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 \((W^{\mathbb{P}},t\geq 0)\) be a \(\mathbb{P}\) standard brownian motion on \((\Omega,\mathcal{F},\mathbb{P})\) and let \(\phi\) be any adapted process. Choose fixed \(T\) and defined the process \(L\) on \([0,T]\) by:
\[ \begin{align*} dL_t = \phi_t L_t dW^{\mathbb{P}}_t \end{align*} \tag{1}\]
\[ \begin{align*} L_0 = 1 \end{align*} \tag{2}\]
that is:
\[ \begin{align*} L_t = \exp\left(\int_0^t \phi_s dW^{\mathbb{P}}_s - \int_0^t \phi^2_s ds\right) \end{align*} \tag{3}\]
Assume that:
\[ \begin{align*} \mathbb{E}^{\mathbb{P}}[L_T] = 1 \end{align*} \tag{4}\]
and define the new probability measure \(\mathbb{Q}\) on \(\mathcal{F}_t\) by:
\[ L_T = \frac{d\mathbb{Q}}{d\mathbb{P}} \tag{5}\]
Then,
\[ dW^{\mathbb{P}}_t = dW^{\mathbb{Q}}_t + \phi_t dt \tag{6}\]
where \(dW^{\mathbb{Q}}_t\) is a \(\mathbb{Q}\)-standard brownian motion.
Proof.
Our claim is that, under the \(\mathbb{Q}\) measure, the increments \((W^{\mathbb{Q}}_t - W^{\mathbb{Q}}_s)\) 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, \(f_x = \frac{1}{x}\), \(f_{xx} = -\frac{1}{x^2}\).
\[ \begin{align*} d(\ln L_t) &= \frac{1}{L_t}dL_t -\frac{1}{2} \frac{1}{L_t^2}(dL_t)^2 \\ &= \frac{1}{L_t}{\phi_t L_t dW^{\mathbb{P}}_t} - \frac{1}{2L_t^2}\phi_t^2 L_t^2 dt\\ &= \phi_t dW^{\mathbb{P}}_t -\frac{1}{2} \phi_t^2 dt \\ L_t &= \exp\left(\int_0^t \phi_s dW^{\mathbb{P}}_s - \frac{1}{2}\int_0^t \phi_s^2 ds \right) \end{align*} \]
To prove our main result, we will now use the MGF of the increments. For \(n \in \mathbb{N}\) and \((t_j,j\leq n)\) a partition of \([0,T]\), with \(t_n = T\), I will show that :
\[ \begin{align*} \mathbb{E}^{\mathbb{Q}}\left[\exp\left(\sum_{j=0}^{n-1}\lambda_j (W^{\mathbb{Q}}_{t_{j+1}} - W^{\mathbb{Q}}_{t_{j}})\right)\right] = \exp\left[\sum_{j=0}^{n-1}\lambda_j^2(t_{j+1}-t_j)\right] \end{align*} \tag{7}\]
This proves that the increments are the ones of standard brownian motion.
