Abstract

In these notes we introduce numeraires and theorems related to change of numeraires along with their applications.

1 Change of Measure

Definition 1 (Radon-Nikodym Derivative).

Consider two equivalent probability measures $\mathbb{P}$ and $\widehat{\mathbb{P}}$ on a measurable space $(\mathrm{\Omega },\mathrm{\Sigma })$. The Radon-Nikodym derivate $d\widehat{\mathbb{P}}/d\mathbb{P}:\mathrm{\Omega }\to ℝ$ is defined such that for any subset $A,\mathrm{\Omega }⫆A\in \mathrm{\Sigma }$,

$${\int }_A𝑑\widehat{\mathbb{P}}={\int }_A(d\widehat{\mathbb{P}}/d\mathbb{P})𝑑\mathbb{P}.$$ (1)

Suppose $(\mathrm{\Omega },ℱ,{ℱ}_t)$ is a filtered probability space, then note from the above definition we have

$${(d\widehat{\mathbb{P}}/d\mathbb{P})|}_{{ℱ}_t}={𝔼}^{\mathbb{P}}[(d\widehat{\mathbb{P}}/d\mathbb{P})|{ℱ}_t]$$ (2)

${(d\widehat{\mathbb{P}}/d\mathbb{P})|}_{{ℱ}_t}$ is also written as ${(d\widehat{\mathbb{P}}/d\mathbb{P})}_t$ and is thus a martingale stochastic process (by iterated conditioning) in $\mathbb{P}$.

Example 1 Let’s consider a simple example to better understand Radom Nikodym derivative. Consider a die roll. We can assign mulitple probability distributions to the outcomes.

$\omega$	$\mathbb{P}$	$\widehat{\mathbb{P}}$	$d\widehat{\mathbb{P}}/d\mathbb{P}$
1	1/6	1/2	3
2	1/6	1/4	3/2
3	1/6	1/8	3/4
4	1/6	1/16	3/8
5	1/6	1/32	3/16
6	1/6	1/32	3/16

The probability in $\widehat{\mathbb{P}}$ of getting an odd number is ${\int }_{\omega \in \{1,3,5\}}𝑑\widehat{\mathbb{P}}=1/2+1/8+1/32=21/32=3\ast 1/6+3/4\ast 1/6+3/16\ast 1/6={\int }_{\omega \in \{1,3,5\}}(d\widehat{\mathbb{P}}/d\mathbb{P})𝑑\mathbb{P}$

Theorem 2 (Abstract Bayes’ Theorem).

Let $\mathbb{P}$ and $\widehat{\mathbb{P}}$ be two measures on measurable space $(\mathrm{\Omega },ℱ)$. Let $𝒢\subset ℱ$ be another sigma algebra on $\mathrm{\Omega }$. Then for any $A\in G$ and random variable X

$${𝔼}^{\widehat{\mathbb{P}}}[X|G]=\frac{{𝔼}^{\mathbb{P}}[(d\widehat{\mathbb{P}}/d\mathbb{P})X|G]}{{𝔼}^{\mathbb{P}}[(d\widehat{\mathbb{P}}/d\mathbb{P})|G]}.$$ (3)

Proof.

We show that for any $A\in G$

$${𝔼}^{\widehat{\mathbb{P}}}[X|G]{𝔼}^{\mathbb{P}}[(d\widehat{\mathbb{P}}/d\mathbb{P})|G]={𝔼}^{\mathbb{P}}[(d\widehat{\mathbb{P}}/d\mathbb{P})X|G]$$

Since the random variables involved are constant over A, we can check equality on integrals over A.

