Fun with numeraires!

Introduction

A proficiency in the change-of-measure technique is useful to the working quant. An excellent summary of the important results is the note Girsanov, Numeraires and all that, by Andrew Lesniewski. In this post, I would like to derive relevant results and then we can enjoy pricing some payoffs together!

Girsanov Theorem

Theorem 1 (Girsanov Theorem) Let $(W^{\mathbb{P}},t\ge 0)$ be a $\mathbb{P}$ standard brownian motion on $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$ and let $\varphi$ be any adapted process. Choose fixed $T$ and defined the process $L$ on $[0,T]$ by:

$$\begin{array}{}\text{(1)}& \begin{array}{r}d{L}_t={\varphi }_t{L}_{t}d{W}_t^{\mathbb{P}}\end{array}\end{array}$$

$$\begin{array}{}\text{(2)}& \begin{array}{r}{L}_0=1\end{array}\end{array}$$

that is:

$$\begin{array}{}\text{(3)}& \begin{array}{r}{L}_t=\exp ({\int }_0^t{\varphi }_{s}d{W}_s^{\mathbb{P}}-{\int }_0^t{\varphi }_s^{2}ds)\end{array}\end{array}$$

Assume that:

$$\begin{array}{}\text{(4)}& \begin{array}{r}{\mathbb{E}}^{\mathbb{P}}[L_T]=1\end{array}\end{array}$$

and define the new probability measure $\mathbb{Q}$ on ${\mathcal{F}}_t$ by:

$$\begin{array}{}\text{(5)}& L_T=\frac{d\mathbb{Q}}{d\mathbb{P}}\end{array}$$

Then,

$$\begin{array}{}\text{(6)}& d{W}_t^{\mathbb{P}}=d{W}_t^{\mathbb{Q}}+{\varphi }_{t}dt\end{array}$$

where $d{W}_t^{\mathbb{Q}}$ is a $\mathbb{Q}$-standard brownian motion.

Proof.

Our claim is that, under the $\mathbb{Q}$ measure, the increments $(W_t^{\mathbb{Q}}-W_s^{\mathbb{Q}})$ are normally distributed with mean $0$ and variance $(t-s)$. We start with the special case $s=0$. Using moment generating functions, it is enough to show that:

It is straightforward to derive Equation 3 using Ito’s lemma. Let $f(x)=\ln x$. Then, $f_x=\frac{1}{x}$, $f_{xx}=-\frac{1}{x^2}$.

$$\begin{array}{rl}d(\ln L_t)& =\frac{1}{L_t}d{L}_t-\frac{1}{2}\frac{1}{L_t^2}(d{L}_t{)}^2\\ & =\frac{1}{L_t}{\varphi }_t{L}_{t}d{W}_t^{\mathbb{P}}-\frac{1}{2L_t^2}{\varphi }_t^2{L}_t^{2}dt\\ & ={\varphi }_{t}d{W}_t^{\mathbb{P}}-\frac{1}{2}{\varphi }_t^{2}dt\\ L_t& =\exp ({\int }_0^t{\varphi }_{s}d{W}_s^{\mathbb{P}}-\frac{1}{2}{\int }_0^t{\varphi }_s^{2}ds)\end{array}$$

To prove our main result, we will now use the MGF of the increments. For $n\in \mathbb{N}$ and $(t_j,j\le n)$ a partition of $[0,T]$, with $t_n=T$, I will show that :

$$\begin{array}{}\text{(7)}& \begin{array}{r}{\mathbb{E}}^{\mathbb{Q}}[\exp (\sum _{j=0}^{n-1}{\lambda }_j(W_{t_{j+1}}^{\mathbb{Q}}-W_{t_j}^{\mathbb{Q}}))]=\exp [\sum _{j=0}^{n-1}{\lambda }_j^2(t_{j+1}-t_j)]\end{array}\end{array}$$

This proves that the increments are the ones of standard brownian motion.

Let $({\mathcal{F}}_{t_j},j\le n)$ be the filtrations of the Brownian motion at the time of the partition. The proof is by successively conditioning from $t_{n-1}$ to $t_1$. We have:

$$\begin{array}{rl}{\mathbb{E}}^{\mathbb{Q}}[\exp (\sum _{j=0}^{n-1}{\lambda }_j(W_{t_{j+1}}^{\mathbb{Q}}-W_{t_j}^{\mathbb{Q}}))]& ={\mathbb{E}}^{\mathbb{P}}\left[{\mathbb{E}}^{\mathbb{P}}[L_{t_n}\exp (\sum _{j=0}^{n-1}{\lambda }_j(W_{t_{j+1}}^{\mathbb{Q}}-W_{t_j}^{\mathbb{Q}}))|{\mathcal{F}}_{t_{n-1}}]\right]\\ & ={\mathbb{E}}^{\mathbb{P}}[\sum _{j=0}^{n-2}{\lambda }_j(W_{t_{j+1}}^{\mathbb{Q}}-W_{t_j}^{\mathbb{Q}}){\mathbb{E}}^{\mathbb{P}}[L_{t_n}\exp ({\lambda }_{n-1}(W_{t_n}^{\mathbb{Q}}-W_{t_{n-1}}^{\mathbb{Q}}))|{\mathcal{F}}_{t_{n-1}}]]\\ & ={\mathbb{E}}^{\mathbb{P}}[\sum _{j=0}^{n-2}{\lambda }_j(W_{t_{j+1}}^{\mathbb{Q}}-W_{t_j}^{\mathbb{Q}}){\mathbb{E}}^{\mathbb{P}}[\exp ({\int }_{t_{n-1}}^{t_n}{\varphi }_{s}d{W}_s^{\mathbb{Q}}+\frac{1}{2}{\int }_{t_{n-1}}^{t_n}{\varphi }_s^{2}ds)\exp \left({\lambda }_{n-1}{\int }_{t_{n-1}}^{t_n}d{W}_s^{\mathbb{Q}}\right)|{\mathcal{F}}_{t_{n-1}}]]\\ & ={\mathbb{E}}^{\mathbb{P}}[\sum _{j=0}^{n-2}{\lambda }_j(W_{t_{j+1}}^{\mathbb{Q}}-W_{t_j}^{\mathbb{Q}}){\mathbb{E}}^{\mathbb{P}}[\exp ({\int }_{t_{n-1}}^{t_n}({\varphi }_s+{\lambda }_{n-1})d{W}_s^{\mathbb{Q}}+\frac{1}{2}{\int }_{t_{n-1}}^{t_n}{\varphi }_s^{2}ds)|{\mathcal{F}}_{t_{n-1}}]]\\ & ={\mathbb{E}}^{\mathbb{P}}[\sum _{j=0}^{n-2}{\lambda }_j(W_{t_{j+1}}^{\mathbb{Q}}-W_{t_j}^{\mathbb{Q}}){\mathbb{E}}^{\mathbb{P}}[\exp ({\int }_{t_{n-1}}^{t_n}({\varphi }_s+{\lambda }_{n-1})(d{W}_s^{\mathbb{P}}-\theta ds)+\frac{1}{2}{\int }_{t_{n-1}}^{t_n}{\varphi }_s^{2}ds)|{\mathcal{F}}_{t_{n-1}}]]\\ & ={\mathbb{E}}^{\mathbb{P}}[\sum _{j=0}^{n-2}{\lambda }_j(W_{t_{j+1}}^{\mathbb{Q}}-W_{t_j}^{\mathbb{Q}})\exp (\frac{1}{2}{\int }_{t_{n-1}}^{t_n}{\varphi }_s^2+{\int }_{t_{n-1}}^{t_n}-{\varphi }_s({\varphi }_s+{\lambda }_{n-1})ds)\exp (\frac{1}{2}{\int }_{t_{n-1}}^{t_n}({\varphi }_s+{\lambda }_{n-1}{)}^{2}ds)]\\ & \{\int Xd{W}_s^{\mathbb{P}}\text{ is a }{\mathcal{N}}^{\mathbb{P}}(0,\int \mathbb{E}\left[X^2\right]ds)\text{gaussian random variable.}\}\\ & ={\mathbb{E}}^{\mathbb{P}}[\sum _{j=0}^{n-2}{\lambda }_j(W_{t_{j+1}}^{\mathbb{Q}}-W_{t_j}^{\mathbb{Q}})\exp \left({\int }_{t_{n-1}}^{t_n}(-\frac{1}{2}{\varphi }_s^2-{\lambda }_{n-1}{\varphi }_s+\frac{1}{2}{\varphi }_s^2+{\lambda }_{n-1}{\varphi }_s+{\lambda }_{n-1}^2)ds\right)]\\ & ={\mathbb{E}}^{\mathbb{P}}[\sum _{j=0}^{n-2}{\lambda }_j(W_{t_{j+1}}^{\mathbb{Q}}-W_{t_j}^{\mathbb{Q}})\exp \left({\lambda }_{n-1}{\int }_{t_{n-1}}^{t_n}ds\right)]\\ & =\exp ({\lambda }_{n-1}(t_n-t_{n-1})\cdot {\mathbb{E}}^{\mathbb{P}}[\sum _{j=0}^{n-2}{\lambda }_j(W_{t_{j+1}}^{\mathbb{Q}}-W_{t_j}^{\mathbb{Q}})]\end{array}$$

Here, I used the fact that $L_{t_{n-1}}$ is ${\mathcal{F}}_{t_{n-1}}$ measurable. I can now condition on ${\mathcal{F}}_{t_{n-2}}$ down to ${\mathcal{F}}_{t_1}$ and proceed as above to obtain the desired result.

The process ${\varphi }_t$ is called the Girsanov kernel.

What is a numeraire?

As Shreve puts it, a numeraire is the unit of account in which other assets are denominated. In practice, we tend to choose numeraires that simply the payoff expression.

Any strictly positive (non-dividend paying) price process can be chosen as a numeraire. A numeraire must be a tradable asset.