Let \((\mathcal{F}_{t_j},j\leq n)\) 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{aligned} \mathbb{E}^{\mathbb{Q}}\left[\exp\left(\sum _{j=0}^{n-1} \lambda _{j} (W_{t_{j+1}}^{\mathbb{Q}} -W_{t_{j}}^{\mathbb{Q}} )\right)\right] & =\mathbb{E}^{\mathbb{P}}\left[\mathbb{E}^{\mathbb{P}}\left[ L_{t_{n}}\exp\left(\sum _{j=0}^{n-1} \lambda _{j} (W_{t_{j+1}}^{\mathbb{Q}} -W_{t_{j}}^{\mathbb{Q}} )\right) |\mathcal{F}_{t_{n-1}}\right]\right]\\ & =\mathbb{E}^{\mathbb{P}}\left[\sum _{j=0}^{n-2} \lambda _{j} (W_{t_{j+1}}^{\mathbb{Q}} -W_{t_{j}}^{\mathbb{Q}} )\mathbb{E}^{\mathbb{P}}\left[ L_{t_{n}}\exp\left( \lambda _{n-1} (W_{t_{n}}^{\mathbb{Q}} -W_{t_{n-1}}^{\mathbb{Q}} )\right) |\mathcal{F}_{t_{n-1}}\right]\right]\\ & =\mathbb{E}^{\mathbb{P}}\left[\sum _{j=0}^{n-2} \lambda _{j} (W_{t_{j+1}}^{\mathbb{Q}} -W_{t_{j}}^{\mathbb{Q}} )\mathbb{E}^{\mathbb{P}}\left[\exp\left(\int _{t_{n-1}}^{t_{n}} \phi _{s} dW_{s}^{\mathbb{Q}} +\frac{1}{2}\int _{t_{n-1}}^{t_{n}} \phi _{s}^{2} ds\right)\exp\left( \lambda _{n-1}\int _{t_{n-1}}^{t_{n}} dW_{s}^{\mathbb{Q}}\right) |\mathcal{F}_{t_{n-1}}\right]\right]\\ & =\mathbb{E}^{\mathbb{P}}\left[\sum _{j=0}^{n-2} \lambda _{j} (W_{t_{j+1}}^{\mathbb{Q}} -W_{t_{j}}^{\mathbb{Q}} )\mathbb{E}^{\mathbb{P}}\left[\exp\left(\int _{t_{n-1}}^{t_{n}}( \phi _{s} +\lambda _{n-1}) dW_{s}^{\mathbb{Q}} +\frac{1}{2}\int _{t_{n-1}}^{t_{n}} \phi _{s}^{2} ds\right) |\mathcal{F}_{t_{n-1}}\right]\right]\\ & =\mathbb{E}^{\mathbb{P}}\left[\sum _{j=0}^{n-2} \lambda _{j} (W_{t_{j+1}}^{\mathbb{Q}} -W_{t_{j}}^{\mathbb{Q}} )\mathbb{E}^{\mathbb{P}}\left[\exp\left(\int _{t_{n-1}}^{t_{n}}( \phi _{s} +\lambda _{n-1})\left( dW_{s}^{\mathbb{P}} -\theta ds\right) +\frac{1}{2}\int _{t_{n-1}}^{t_{n}} \phi _{s}^{2} ds\right) |\mathcal{F}_{t_{n-1}}\right]\right]\\ & =\mathbb{E}^{\mathbb{P}}\left[\sum _{j=0}^{n-2} \lambda _{j} (W_{t_{j+1}}^{\mathbb{Q}} -W_{t_{j}}^{\mathbb{Q}} )\exp\left(\frac{1}{2}\int _{t_{n-1}}^{t_{n}} \phi _{s}^{2} +\int _{t_{n-1}}^{t_{n}} -\phi _{s}( \phi _{s} +\lambda _{n-1}) ds\right) \ \exp\left(\frac{1}{2}\int _{t_{n-1}}^{t_{n}}( \phi _{s} +\lambda _{n-1})^{2} ds\right)\right]\\ & \left\{\ \int XdW_{s}^{\mathbb{P}} \ \text{ is a }\mathcal{N}^{\mathbb{P}}\left( 0,\int \mathbb{E}\left[ X^{2}\right] ds\right) \ \text{gaussian random variable.