${\displaystyle {\int }_A}{𝔼}^{\widehat{\mathbb{P}}}[X|G]{𝔼}^{\mathbb{P}}[(d\widehat{\mathbb{P}}/d\mathbb{P})|G]𝑑\mathbb{P}$ $={\displaystyle {\int }_A}{𝔼}^{\mathbb{P}}[(d\widehat{\mathbb{P}}/d\mathbb{P}){𝔼}^{\widehat{\mathbb{P}}}[X|G]|G]𝑑\mathbb{P}$ $({𝔼}^{\widehat{\mathbb{P}}}[X|G]\text{ is G measurable})$

$={\displaystyle {\int }_A}(d\widehat{\mathbb{P}}/d\mathbb{P}){𝔼}^{\widehat{\mathbb{P}}}[X|G]𝑑\mathbb{P}$ $(\text{definition of conditional expectation})$

$={\displaystyle {\int }_A}{𝔼}^{\widehat{\mathbb{P}}}[X|G]𝑑\widehat{\mathbb{P}}$ $(\text{definition of Radon Nikodym derivative})$

$={\displaystyle {\int }_A}X𝑑\widehat{\mathbb{P}}$ $(\text{definition of conditional expectation})$

$={\displaystyle {\int }_A}(d\widehat{\mathbb{P}}/d\mathbb{P})X𝑑\mathbb{P}$ $(\text{definition of Radon Nikodym derivative})$

$={\displaystyle {\int }_A}{𝔼}^{\mathbb{P}}[(d\widehat{\mathbb{P}}/d\mathbb{P})X|G]𝑑\mathbb{P}$ $(\text{definition of conditional expectation}).$

∎

Remark.

Taking $X=V_T$ where $V_s$ is ${ℱ}_s$ adapted and $G={ℱ}_t$ for in the above theorem, we get the very useful formula for measure change for conditional expectations on filtered spaces,

$${𝔼}^{\widehat{\mathbb{P}}}[V_T|{ℱ}_t]={𝔼}^{\mathbb{P}}[\frac{{(d\widehat{\mathbb{P}}/d\mathbb{P})}_T}{{(d\widehat{\mathbb{P}}/d\mathbb{P})}_t}{V}_T|{ℱ}_t]$$ (4)

Proof.

${𝔼}^{\widehat{\mathbb{P}}}[V_T|{ℱ}_t]$ $={\displaystyle \frac{{𝔼}^{\mathbb{P}}[(d\widehat{\mathbb{P}}/d\mathbb{P})V_T|{ℱ}_t]}{{𝔼}^{\mathbb{P}}[(d\widehat{\mathbb{P}}/d\mathbb{P})|{ℱ}_t]}}$ $(\text{abstract Bayes’ theorem})$

$={\displaystyle \frac{{𝔼}^{\mathbb{P}}[{𝔼}^{\mathbb{P}}[(d\widehat{\mathbb{P}}/d\mathbb{P})V_T|{ℱ}_T]|{ℱ}_t]}{{𝔼}^{\mathbb{P}}[(d\widehat{\mathbb{P}}/d\mathbb{P})|{ℱ}_t]}}$ $(\text{iterated conditioning})$

$={\displaystyle \frac{{𝔼}^{\mathbb{P}}[{𝔼}^{\mathbb{P}}[(d\widehat{\mathbb{P}}/d\mathbb{P})|{ℱ}_T]V_T|{ℱ}_t]}{{𝔼}^{\mathbb{P}}[(d\widehat{\mathbb{P}}/d\mathbb{P})|{ℱ}_t]}}$ $(V_T\text{ is }{ℱ}_T\text{ measureable})$

$={\displaystyle \frac{{𝔼}^{\mathbb{P}}[{(d\widehat{\mathbb{P}}/d\mathbb{P})}_T{V}_T|{ℱ}_t]}{{(d\widehat{\mathbb{P}}/d\mathbb{P})}_t}}$ $(\text{martigale property of Radon Nikodym derivative})$

$={𝔼}^{\mathbb{P}}[{\displaystyle \frac{{(d\widehat{\mathbb{P}}/d\mathbb{P})}_T}{{(d\widehat{\mathbb{P}}/d\mathbb{P})}_t}}{V}_T|{ℱ}_t]$ $({(d\widehat{\mathbb{P}}/d\mathbb{P})}_t\text{ is }{ℱ}_t\text{ measurable}).$