Consider a unit of stock worth $S_t$. It can be used as numeraire, because the price process $e^{-rt}{S}_t$ (assume a constant short rate) is a martingale under risk-neutral measure ${\mathbb{Q}}^M$. Powers of the stock price $S_t^{\alpha }$ cannot be used as numeraires, because their discounted values are not martingales under the risk-neutral measure. Clearly, set the short rate $r=0$, then ${\mathbb{E}}^{{\mathbb{Q}}^M}[S_T^2]\ge ({\mathbb{E}}^{{\mathbb{Q}}^M}[S_T]{)}^2=S_0^2$ by the Jensen’s inequality.

The price-process $V_t$ of a derivative contract that pays $V_T=S_T^2$ is a martingale under $\mathbb{Q}$ and can be used as a numeraire.

Consider the price of a contract that pays a unit sum $1$ at maturity $T$. This instrument is the zero-coupon bond. Its an observable and tradable asset. Its price process $P(t,T)={\mathbb{E}}^{{\mathbb{Q}}^M}[1/M_T]$ can be used as a numeraire. ${\mathbb{Q}}^T$ is called the $T$-forward measure.

Abstract Bayes Formula

Theorem 2 (Abstract Bayes Formula) Let $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$ be a probability space and let $\mathbb{Q}$ be any other probability measure on it. By the Radon-Nikodym theorem, $\mathrm{\exists }L=\frac{d\mathbb{Q}}{d\mathbb{P}}$, $L\ge 0$ with ${\mathbb{E}}^{\mathbb{P}}[L]=1$. Then we have:

$$\begin{array}{}\text{(8)}& \begin{array}{r}{\mathbb{E}}^{\mathbb{Q}}[X|\mathcal{G}]=\frac{{\mathbb{E}}^{\mathbb{P}}[LX|\mathcal{G}]}{{\mathbb{E}}^{\mathbb{P}}[L|\mathcal{G}]}\end{array}\end{array}$$

Proof.

By the definition of conditional expectations, recall that if $W$ is any $\mathcal{G}$-measurable random variable, then the conditional expectation must satisfy the relationship:

$$\mathbb{E}[WX]=\mathbb{E}[W\mathbb{E}[X|\mathcal{G}]]$$

It is sufficient to prove that:

$${\mathbb{E}}^{\mathbb{P}}[X|\mathcal{G}]{\mathbb{E}}^{\mathbb{Q}}[L|\mathcal{G}]={\mathbb{E}}^{\mathbb{P}}[LX|\mathcal{G}]$$

Let $G$ be an arbitrary set in $\mathcal{G}$. We have:

$$\begin{array}{rl}& {\int }_G{\mathbb{E}}^{\mathbb{Q}}[X|\mathcal{G}]{\mathbb{E}}^{\mathbb{P}}[L|\mathcal{G}]d\mathbb{P}\\ & ={\int }_G{\mathbb{E}}^{\mathbb{P}}[L\cdot {\mathbb{E}}^{\mathbb{Q}}[X|\mathcal{G}]|\mathcal{G}]d\mathbb{P}\\ & \{{\mathbb{E}}^{\mathbb{Q}}[X|\mathcal{G}]\text{ is }\mathcal{G}\text{-measurable }\}\\ & ={\int }_{G}L\cdot {\mathbb{E}}^{\mathbb{Q}}[X|\mathcal{G}]d\mathbb{P}\\ & ={\int }_G\frac{d\mathbb{Q}}{d\mathbb{P}}\cdot {\mathbb{E}}^{\mathbb{Q}}[X|\mathcal{G}]d\mathbb{P}\\ & ={\int }_G{\mathbb{E}}^{\mathbb{Q}}[X|\mathcal{G}]d\mathbb{Q}\\ & ={\int }_{G}Xd\mathbb{Q}\end{array}$$

Also, we have:

$$\begin{array}{rl}{\int }_G{\mathbb{E}}^{\mathbb{P}}[LX|\mathcal{G}]d\mathbb{P}& ={\int }_{G}LXd\mathbb{P}\\ & ={\int }_G\frac{d\mathbb{Q}}{d\mathbb{P}}Xd\mathbb{P}\\ & ={\int }_{G}Xd\mathbb{Q}\end{array}$$

Hence, proved.

Note that, the filtration $\mathcal{G}$ is the same irrespective of what probability measure we construct on $\mathrm{\Omega }$.

Martingale property

Proposition 1 Assume that there exists a numeraire $M$ and a probability measure ${\mathbb{Q}}^M$, such that the price of any traded asset $X$ (without intermediate payments) relative to $M$ is a martingale under ${\mathbb{Q}}^M$.That is:

$$\frac{X_t}{M_t}={\mathbb{E}}^{{\mathbb{Q}}^M}\{\frac{X_T}{M_T}|{\mathcal{F}}_t\}$$

Let $N_t$ be an arbitrary numeraire. Then, there exists a probability measure ${\mathbb{Q}}^N$ such that the price of $X$ normalized by $N$ is a martingale under ${\mathbb{Q}}^N$.

$$\frac{X_t}{N_t}={\mathbb{E}}^{{\mathbb{Q}}^N}\{\frac{X_T}{N_T}|{\mathcal{F}}_t\}$$

Moreover, the Radon-Nikodym derivative defining the measure ${\mathbb{Q}}^N$ is given by:

$$\frac{d{\mathbb{Q}}^N}{d{\mathbb{Q}}^M}=\frac{N_T/N_0}{M_T/M_0}$$

Proof.