}\right\} \ \\ & =\mathbb{E}^{\mathbb{P}}\left[\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}}\left( -\frac{1}{2} \phi _{s}^{2} -\lambda _{n-1} \phi _{s} +\frac{1}{2} \phi _{s}^{2} +\lambda _{n-1} \phi _{s} +\lambda _{n-1}^{2}\right) ds\right)\right]\\ & =\mathbb{E}^{\mathbb{P}}\left[\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)\right]\\ & =\exp( \lambda _{n-1}( t_{n} -t_{n-1}) \cdot \mathbb{E}^{\mathbb{P}}\left[\sum _{j=0}^{n-2} \lambda _{j} (W_{t_{j+1}}^{\mathbb{Q}} -W_{t_{j}}^{\mathbb{Q}} )\right] \end{aligned} \]
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 \(\phi_t\) 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 \(\mathbb{E}^{\mathbb{Q}^M}[S_T^2] \geq (\mathbb{E}^{\mathbb{Q}^M}[S_T])^2 =S_0^2\) 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 \((\Omega,\mathcal{F},\mathbb{P})\) be a probability space and let \(\mathbb{Q}\) be any other probability measure on it. By the Radon-Nikodym theorem, \(\exists L = \frac{d\mathbb{Q}}{d\mathbb{P}}\), \(L \geq 0\) with \(\mathbb{E}^{\mathbb{P}}[L]=1\). Then we have:
\[ \begin{align*} \mathbb{E}^{\mathbb{Q}}[X|\mathcal{G}] = \frac{\mathbb{E}^{\mathbb{P}}[LX|\mathcal{G}]}{\mathbb{E}^{\mathbb{P}}[L|\mathcal{G}]} \end{align*} \tag{8}\]
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:
\[ \begin{align*} & \int_G \mathbb{E}^{\mathbb{Q}}[X|\mathcal{G}]\mathbb{E}^{\mathbb{P}}[L|\mathcal{G}] d\mathbb{P} \\ &= \int_G \mathbb{E}^{\mathbb{P}}[L\cdot \mathbb{E}^{\mathbb{Q}}[X|\mathcal{G}]|\mathcal{G}] d\mathbb{P} \\ & \quad \{ \mathbb{E}^{\mathbb{Q}}[X|\mathcal{G}] \text{ is }\mathcal{G}\text{-measurable } \}\\ &= \int_G L\cdot \mathbb{E}^{\mathbb{Q}}[X|\mathcal{G}] d\mathbb{P}\\ &= \int_G \frac{d\mathbb{Q}}{d\mathbb{P}}\cdot \mathbb{E}^{\mathbb{Q}}[X|\mathcal{G}] d\mathbb{P}\\ &= \int_G \mathbb{E}^{\mathbb{Q}}[X|\mathcal{G}] d\mathbb{Q}\\ &= \int_G X d\mathbb{Q} \end{align*} \]
Also, we have:
\[ \begin{align*} \int_{G} \mathbb{E}^{\mathbb{P}}[LX|\mathcal{G}]d\mathbb{P} &= \int_G LX d\mathbb{P}\\ &= \int_G \frac{d\mathbb{Q}}{d\mathbb{P}} X d\mathbb{P}\\ &= \int_G X d\mathbb{Q} \end{align*} \]
Hence, proved.
Note that, the filtration \(\mathcal{G}\) is the same irrespective of what probability measure we construct on \(\Omega\).
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:
\[ \frac{X_t}{M_t} = \mathbb{E}^{\mathbb{Q}^M} \left\{\frac{X_T}{M_T}|\mathcal{F}_t\right\} \]
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 \(\mathbb{Q}^N\).
