Let’s quickly change the numéraire

1 The numéraire change

In finance a common technique of simplifying computations (especially those involving expectation(s)) is a so-called change of numéraire. Wise choice of numéraire can lead to a dramatically simpler formulas, in particular in pricing.

To compute prices of claims using the risk neutral valuation formula, asset prices divided by numéraire asset \(N\) must be martingales wrt the measure associated with this numéraire. That said, if \(A\) is any (non-dividend) asset price, then \(A/N\) must be a driftless process under the measure \(\mathbb {Q}_N\). In many applications, switching to a measure \(\mathbb {Q}_U\) associated with different numéraire \(U\) leads to much simpler formulas, involving expectation \(\mathbb {E}^ \bullet \).

To illustrate the complexity on a numéraire change, if a change from measure \(\mathbb {Q}_N\) to \(\mathbb {Q}_U\) is to be done, the following steps need to be taken:

• • start by being given the dynamics of all processes under \(\mathbb {Q}_N\),

• • find the Radon-Nikodym derivative to change measure from \(\mathbb {Q}_N\) to \(\mathbb {Q}_U\),

• • compute the dynamics of the Radon-Nikodym derivative (process) using Ito’s lemma and obtain the transformation kernel \(\varphi \) to switch from \(\mathbb {Q}_N\) to \(\mathbb {Q}_U\),

• • take the transformation kernel \(\varphi \) and use the result from the Girsanov theorem which states \(d{W^{{\mathbb {Q}_N}}}(t) = \varphi (t)dt + d{W^{{\mathbb {Q}_U}}}(t)\),

• • substitute the previous result for \(d{W^{{\mathbb {Q}_N}}}(t)\) in the original equations of dynamics to obtain the dynamics under the new numéraire measure \(\mathbb {Q}_U\).

Eventually, the change of numéraire leads to changes in drifts of processes, i.e. when a process had a drift \(\mu _N\) under the numéraire \(N\), it gets a new drift \(\mu _U\) under the measure \(\mathbb {Q}_U\).

The 5 steps above can be time-consuming, especially the 3rd step. A natural question is: can we fast-forward and get the new drift \(\mu _U\) directly without having to go through all the 5 steps above?

The answer is yes. There is a (somewhat underrated) shortcut in (Brigo and Mercurio, 2006, Section 2.3) which we rephrase here.

To able to apply this shortcut, let’s first define a general setup. Consider a system of SDEs entirely driven by a \(d\)–dimensional Wiener process; in which:

• • there is a vector-valued process \(X\) which has the following dynamics under \(\mathbb {Q}_N\)

  \(\seteqnumber{0}{1.}{0}\)

  \begin{equation*} dX(t) = {\mu _N}(t,X(t))dt + \sigma (t,X(t))Cd{W^{{\mathbb {Q}_N}}}(t). \end{equation*}

  Here, \(X\) is \(d\)–dimensional, \(\mu _N:[0,T]\times \mathbb {R}^d \rightarrow \mathbb {R}^d\), \(\sigma : [0,T]\times \mathbb {R}^d \rightarrow \mathbb {R}^d \times \mathbb {R}^d\) is diagonal volatility matrix and \(C\) is \(\mathbb {R}^d \times \mathbb {R}^d\) Cholesky matrix.

• • scalar process of the current numéraire \(N\):

  \(\seteqnumber{0}{1.}{0}\)

  \begin{equation*} dN(t) = [...]dt + {\sigma _N}(t,N(t))Cd{W^{{\mathbb {Q}_N}}}(t), \end{equation*}

  where \(\sigma _N:[0,T] \times \mathbb {R} \rightarrow \mathbb {R} \times \mathbb {R}^d\) is a row vector,

• • scalar process of the new numéraire \(U\):

  \(\seteqnumber{0}{1.}{0}\)

  \begin{equation*} dU(t) = [...]dt + {\sigma _U}(t,U(t))Cd{W^{{\mathbb {Q}_N}}}(t), \end{equation*}

  where \(\sigma _U:[0,T] \times \mathbb {R} \rightarrow \mathbb {R} \times \mathbb {R}^d\) is a row vector.

It is customary to make \(N\) and \(U\) components of the vector-valued \(X\) process (which we also use below in the example).

Given this setup, the switch of the numéraire from \(N\) to \(U\), is direct and it implies \(X\) under the new measure \(\mathbb {Q}^U\) has the new drift \(\mu _U\) defined as

\begin{equation} \label {eq:new_drift} {\mu _U}(t,X(t)) = {\mu _N}(t,X(t)) - \sigma (t,X(t))\rho {\left ( {\frac {{{\sigma _N}(t,N(t))}}{{N(t)}} - \frac {{{\sigma _U}(t,U(t))}}{{U(t)}}} \right )^{\rm {T}}}, \end{equation}

where \(\rho = CC^\text {T}\).

We don’t provide a proof, for which the reader is referred to the original source. We, however, provide an example instead.