We have:

$$X_0=M_0{\mathbb{E}}^{{\mathbb{Q}}^M}\left[\frac{X_T}{M_T}\right]$$

Imposing the simple fact that, the price of the derivative contract should be the same, even if we switch numeraires from $M$ to $N$, we should have:

$$X_0=N_0{\mathbb{E}}^{{\mathbb{Q}}^N}\left[\frac{X_T}{N_T}\right]$$

Thus,

$$\begin{array}{rl}{N}_0{\mathbb{E}}^{{\mathbb{Q}}^N}\left[\frac{X_T}{N_T}\right]& =M_0{\mathbb{E}}^{{\mathbb{Q}}^M}\left[\frac{X_T}{M_T}\right]\\ \frac{N_T}{N_0}\times N_0{\mathbb{E}}^{{\mathbb{Q}}^N}\left[\frac{X_T}{N_T}\right]& =\frac{N_T}{N_0}\times M_0{\mathbb{E}}^{{\mathbb{Q}}^M}\left[\frac{X_T}{M_T}\right]\left\{\text{Multiplying both sides by }\frac{N_T}{N_0}\right\}\\ ⟹{\mathbb{E}}^{{\mathbb{Q}}^N}[X_T]& ={\mathbb{E}}^{{\mathbb{Q}}^M}\left[\frac{N_T/N_0}{M_T/M_0}{X}_T\right]\end{array}$$

But, we know that:

$${\mathbb{E}}^{{\mathbb{Q}}^N}[X_T]={\mathbb{E}}^{{\mathbb{Q}}^M}\left[\frac{d{\mathbb{Q}}^N}{d{\mathbb{Q}}^M}{X}_T\right]$$

Consequently, our candidate for the Radon-Nikodym derivative should be:

$$L_T=\frac{d{\mathbb{Q}}^N}{d{\mathbb{Q}}^M}=\frac{N_T/N_0}{M_T/M_0}$$

Further $(X_t/N_t)$ is a martingale under ${\mathbb{Q}}^N$. Its easy to see that:

$$\begin{array}{rl}{\mathbb{E}}^{{\mathbb{Q}}^N}[\frac{X_T}{N_T}|{\mathcal{F}}_t]& =\frac{{\mathbb{E}}^{{\mathbb{Q}}^M}[L_T\cdot \frac{X_T}{N_T}|{\mathcal{F}}_t]}{{\mathbb{E}}^{{\mathbb{Q}}^M}[L_T|{\mathcal{F}}_t]}\left\{\text{ Abstract bayes formula }\right\}\\ & =\frac{{\mathbb{E}}^{{\mathbb{Q}}^M}[L_T\cdot \frac{X_T}{N_T}|{\mathcal{F}}_t]}{L_t}\\ & =\frac{{\mathbb{E}}^{{\mathbb{Q}}^M}[\frac{N_T}{N_0}\cdot \frac{M_0}{M_T}\cdot \frac{X_T}{N_T}|{\mathcal{F}}_t]}{\frac{N_t}{N_0}\cdot \frac{M_0}{M_t}}\\ & =\frac{{\mathbb{E}}^{{\mathbb{Q}}^M}[\frac{N_T}{N_0}\cdot \frac{M_0}{M_T}\cdot \frac{X_T}{N_T}|{\mathcal{F}}_t]}{\frac{N_t}{N_0}\cdot \frac{M_0}{M_t}}\\ & =\frac{M_t}{N_t}\cdot {\mathbb{E}}^{{\mathbb{Q}}^M}\left[\frac{X_T}{M_T}\right]\\ & =\frac{M_t}{N_t}\cdot \frac{X_t}{M_t}\\ & =\frac{X_t}{N_t}\end{array}$$

Since we determined the relevant likelihood process, it is easy to find the Girsanov Kernel.

Drift transformation under change of numeraire

Suppose we are interested in the dynamics of the stochastic process $(X_t,t\ge 0)$. Under ${\mathbb{Q}}^M$ measure, its dynamics reads:

$$\begin{array}{}\text{(9)}& \begin{array}{r}dX(t)={\mu }_X^{{\mathbb{Q}}^M}(t)dt+c_X(t)d{W}_t^{{\mathbb{Q}}^M}\end{array}\end{array}$$

I supressed ${\mu }_X^{{\mathbb{Q}}^M}(t,X_t)$ as ${\mu }_X^{{\mathbb{Q}}^M}(t)$ for brevity.