∎

We consider processes until terminal time S i.e. $ℱ={ℱ}_S$ and $(d\widehat{\mathbb{P}}/\mathbb{P})={(d\widehat{\mathbb{P}}/\mathbb{P})}_S$. We take a strictly positive martingale process to be the Random Nikodym derivative:

$$d{f}_t=f_t\sigma (t)d{W}_t;f_t=e^{-{\int }_0^t\frac{1}{2}{\sigma }^2(s)𝑑s+{\int }_0^t\sigma (s)𝑑W_s}.$$ (5)

Theorem 3 (Girsanov Theorem).

If $W_{1,t}$ is a Brownian motion in $\mathbb{P}$ and the Radon-Nikodym derivative ${(d\widehat{\mathbb{P}}/d\mathbb{P})}_t=f_t$ is given by $f_t=e^{-{\int }_0^t\frac{1}{2}{\sigma }^2(s)𝑑s+{\int }_0^t\sigma (s)𝑑W_{2,s}^{\mathbb{P}}}$ then $W_{1,t}-{\int }_0^t\sigma (s)𝑑W_{1,s}𝑑W_{2,s}$ is a Brownian motion in $\widehat{\mathbb{P}}$.

Proof.

We show that $X_t=W_{1,t}-{\int }_0^t\sigma (s)𝑑W_{1,s}𝑑W_{2,s}$ follows normal distribution with mean 0 and variance t in $\widehat{\mathbb{P}}$. Other required properties can be verified easily. We show that the moment generating function of $X_t$ is same as that of normal distribution with mean 0 and variance t.

${𝔼}^{\widehat{\mathbb{P}}}[e^{-y{X}_t}]$ $={𝔼}^{\mathbb{P}}[{(d\widehat{\mathbb{P}}/d\mathbb{P})}_t{e}^{-y{X}_t}]$

$={𝔼}^{\mathbb{P}}[e^{-{\int }_0^t\frac{1}{2}{\sigma }^2(s)𝑑s+{\int }_0^t\sigma (s)𝑑W_{2,s}}{e}^{-y{X}_t}]$

$=e^{-{\int }_0^t\frac{1}{2}{\sigma }^2(s)𝑑s}{𝔼}^{\mathbb{P}}[e^{{\int }_0^t\sigma (s)𝑑W_{2,s}-y{X}_t}]$

$=e^{-{\int }_0^t\frac{1}{2}{\sigma }^2(s)𝑑s}{𝔼}^{\mathbb{P}}[e^{{\int }_0^t\sigma (s)𝑑W_{2,s}-y(W_{1,t}-{\int }_0^t\sigma (s)𝑑W_{1,s}𝑑W_{2,s})}]$

$=e^{-{\int }_0^t\frac{1}{2}{\sigma }^2(s)𝑑s}{𝔼}^{\mathbb{P}}[e^{{\int }_0^t(\sigma (s)d{W}_{2,s}-yd{W}_{1,s}+y\sigma (s)d{W}_{1,s}d{W}_{2,s})}].$

Define the martingale $Z_{y,t}={\int }_0^t(\sigma (s)d{W}_{2,s}-yd{W}_{1,s})$. Then $e^{Z_{y,t}-\frac{1}{2}{\int }_0^t𝑑Z_{y,s}𝑑Z_{y,s}}$ is a martingale as well, with

$$d{Z}_{y,s}d{Z}_{y,s}=({\sigma }^2(s)+y^2)ds-2y\sigma d{W}_{1,s}d{W}_{2,s}.$$

We then have

${𝔼}^{\widehat{\mathbb{P}}}[e^{-y{X}_t}]$ $=e^{-{\int }_0^t\frac{1}{2}{\sigma }^2(s)𝑑s}{𝔼}^{\mathbb{P}}[e^{Z_{y,t}-\frac{1}{2}{\int }_0^t𝑑Z_{y,s}𝑑Z_{y,s}+\frac{1}{2}{\int }_0^t({\sigma }^2(s)+y^2)𝑑s}]$