\[ \frac{X_t}{N_t} = \mathbb{E}^{\mathbb{Q}^N} \left\{\frac{X_T}{N_T}|\mathcal{F}_t\right\} \]
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,
\[ \begin{aligned} N_{0}\mathbb{E}^{\mathbb{Q}^{N}}\left[\frac{X_{T}}{N_{T}}\right] & =M_{0}\mathbb{E}^{\mathbb{Q}^{M}}\left[\frac{X_{T}}{M_{T}}\right]\\ \frac{N_{T}}{N_{0}} \times N_{0}\mathbb{E}^{\mathbb{Q}^{N}}\left[\frac{X_{T}}{N_{T}}\right] & =\frac{N_{T}}{N_{0}} \times M_{0} \ \mathbb{E}^{\mathbb{Q}^{M}}\left[\frac{X_{T}}{M_{T}}\right] \quad \left\{\text{Multiplying both sides by }\frac{N_{T}}{N_{0}}\right\}\\ \Longrightarrow \mathbb{E}^{\mathbb{Q}^{N}}[ X_{T}] & =\mathbb{E}^{\mathbb{Q}^{M}}\left[\frac{N_{T} /N_{0}}{M_{T} /M_{0}} X_{T}\right] \end{aligned} \]
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:
\[ \begin{aligned} \mathbb{E}^{\mathbb{Q}^{N}}\left[\frac{X_{T}}{N_{T}} |\mathcal{F}_{t}\right] & =\frac{\mathbb{E}^{\mathbb{Q}^{M}}\left[ L_{T} \cdot \frac{X_{T}}{N_{T}} |\mathcal{F}_{t}\right]}{\mathbb{E}^{\mathbb{Q}^{M}}[ L_{T} |\mathcal{F}_{t}]} \quad \left\{\text{ Abstract bayes formula }\right\}\\ & =\frac{\mathbb{E}^{\mathbb{Q}^{M}}\left[ L_{T} \cdot \frac{X_{T}}{N_{T}} |\mathcal{F}_{t}\right]}{L_{t}}\\ & =\frac{\mathbb{E}^{\mathbb{Q}^{M}}\left[\frac{N_{T}}{N_{0}} \cdot \frac{M_{0}}{M_{T}} \cdot \frac{X_{T}}{N_{T}} |\mathcal{F}_{t}\right]}{\frac{N_{t}}{N_{0}} \cdot \frac{M_{0}}{M_{t}}}\\ & =\frac{\mathbb{E}^{\mathbb{Q}^{M}}\left[\frac{N_{T}}{N_{0}} \cdot \frac{M_{0}}{M_{T}} \cdot \frac{X_{T}}{N_{T}} |\mathcal{F}_{t}\right]}{\frac{N_{t}}{N_{0}} \cdot \frac{M_{0}}{M_{t}}}\\ & =\frac{M_{t}}{N_{t}} \cdot \mathbb{E}^{\mathbb{Q}^{M}}\left[\frac{X_{T}}{M_{T}}\right]\\ & =\frac{M_{t}}{N_{t}} \cdot \frac{X_{t}}{M_{t}}\\ & =\frac{X_{t}}{N_{t}} \end{aligned} \]
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 \((X_t,t\geq 0)\). Under \(\mathbb{Q}^M\) measure, its dynamics reads:
\[ \begin{aligned} dX(t) = \mu_X^{\mathbb{Q}^M}(t) dt + c_X(t)dW^{\mathbb{Q}^M}_t \end{aligned} \tag{9}\]
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:
\[ \begin{aligned} dX(t) = \mu_X^{\mathbb{Q}^N}(t) dt + c_X(t)dW^{\mathbb{Q}^N}_t \end{aligned} \tag{10}\]
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:
\[ dM(t) = \mu_M(t)dt + c_M(t)dW^{\mathbb{Q}^M} \]
and that the numeraire \(N\) has \(\mathbb{Q}^M\) dynamics:
\[ dN(t) = \mu_N(t)dt + c_N(t)dW^{\mathbb{Q}^N} \]
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:
\[ dL_t = L(t)\theta(t)dW^{\mathbb{Q}^M}_t \]
Explicitly, the likelihood process \(L(t)\) is given by the stochastic exponential of the martingale \(\int_0^t \theta_s dW^{\mathbb{Q}^M}_s\):
\[ L(t) = \exp\left(\int_0^t \theta_s dW^{\mathbb{Q}^M}_s - \frac{1}{2}\int_0^t \theta^2_s ds \right) \]
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:
\[ \begin{aligned} dL_{t} & =\frac{M_{0}}{N_{0}} d\left(\frac{N_{t}}{M_{t}}\right)\\ & =\frac{M_{0}}{N_{0}}\left( -\frac{N_{t}}{M_{t}^{2}} dM_{t} +\frac{1}{M_{t}} dN_{t} +\frac{1}{2} \cdot \frac{2N_{t}}{M_{t}^{3}}( dM_{t})^{2} -\frac{1}{M_{t}^{2}}( dM_{t} \cdot dN_{t})\right)\\ & \begin{array}{l} =\frac{M_{0}}{N_{0}}( -\frac{N_{t}}{M_{t}^{2}}\left( \mu _{M}( t) dt+c_{M}( t) dW_{t}^{\mathbb{Q}^{M}}\right) +\frac{1}{M_{t}}\left( \mu _{N}( t) dt+c_{N}( t) dW_{t}^{\mathbb{Q}^{M}}\right)\\ +\frac{N_{t}}{M_{t}^{3}} c_{M}^{2}( t) dt-\frac{1}{M_{t}^{2}} c_{M}( t) c_{N}( t) dt \end{array} \end{aligned} \]
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:
\[ \begin{aligned} dL_{t} & =\frac{M_{0}}{N_{0}}\left( -\frac{N_{t}}{M_{t}^{2}} c_{M}( t) +\frac{1}{M_{t}} c_{N}( t)\right) dW{_{t}^{\mathbb{Q}}}^{M}\\ & =\frac{N_{t} /N_{0}}{M_{t} /M_{0}}\left(\frac{c_{N}( t)}{N_{t}} -\frac{c_{M}( t)}{M_{t}}\right) dW{_{t}^{\mathbb{Q}}}^{M}\\ & =L_{t}\left(\frac{c_{N}( t)}{N_{t}} -\frac{c_{M}( t)}{M_{t}}\right) dW{_{t}^{\mathbb{Q}}}^{M} \end{aligned} \]
Comparing this, we can infer that:
\[ \theta_t = \frac{c_N(t)}{N_t} - \frac{c_M(t)}{M_t} \]
Since we can write:
\[ \begin{align*} dX_t = \mu^{\mathbb{P}}_X(t) dt + c_X(t)dW^{\mathbb{P}}(t) &= \mu^{\mathbb{Q}}_X(t) + c_X(t)dW^{\mathbb{Q}}(t)\\ dW^{\mathbb{P}}(t) &= \frac{\mu^{\mathbb{Q}}_X(t) - \mu^{\mathbb{P}}_X(t)}{c_X(t)} + dW^{\mathbb{Q}}(t) \end{align*} \tag{11}\]
Using Equation 11, we conclude that the change of drift accompanying a change of numeraire is given by:
\[ \begin{align*} \mu_X^{\mathbb{Q}}(t) - \mu_X^{\mathbb{P}}(t) = c_X(t)\left(\frac{c_M(t)}{M(t)} - \frac{c_N(t)}{N(t)}\right) \end{align*} \tag{12}\]
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 \(t \leq s\).
A zero-coupon bond settling at time \(T_0\) and maturing at time \(T > T_0\) is the process:
\[ P(t,T_0,T) = \exp\left(-\int_{T_0}^Tf(t,s)ds\right) \]
for \(t \leq T_0\). In other words, it is the time \(T_0\) value of 1\(\$\) (without the risk of default) at \(T\), as observed at time \(t \leq T_0\). Its current value is given by:
Bank-account numeraire
The bank-account(money-market account) numeraire is simply the value of \(1\$\) 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:
\[ \begin{align*} M(t) = \exp(\int_0^t r(s)ds) \end{align*} \]
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 \(1\$\) 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:
\[ \begin{align*} P(t,T_0,T) = \frac{1}{1 + \delta F(t,T_0,T)} \end{align*} \]
where \(\delta\) is the day-count fraction for the period \([T_0,T]\). Re-arranging, we have:
\[ \begin{align*} F(t,T_0,T) &= \frac{1}{\delta}\frac{P(t,T,T) - P(t,T_0, T)}{P(t,T_0,T)} F(t,T_0,T)P(t,T_0,T) &= \frac{1}{\delta}(P(t,T,T) - P(t,T_0,T)) \end{align*} \]
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 \(0 \leq u \leq t \leq S < T\). 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.
\[ \mathbb{E}^{\mathbb{Q}^T}[L(S,T)|\mathcal{F}_t] = F(t;S,T) \tag{13}\]
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, \[ \begin{align*} \mathbb{E}^T{r_T|\mathcal{F}_t} = f(t,T) \end{align*} \]
for each \(0 \leq t \leq T\).