Under the ${\mathbb{Q}}^N$ measure, its dynamics reads:

$$\begin{array}{}\text{(10)}& \begin{array}{r}dX(t)={\mu }_X^{{\mathbb{Q}}^N}(t)dt+c_X(t)d{W}_t^{{\mathbb{Q}}^N}\end{array}\end{array}$$

Remember that the diffusion coefficients in these equations are unaffected by the change of measure! We assume that ${\mathbb{Q}}^M$ is associated with the numeraire $M(t)$ whose dynamics is given by:

$$dM(t)={\mu }_M(t)dt+c_M(t)d{W}^{{\mathbb{Q}}^M}$$

and that the numeraire $N$ has ${\mathbb{Q}}^M$ dynamics:

$$dN(t)={\mu }_N(t)dt+c_N(t)d{W}^{{\mathbb{Q}}^N}$$

According to the Girsanov theorem, the likelihood process $L(t)$ accompanying this change of measure is a martingale under the measure ${\mathbb{Q}}^M$ measure and satisfies the stochastic differential equation:

$$d{L}_t=L(t)\theta (t)d{W}_t^{{\mathbb{Q}}^M}$$

Explicitly, the likelihood process $L(t)$ is given by the stochastic exponential of the martingale ${\int }_0^t{\theta }_{s}d{W}_s^{{\mathbb{Q}}^M}$:

$$L(t)=\exp ({\int }_0^t{\theta }_{s}d{W}_s^{{\mathbb{Q}}^M}-\frac{1}{2}{\int }_0^t{\theta }_s^{2}ds)$$

On the other hand, from Proposition 1, we have:

$$L_t=\frac{N_t/N_0}{M_t/M_0}$$

Differentiating using Ito’s lemma, we have:

$$\begin{array}{rl}d{L}_t& =\frac{M_0}{N_0}d\left(\frac{N_t}{M_t}\right)\\ & =\frac{M_0}{N_0}(-\frac{N_t}{M_t^2}d{M}_t+\frac{1}{M_t}d{N}_t+\frac{1}{2}\cdot \frac{2N_t}{M_t^3}(d{M}_t{)}^2-\frac{1}{M_t^2}(d{M}_t\cdot d{N}_t))\\ & \begin{array}{c}=\frac{M_0}{N_0}(-\frac{N_t}{M_t^2}({\mu }_M(t)dt+c_M(t)d{W}_t^{{\mathbb{Q}}^M})+\frac{1}{M_t}({\mu }_N(t)dt+c_N(t)d{W}_t^{{\mathbb{Q}}^M})\\ +\frac{N_t}{M_t^3}{c}_M^2(t)dt-\frac{1}{M_t^2}{c}_M(t)c_N(t)dt\end{array}\end{array}$$

But since $L_t$ is driftless, we can ignore the $dt$ terms (whatever they are, they are bound to cancel out) and only look at the diffusion coefficient. So, we can write:

$$\begin{array}{rl}d{L}_t& =\frac{M_0}{N_0}(-\frac{N_t}{M_t^2}{c}_M(t)+\frac{1}{M_t}{c}_N(t))dW{{}_t^{\mathbb{Q}}}^M\\ & =\frac{N_t/N_0}{M_t/M_0}(\frac{c_N(t)}{N_t}-\frac{c_M(t)}{M_t})dW{{}_t^{\mathbb{Q}}}^M\\ & =L_t(\frac{c_N(t)}{N_t}-\frac{c_M(t)}{M_t})dW{{}_t^{\mathbb{Q}}}^M\end{array}$$

Comparing this, we can infer that:

$${\theta }_t=\frac{c_N(t)}{N_t}-\frac{c_M(t)}{M_t}$$

Since we can write:

$$\begin{array}{}\text{(11)}& \begin{array}{rl}d{X}_t={\mu }_X^{\mathbb{P}}(t)dt+c_X(t)d{W}^{\mathbb{P}}(t)& ={\mu }_X^{\mathbb{Q}}(t)+c_X(t)d{W}^{\mathbb{Q}}(t)\\ d{W}^{\mathbb{P}}(t)& =\frac{{\mu }_X^{\mathbb{Q}}(t)-{\mu }_X^{\mathbb{P}}(t)}{c_X(t)}+d{W}^{\mathbb{Q}}(t)\end{array}\end{array}$$

Using Equation 11, we conclude that the change of drift accompanying a change of numeraire is given by:

$$\begin{array}{}\text{(12)}& \begin{array}{r}{\mu }_X^{\mathbb{Q}}(t)-{\mu }_X^{\mathbb{P}}(t)=c_X(t)(\frac{c_M(t)}{M(t)}-\frac{c_N(t)}{N(t)})\end{array}\end{array}$$

Examples of numeraires

The basic component of an interest rate model is an instantaneous forward rate process $f(t,s)$. Its value is the future instantaneous interest rate at a future time $s$, that is the rate for the infinitesimally short term $[s,s+ds]$ observed at time $t\le s$.

A zero-coupon bond settling at time $T_0$ and maturing at time $T>T_0$ is the process:

$$P(t,T_0,T)=\exp (-{\int }_{T_0}^{T}f(t,s)ds)$$

for $t\le T_0$. In other words, it is the time $T_0$ value of 1$\mathrm{\$}$ (without the risk of default) at $T$, as observed at time $t\le T_0$. Its current value is given by:

Bank-account numeraire

The bank-account(money-market account) numeraire is simply the value of $1\mathrm{\$}$ deposited in a bank and accruing the (credit-riskless) instantaneous interest rate. In reality, the bank credits interest to the account daily, but this can very well be approximated to a continous process. The associated stochastic price process $M(t)$ is given by:

$$\begin{array}{r}M(t)=\exp ({\int }_0^{t}r(s)ds)\end{array}$$

Here, the spot rate $r(t)$ is the instantaneous forward observed at the time it settles. That is,

$$r(t)=f(t,t)$$

Forward numeraire

A zero-coupon bond(ZCB) is a simple contract with unit payoff $1\mathrm{\$}$ at maturity $T$. By the risk-neutral valuation formula:

$$P(t,t,T)=V(t)=M(t){\mathbb{E}}^{{\mathbb{Q}}^M}\left[\frac{1}{M(T)}\right]$$

So, a $T$-maturity ZCB is a tradable asset and its price $P(t,T)$ can be used as a numeraire. The associated measure is called the $T$-foward measure ${\mathbb{Q}}^T$.