$=e^{\frac{1}{2}{y}^{2}t}{𝔼}^{\mathbb{P}}[e^{Z_{y,t}-\frac{1}{2}{\int }_0^t𝑑Z_{y,s}𝑑Z_{y,s}}]$

$={e^{\frac{1}{2}{y}^{2}t}(e^{Z_{y,t}-\frac{1}{2}{\int }_0^t𝑑Z_{y,s}𝑑Z_{y,s}})|}_{t=0}$

$=e^{\frac{1}{2}{y}^{2}t}.$

∎

2 Numeraires

Definition 4 (Numeraire).

A Numeraire is a strictly positive price process of a tradable relative to which prices of all other tradables are expressed.

A savings account which earns interest at the instantaneous interest rate can be taken as a numeraire. The value of the savings account at any point is given by

$$A_t=e^{{\int }_0^{t}r(t)𝑑t}=1/D(t)$$ (6)

where $D(t)$ is the discount factor.

The fact that discounted trade prices are martingales in risk-neutral measure can then also be stated as: tradable prices in savings account (or money market) numeraire are martingales.

Consider another numeraire $N_t$. Since the numeraire is itself a price process, $D_t{N}_t$ is a martingale and we can take it as a Radon-Nikodym derivative, with a suitable normalization so that ${𝔼}^{\mathbb{P}}[(d\mathbb{N}/d\mathbb{P})]=1$, giving

$${𝔼}^{\mathbb{N}}[\frac{X_T}{N_T}|{ℱ}_t]={𝔼}^{\mathbb{P}}[\frac{{(d\mathbb{N}/d\mathbb{P})}_T}{{(d\mathbb{N}/d\mathbb{P})}_t}\frac{X_T}{N_T}|{ℱ}_t]={𝔼}^{\mathbb{P}}[\frac{D_T{N}_T}{D_t{N}_t}\frac{X_T}{N_T}|{ℱ}_t]=\frac{1}{D_t{N}_t}{𝔼}^{\mathbb{P}}[D_T{X}_T|{ℱ}_t]=\frac{X_t}{N_t}$$ (7)

where $\mathbb{P}$ is the risk neutral measure and $\mathbb{N}$ is the measure corresponding to the numeraire $N_t$.

Remark.

The above equation shows that tradable prices with respect to a numeraire are martingales in the measure corresponding to the numeraire.

Risk neutral measure corresponds to the savings account (or money market) numeraire.

Definition 5 (T-forward measure).

If we take $N_t$ to be the price of a riskless bond maturing at time T, the corresponding measure as the T-forward measure.

$$X_t/B(t,T)={𝔼}^T[X_T/B(T,T)]={𝔼}^T[X_T]$$ (8)

We see that that expectation of $X_T$ in the T-forward measure gives the forward price $X_t/B(t,T)$ of the trade with expiration date T, hence the name T-forward measure. From the above equation we also notice that expiration T forward price process on an asset is martingale in T-forward measure.

Example 2 Option Pricing To see how measure changes can be used in pricing, let’s take the example of Black Scholes option pricing. For an option on stock with expiry at T and strike K, we have the valuation forumula

$$V_t=\frac{1}{D_t}{𝔼}^{\mathbb{P}}[D_T\max (S_T-K,0)|{ℱ}_t]$$

where $\mathbb{P}$ is the risk neutral probability measure, $D_t$ is the discount factor and $S_t$ is the stock price which follows the Black Scholes diffusion

$$\frac{d{S}_t}{S_t}=rdt+\sigma d{W}_t$$

where $r$ is the riskfree interest rate and $W_t$ is a Brownian motion in $\mathbb{P}$.