2 Example

Consider the following problem: We know the dynamics of the short rate \(r\) in Vasicek model under the spot risk neutral measure \(\mathbb {Q}^N\) (money market account \(N\) used as numéraire) and we would like to change this to a \(T\)–forward measure \(\mathbb {Q}^U\), associated with \(T\)–bond \(U(\cdot , T)\). The motivation for this change, typically, might be the simplification of bond option formula.

In the initial setup under \(\mathbb {Q}^N\) – when \(N\) is the numéraire, we observe the following dynamics

\begin{eqnarray*} dr(t) &=& (b - ar(t))dt + \sigma d{W^{{\mathbb {Q}_N}}}(t)\\ dN(t) &=& rN(t)dt\\ dU(t,T) &=& [...]dt - \sigma b(t,T)U(t,T)d{W^{{\mathbb {Q}_N}}}(t), \end{eqnarray*}

where all elements in these SDEs are scalar, including the Wiener (note that both the short rate \(r\) and the bond \(U(\cdot , T)\) are driven by the same (scalar) Wiener).

To use our vector \(X\)–process notation, let us set

\begin{eqnarray*} {X_1} &\equiv & r\\ {X_2} &\equiv & N\\ {X_3} &\equiv & U. \end{eqnarray*}

Given that we only have one Wiener process involved, and we need to represent this by a system of three Wiener processes, we set up the Cholesky matrix in a way that all Wiener processes are perfectly correlated. So we set

\begin{equation*} C = \left [ {\begin{array}{*{20}{c}} 1&0&0\\ 1&0&0\\ 1&0&0 \end {array}} \right ], \end{equation*}

and

\begin{equation*} d{W^{{\mathbb {Q}_N}}}(t) = {[dW_1^{{\mathbb {Q}_N}}(t),dW_2^{{\mathbb {Q}_N}}(t),dW_3^{{\mathbb {Q}_N}}(t)]^{\rm {T}}}, \end{equation*}

where the individual Wiener processes are mutually independent.

The dynamics of \(X\) can be written as

\begin{equation*} \left [ {\begin{array}{*{20}{c}} {d{X_1}(t)}\\ {d{X_2}(t)}\\ {d{X_3}(t)} \end {array}} \right ] = \left [ {\begin{array}{*{20}{c}} {b - a{X_1}(t)}\\ {r{X_2}(t)}\\ {[...]} \end {array}} \right ]dt + \left [ {\begin{array}{*{20}{c}} \sigma &0&0\\ 0&0&0\\ 0&0&{ - \sigma b(t,T){X_3}(t)} \end {array}} \right ]\left [ {\begin{array}{*{20}{c}} 1&0&0\\ 1&0&0\\ 1&0&0 \end {array}} \right ]\left [ {\begin{array}{*{20}{c}} {dW_1^{{\mathbb {Q}_N}}(t)}\\ {dW_2^{{\mathbb {Q}_N}}(t)}\\ {dW_3^{{\mathbb {Q}_N}}(t)} \end {array}} \right ] \end{equation*}

Having the dynamics of \(X\) formulated, we can use the formula (1.1) for new drift (for dynamics under \(\mathbb {Q}^U\))

\begin{equation*} {\mu _U}(t,X(t)) = \left [ {\begin{array}{*{20}{c}} {b - a{X_1}(t)}\\ {r{X_2}(t)}\\ {[...]} \end {array}} \right ] - \left [ {\begin{array}{*{20}{c}} \sigma &0&0\\ 0&0&0\\ 0&0&{ - \sigma b(t,T){X_3}(t)} \end {array}} \right ]\left [ {\begin{array}{*{20}{c}} 1&1&1\\ 1&1&1\\ 1&1&1 \end {array}} \right ]{\left [ {\frac {{[0,0,0]}}{{{X_2}(t)}} - \frac {{[0,0, - \sigma b(t,T){X_3}(t)]}}{{{X_3}(t)}}} \right ]^{\rm {T}}}. \end{equation*}

After simplification the new drift thus reads

\begin{equation*} {\mu _U}(t,X(t)) = \left [ {\begin{array}{*{20}{c}} {b - a{X_1}(t) - {\sigma ^2}b(t,T)}\\ {r{X_2}(t)}\\ {[...] + {\sigma ^2}{b^2}(t,T){X_3}(t)} \end {array}} \right ] \end{equation*}

The dynamics under \(\mathbb {Q}^U\) then equals

\begin{eqnarray*} dr(t) &=& (b - a{X_1}(t) - {\sigma ^2}b(t,T))dt + \sigma d{W^{{\mathbb {Q}_U}}}(t)\\ dN(t) &=& rN(t)dt\\ dU(t,T) &=& ([...] + {\sigma ^2}{b^2}(t,T)U(t,T))dt - \sigma b(t,T)U(t,T)d{W^{{\mathbb {Q}_U}}}(t). \end{eqnarray*}

References

• Brigo, Damiano and Fabio Mercurio (2006). Interest Rate Models - Theory and Practice: With Smile, Inflation and Credit. 2nd. 1037 p. Heidelberg, Germany: Springer. isbn: 978-3540221494.