The term(e.g. 3 months) forward rates for settlement at $T_0$ and maturity at $T$ are defined by the equation:

$$\begin{array}{r}P(t,T_0,T)=\frac{1}{1+\delta F(t,T_0,T)}\end{array}$$

where $\delta$ is the day-count fraction for the period $[T_0,T]$. Re-arranging, we have:

$$\begin{array}{rlr}F(t,T_0,T)& =\frac{1}{\delta }\frac{P(t,T,T)-P(t,T_0,T)}{P(t,T_0,T)}F(t,T_0,T)P(t,T_0,T)& =\frac{1}{\delta }(P(t,T,T)-P(t,T_0,T))\end{array}$$

Clearly, it is a multiple of a difference $P(t,T,T)$ and $P(t,T_0,T)$ normalized by $T$-maturity zero coupon bond price $P(t,T_0,T)$. So, the forward iBOR-rate must be a martingale under the $T$-forward measure $Q^T$.

Proposition 2 (Forward rates are ${\mathbb{Q}}^T$ expectations of future spot rates.) Any simply compounded forward rate spanning a time interval ending in $T$ is a martingale under the $T$-forward measure.

$${\mathbb{E}}^{{\mathbb{Q}}^T}[F(t;S,T)|{\mathcal{F}}_u]=F(u;S,T)$$

for each $0\le u\le t\le S<T$. In particular, the forward rate spanning the interval $[S,T]$ is the ${\mathbb{Q}}^T$ expectation of the future simply-compounded spot rate at time $S$ for the maturity T.

$$\begin{array}{}\text{(13)}& {\mathbb{E}}^{{\mathbb{Q}}^T}[L(S,T)|{\mathcal{F}}_t]=F(t;S,T)\end{array}$$

The expected value of any future instantaneous spot interest rate, under the corresponding forward measure, is equal to the related instantaneuous forward rate. That is, $$\begin{array}{r}{\mathbb{E}}^T{r}_T|{\mathcal{F}}_t=f(t,T)\end{array}$$

for each $0\le t\le T$.

Proof.

We have:

$$\begin{array}{rl}{\mathbb{E}}^{{\mathbb{Q}}^T}[r_T|{\mathcal{F}}_t]& =\frac{1}{P(t,T)}\mathbb{E}[r_{T}P(t,T)|{\mathcal{F}}_t]\\ & =-\frac{1}{P(t,T)}\mathbb{E}[r_T{e}^{-{\int }_t^{T}r(s)ds}|{\mathcal{F}}_t]\\ & =-\frac{1}{P(t,T)}\mathbb{E}[\frac{\partial }{\partial T}{e}^{-{\int }_t^{T}r(s)ds}|{\mathcal{F}}_t]\\ & =-\frac{1}{P(t,T)}\frac{\partial }{\partial T}\mathbb{E}[e^{-{\int }_t^{T}r(s)ds}|{\mathcal{F}}_t]\\ & =-\frac{1}{P(t,T)}\frac{\partial }{\partial T}P(t,T)\\ & =f(t,T)\end{array}$$

Pricing an IRS

Consider a forward starting interest rate swap(IRS) which settles in $T_0$ and matures in $T_N$ years from now. An IRS is a transaction between two counterparties who exchange interest rate payments on an agreed notional principal.

On a vanilla swap, a fixed-coupon interest payments are exchanged for floating rate payments. For the sake of simplicity, we assume that the payment dates on the fixed and floating leg of the swap are the same, and that the floating rate is the same as the discounting rate. The former of these assumptions is a minor simplification, made to lighten up the notation only. The latter is an important simplification, as the basis between the floating rate and the discounting rate may exhibit a complex dynamics.

Let $S$ be the fixed-rate on the swap. By the risk-neutral valuation formula, the fixed leg value at time $t$ can be expressed as:

$$\begin{array}{rl}{V}_{fixed}(t)& =N\cdot \sum _{i=1}^N{\mathbb{E}}^{{\mathbb{Q}}^{\mathbb{M}}}[e^{-r(T_i-t)}S\tau (T_{i-1},T_i)|{\mathcal{F}}_t]\\ & =S\cdot N\cdot \sum _{i=1}^N{\mathbb{E}}^{\mathbb{Q}}[e^{-r(T_i-t)}\tau (T_{i-1},T_i)|{\mathcal{F}}_t]\\ & =S\cdot N\cdot (\sum _{i=1}^{N}P(t,T_i)\tau (T_{i-1},T_i))\end{array}$$