Denote by ${𝕀}_{S_T>K}$ the indicator variable which takes value 1 is $S_T>K$ and 0 otherwise. We then have

$V_t$ $={\displaystyle \frac{1}{D_t}}{𝔼}^{\mathbb{P}}[D_T{𝕀}_{S_T>K}(S_T-K)|{ℱ}_t]$

$={\displaystyle \frac{1}{D_t}}{𝔼}^{\mathbb{P}}[D_T{𝕀}_{S_T>K}{S}_T|{ℱ}_t]-{\displaystyle \frac{1}{D_t}}{𝔼}^{\mathbb{P}}[D_T{𝕀}_{S_T>K}K|{ℱ}_t]$

Taking $𝕊$ to be the measure corresponding to stock price as numeraire, and taking $𝕋$ to be the measure corresponding to T-expiry bond as numeraire, we have

$V_t$ $={\displaystyle \frac{1}{D_t}}{𝔼}^{𝕊}[D_T{\displaystyle \frac{{(d\mathbb{P}/d𝕊)}_T}{{(d\mathbb{P}/d𝕊)}_t}}{𝕀}_{S_T>K}{S}_T|{ℱ}_t]-{\displaystyle \frac{1}{D_t}}{𝔼}^{𝕋}[D_T{\displaystyle \frac{{(d\mathbb{P}/d𝕋)}_T}{{(d\mathbb{P}/d𝕋)}_t}}{𝕀}_{S_T>K}K|{ℱ}_t]$

$={\displaystyle \frac{1}{D_t}}{𝔼}^{𝕊}[D_T{\displaystyle \frac{{(d𝕊/d\mathbb{P})}_t}{{(d𝕊/d\mathbb{P})}_T}}{𝕀}_{S_T>K}{S}_T|{ℱ}_t]-{\displaystyle \frac{1}{D_t}}{𝔼}^{𝕋}[D_T{\displaystyle \frac{{(d𝕋/d\mathbb{P})}_t}{{(d𝕋/d\mathbb{P})}_T}}{𝕀}_{S_T>K}K|{ℱ}_t]$

$={\displaystyle \frac{1}{D_t}}{𝔼}^{𝕊}[D_T{\displaystyle \frac{D_t{S}_t}{D_T{S}_T}}{𝕀}_{S_T>K}{S}_T|{ℱ}_t]-{\displaystyle \frac{1}{D_t}}{𝔼}^{𝕋}[D_T{\displaystyle \frac{D_t(B(t,T))}{D_{T}B(T,T)}}{𝕀}_{S_T>K}K|{ℱ}_t]$

$=S_t{𝔼}^{𝕊}[{𝕀}_{S_T>K}|{ℱ}_t]-KB(t,T){𝔼}^{𝕋}[{𝕀}_{S_T>K}|{ℱ}_t]$

where $B(t,T)$ is the bond price with the diffusion

$$dB(t,T)/B(t,T)=rdt+{\sigma }_{2}d{W}_{2,t}$$

where $W_{2,t}$ is another Brownian motion in $\mathbb{P}$ such that $d{W}_{t}d{W}_{2,t}=\rho dt$.

Using Girsaonov’s theorem, we have

$$\frac{d{S}_t}{S_t}=rdt+\sigma (d{W}_t^{𝕊}+\sigma dt)=rdt+\sigma (d{W}_t^{𝕋}+{\sigma }_2\rho dt)$$ (9)

where $d{W}_t^{𝕊}=d{W}_t-\sigma dt$ is a Brownian motion in $𝕊$ and $d{W}_t^{𝕋}=d{W}_t^{𝕊}-{\sigma }_2\rho dt$ is a Brownian motion in $𝕋$. Rearranging,

$$\frac{d{S}_t}{S_t}=(r+{\sigma }^2)dt+\sigma d{W}_t^{𝕊}=(r+\sigma {\sigma }_2\rho )dt+\sigma d{W}_t^{𝕋}$$ (10)

$$S_T=S_t{e}^{(r+\frac{1}{2}{\sigma }^2)\tau +\sigma W_{\tau }^{𝕊}}=S_t{e}^{(r+\sigma ({\sigma }_2\rho -\frac{1}{2}\sigma )\tau +\sigma W_{\tau }^{𝕋}}$$ (11)

where $\tau =T-t$.

