb{Q}}_t$ has under $\mathbb{Q}^{T}$ the dynamics

The volatility is unchanged and the drift moves by the instantaneous covariation of $X$ with the log bond price. In the Heath-Jarrow-Morton framework the bond loading is minus the integrated forward-rate volatility, so the adjustment there carries a minus sign. Working out the loading and the density convention first is safer than remembering a drift sign.