The floating leg can be written as:

$$\begin{array}{rl}{V}_{float}(t)& =N\cdot \sum _{i=1}^{N}P(t,T_i){\mathbb{E}}^{T_i}[L(T,T_{i-1},T_i)|{\mathcal{F}}_t]\tau (T_{i-1},T_i)\\ & =N\cdot \sum _{i=1}^{N}P(t,T_i)L(t,T_{i-1},T_i)\tau (T_{i-1},T_i)\\ & =N\cdot \sum _{i=1}^{N}P(t,T_i)\frac{1}{\tau (T_{i-1},T_i)}\cdot (\frac{P(t,T_{i-1})}{P(t,T_i)}-1)\tau (T_{i-1},T_i)\\ & =N\cdot \sum _{i=1}^N(P(t,T_i)-P(t,T_{i-1}))\\ & =-NP(t,T_0)+NP(t,T_N)\end{array}$$

where the expectations are under the $T_i$-forward measure. Note that, I used the fact that the iBOR-rates are martingales under the forward measure.

The par-swap rate $S$ is the fixed-rate which renders the value of the swap zero at the contract start date $t$.

$$\begin{array}{rl}-V_{fix}+V_{floating}& =0\\ S\cdot N\cdot (\sum _{i=1}^{N}P(t,T_i)\tau (T_{i-1},T_i))& =-NP(t,T_0)+NP(t,T_N)\\ S(t,T_{0:N})& =\frac{-P(t,T_0)+P(t,T_N)}{(\sum _{i=1}^{N}P(t,T_i)\tau (T_{i-1},T_i))}\end{array}$$

The Annuity Measure

The annuity is an asset that pays $1\mathrm{\$}$ on each coupon payment day of the swap, accrued according to the swap’s day count convention.

$$A(t,T_{0:N})=(\sum _{i=1}^{N}P(t,T_i)\tau (T_{i-1},T_i))$$

Since, it is a portfolio of zero coupon bonds, it is a tradable asset and its price $A(t,T_{0:N})$ can be used as numeraire. This is called the Annuity numeraire and the measure ${\mathbb{Q}}^{T_{0:N}}$ associated with this numeraire is called the (forward) swap measure. The annuity numeraire arises as the natural numeraire when valuing swaptions. It is the mechanism that allows us to link the swaption as an option on a swap to the option on the corresponding swap rate.

The forward swap rate $S(t,T_{0:N})$ is a martingale in the annuity measure ${\mathbb{Q}}^{T_{0:N}}$.

Pricing the payoff $V(T)=S_T(S_T-K{)}^{+}$

Under the risk-neutral measure ${\mathbb{Q}}^M$ associated with the money-market account numeraire $M(t)$, the stock price $S(t)$ evolves according to:

$$\begin{array}{}\text{(14)}& \begin{array}{r}d{S}_t=r{S}_{t}dt+\sigma S_{t}d{W}^{{\mathbb{Q}}^M}\end{array}\end{array}$$

which has the solution:

$$\begin{array}{}\text{(15)}& \begin{array}{r}{S}_t=S_0\exp [(r-\frac{{\sigma }^2}{2})t+\sigma W^{{\mathbb{Q}}^M}(t)]\end{array}\end{array}$$

By the risk-neutral pricing formula, we have:

$$\begin{array}{}\text{(16)}& \begin{array}{rl}V(0)& ={\mathbb{E}}^{{\mathbb{Q}}^M}\left[\frac{V(T)}{M(T)}\right]\\ & ={\mathbb{E}}^{{\mathbb{Q}}^M}[\frac{S_T}{M_T}(S_T-K{)}^{+}]\end{array}\end{array}$$

By the change-of-measure formula, if $N$ is any other numeraire with associated probability measure ${\mathbb{Q}}^N$, we know that:

$$\begin{array}{}\text{(17)}& \begin{array}{r}{\mathbb{E}}^{{\mathbb{Q}}^N}[V(T)]={\mathbb{E}}^{{\mathbb{Q}}^M}[\frac{d{\mathbb{Q}}^N}{d{\mathbb{Q}}^M}V(T)]\end{array}\end{array}$$

We switch to the stock numeraire $S_t$. The Radon-Nikodym derivative $L_T=d{\mathbb{Q}}^S/d{\mathbb{Q}}^M$ is simply:

$$\begin{array}{}\text{(18)}& \begin{array}{r}{L}_T=\frac{d{\mathbb{Q}}^S}{d{\mathbb{Q}}^M}=\frac{S(T)/S(0)}{M(T)/M(0)}=\frac{1}{S(0)}\cdot \frac{S(T)}{M(T)}\end{array}\end{array}$$

The dynamics of $L_t$ under the ${\mathbb{Q}}_M$ measure is given by:

$$d{L}_t={\varphi }_t{L}_{t}d{W}^{{\mathbb{Q}}^M}$$

where the Girsanov kernel ${\varphi }_t=\sigma$. Then, the Girsanov transformation is:

$$\begin{array}{}\text{(19)}& W_T^{{\mathbb{Q}}^M}=W_T^{{\mathbb{Q}}^S}+\sigma T\end{array}$$