${𝔼}^{𝕊}[{𝕀}_{S_T>K}|{ℱ}_t]$ $={𝔼}^{𝕊}[{𝕀}_{S_t{e}^{(r+\frac{1}{2}{\sigma }^2)\tau +\sigma W_{\tau }^{𝕊}}>K}|{ℱ}_t]$

$={𝔼}^{𝕊}[{𝕀}_{(r+\frac{1}{2}{\sigma }^2)\tau +\sigma W_{\tau }^{𝕊}>\log (K/S_t)}|{ℱ}_t]$

$={𝔼}^{𝕊}[{𝕀}_{\sigma W_{\tau }^{𝕊}>\log (K/S_t)-(r+\frac{1}{2}{\sigma }^2)\tau }|{ℱ}_t]$

$={𝔼}^{𝕊}[{𝕀}_{\frac{1}{\sqrt{\tau }}{W}_{\tau }^{𝕊}>\frac{1}{\sigma \sqrt{\tau }}(\log (K/S_t)-(r+\frac{1}{2}{\sigma }^2)\tau )})|{ℱ}_t]$

$=N(-{\displaystyle \frac{1}{\sigma \sqrt{\tau }}}(\log (K/S_t)-(r+{\displaystyle \frac{1}{2}}{\sigma }^2)\tau ))$

$=N(-d_1+\sigma \sqrt{\tau })$

where $N$ is the cumulative normal distribution, and $d_1=\frac{1}{\sigma \sqrt{\tau }}(\log (K/S_t)-(r-\frac{1}{2}{\sigma }^2)\tau )$.

Similarly we have,

${𝔼}^{𝕋}[{𝕀}_{S_T>K}|{ℱ}_t]$ $={𝔼}^{𝕋}[{𝕀}_{S_t{e}^{(r+\sigma ({\sigma }_2\rho -\frac{1}{2}\sigma )\tau +\sigma W_{\tau }^{𝕊}}>K}|{ℱ}_t]$

$={𝔼}^{𝕋}[{𝕀}_{(r+\sigma ({\sigma }_2\rho -\frac{1}{2}\sigma )\tau +\sigma W_{\tau }^{𝕊}>\log (K/S_t)}|{ℱ}_t]$

$={𝔼}^{𝕋}[{𝕀}_{\sigma W_{\tau }^{𝕊}>\log (K/S_t)-(r+\sigma ({\sigma }_2\rho -\frac{1}{2}\sigma )\tau }|{ℱ}_t]$

$={𝔼}^{𝕋}[{𝕀}_{\frac{1}{\sqrt{\tau }}{W}_{\tau }^{𝕊}>\frac{1}{\sigma \sqrt{\tau }}(\log (K/S_t)-(r+\sigma ({\sigma }_2\rho -\frac{1}{2}\sigma )\tau )}|{ℱ}_t]$

$=N(-{\displaystyle \frac{1}{\sigma \sqrt{\tau }}}(\log (K/S_t)-(r+\sigma ({\sigma }_2\rho -{\displaystyle \frac{1}{2}}\sigma )\tau ))$

$=N(-d_1+{\sigma }_2\rho \sqrt{\tau }))$

And finally we have,

$$V_t=S_{t}N(-d_1+\sigma \sqrt{\tau })-KB(t,T)N(-d_1+{\sigma }_2\rho \sqrt{\tau })$$

Note that $\sigma d\tau$ and ${\sigma }_2\rho d\tau$ are the drift correction terms we get to stock price Brownian motion using Girsanov’s theorem to change measure to stock and bond numeraires respectively.