So, the ${\mathbb{Q}}^S$ dynamics of the stock price is:

$$\begin{array}{}\text{(20)}& \begin{array}{rl}{S}_T& =S_0\exp [(r-\frac{{\sigma }^2}{2})T+\sigma (W_T^{{\mathbb{Q}}^S}+\sigma T)]\\ & =S_0\exp [(r+\frac{{\sigma }^2}{2})T+\sigma (W_T^{{\mathbb{Q}}^S})]\end{array}\end{array}$$

We can develop the expression in Equation 16 to be:

$$\begin{array}{}\text{(21)}& \begin{array}{rl}V(0)& =S_0{\mathbb{E}}^{{\mathbb{Q}}^M}[\frac{1}{S_0}\frac{S_T}{M_T}(S_T-K{)}^{+}]\end{array}\end{array}$$

Applying the change-of-measure (abstract Baye’s formula),we have:

$$\begin{array}{}\text{(22)}& \begin{array}{rl}V(0)& =S_0{\mathbb{E}}^{{\mathbb{Q}}^S}[(S_T-K{)}^{+}]\end{array}\end{array}$$

Now, we have:

$$\begin{array}{}\text{(23)}& \begin{array}{rl}{\mathbb{Q}}^S(S_T>K)& ={\mathbb{Q}}^S(S_0\exp [(r+\frac{{\sigma }^2}{2})T+\sigma (W_T^{{\mathbb{Q}}^S})]>K)\\ & ={\mathbb{Q}}^S(S_0\exp [(r+\frac{{\sigma }^2}{2})T+\sigma (-\sqrt{T}Z)]>K)\\ & ={\mathbb{Q}}^S(\log \frac{S_0}{K}+(r+\frac{{\sigma }^2}{2})T>\sigma \sqrt{T}Z)\end{array}\end{array}$$

where we subbed $-\sqrt{T}Z=W_T^{{\mathbb{Q}}^S}$. Recall, the standard normals $Z$ and $-Z$ have the same distribution by symmetry.

Define

$$\begin{array}{}\text{(24)}& d_{-}=\frac{\log \frac{S_0}{K}+(r+\frac{{\sigma }^2}{2})T}{\sigma \sqrt{T}}\end{array}$$

Then, since $Z\sim {\mathcal{N}}^{{\mathbb{Q}}^S}(0,1)$:

$$\begin{array}{}\text{(25)}& {\mathbb{Q}}^S(S_T>K)={\mathbb{Q}}^S(Z<d_{-})=\mathrm{\Phi }(d_{-})\end{array}$$

So, we can expand the expectation in Equation 22. The indicator random variable $1_{\{S_T>K\}}$ is $1$ for all points $Z<d_{-}$. So, the limits of integration will be $-\mathrm{\infty }$ to $d_{-}$.

$$\begin{array}{rl}V(0)& =S_0[{\int }_{-\mathrm{\infty }}^{d_{-}}{S}_{T}d{\mathbb{Q}}^S-K{\int }_{-\mathrm{\infty }}^{d_{-}}d{Q}^S]\\ & =S_0[{\int }_{-\mathrm{\infty }}^{d_{-}}{S}_T{f}_Z^{{\mathbb{Q}}^S}(z)dz-K{\int }_{-\mathrm{\infty }}^{d_{-}}{f}_Z^{{\mathbb{Q}}^S}(z)dz]\\ & =S_0[\frac{1}{\sqrt{2\pi }}{\int }_{-\mathrm{\infty }}^{d_{-}}{S}_0\exp [(r+\frac{{\sigma }^2}{2})T-\sigma \sqrt{T}z]\exp (-\frac{z^2}{2})dz-K\mathrm{\Phi }(d_{-})]\end{array}$$

Completing the square, we have:

$$\begin{array}{rl}V(0)& =S_0[\frac{1}{\sqrt{2\pi }}{\int }_{-\mathrm{\infty }}^{d_{-}}{S}_0\exp [-\frac{1}{2}(z^2+2\sigma \sqrt{T}z+(\sigma \sqrt{T}{)}^2)]\exp \left[(r+{\sigma }^2)T\right]dz-K\mathrm{\Phi }(d_{-})]\\ & =S_0[S_0\frac{e^{(r+{\sigma }^2)T}}{\sqrt{2\pi }}{\int }_{-\mathrm{\infty }}^{d_{-}}\exp (-\frac{1}{2}(z+\sigma \sqrt{T}{)}^2)dz-K\mathrm{\Phi }(d_{-})]\end{array}$$

Let $u=z+\sigma \sqrt{T}$. And define:

$$d_{+}=d_{-}+\sigma \sqrt{T}=\frac{\log \frac{S_0}{K}+(r+\frac{3}{2}{\sigma }^2)}{\sigma \sqrt{T}}$$

We have the closed-form formula:

$$V(0)=S_0^2{e}^{(r+{\sigma }^2)T}\mathrm{\Phi }(d_{+})-K{S}_0\mathrm{\Phi }(d_{-})$$

References

• Girsanov, Numeraires and all that, Andrew Lesniewski.
• Interest-rate Models - Theory and Practice, Damiano Brigo and Fabio Mercurio.