Proof.
We have:
\[ \begin{align*} \mathbb{E}^{\mathbb{Q}^T} [r_T|\mathcal{F}_t] &= \frac{1}{P(t,T)}\mathbb{E}\left[r_TP(t,T)|\mathcal{F}_t\right]\\ &= - \frac{1}{P(t,T)}\mathbb{E}\left[r_Te^{-\int_t^T r(s)ds}|\mathcal{F}_t\right]\\ &= - \frac{1}{P(t,T)}\mathbb{E}\left[\frac{\partial}{\partial T}e^{-\int_t^T r(s)ds}|\mathcal{F}_t\right]\\ &= - \frac{1}{P(t,T)}\frac{\partial}{\partial T}\mathbb{E}\left[e^{-\int_t^T r(s)ds}|\mathcal{F}_t\right]\\ &= - \frac{1}{P(t,T)}\frac{\partial}{\partial T}P(t,T)\\ &= f(t,T) \end{align*} \]
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:
\[ \begin{align*} V_{fixed}(t) &= N \cdot \sum_{i=1}^N \mathbb{E}^{\mathbb{Q^M}}[e^{-r(T_i-t)}S \tau(T_{i-1},T_i)|\mathcal{F}_t]\\ &=S \cdot N \cdot \sum_{i=1}^N \mathbb{E}^{\mathbb{Q}}[e^{-r(T_i-t)}\tau(T_{i-1},T_i)|\mathcal{F}_t]\\ &=S \cdot N \cdot \left(\sum_{i=1}^N P(t,T_i)\tau(T_{i-1},T_i)\right) \end{align*} \]
The floating leg can be written as:
\[ \begin{align*} V_{float}(t) & = N \cdot \sum_{i=1}^N P(t,T_i)\mathbb{E}^{T_i}\left[L(T,T_{i-1}, T_i)|\mathcal{F}_t\right]\tau(T_{i-1},T_i)\\ &= N \cdot \sum_{i=1}^N P(t,T_i)L(t,T_{i-1}, T_i)\tau(T_{i-1},T_i)\\ &= N \cdot \sum_{i=1}^N P(t,T_i)\frac{1}{\tau(T_{i-1},T_i)}\cdot\left(\frac{P(t,T_{i-1})}{P(t,T_{i})}-1\right)\tau(T_{i-1},T_i)\\ &= N \cdot \sum_{i=1}^N (P(t,T_i) - P(t,T_{i-1}))\\ &= -N P(t,T_0) + NP(t,T_N) \end{align*} \]
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 \(t\).
\[ \begin{align*} -V_{fix} + V_{floating} &= 0\\ S \cdot N \cdot \left(\sum_{i=1}^N P(t,T_i)\tau(T_{i-1},T_i)\right) &= -N P(t,T_0) + NP(t,T_N) \\ S(t,T_{0:N}) &= \frac{- P(t,T_0) + P(t,T_N)}{\left(\sum_{i=1}^N P(t,T_i)\tau(T_{i-1},T_i)\right)} \end{align*} \]
The Annuity Measure
The annuity is an asset that pays \(1\$\) on each coupon payment day of the swap, accrued according to the swap’s day count convention.
\[ A(t,T_{0:N}) = \left( \sum_{i=1}^N P(t,T_i)\tau(T_{i-1},T_i) \right) \]
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 \(V(T) = [S_T(S_T - K)]^{+}\)
Sell-side quant interviews are known to ask puzzles to price tricky payoffs like the power option \(V_T=(S_T^2 - K)1_{S_T > K}\), exchange options \(V_T = (S_2(T) - S_1(T))^{+}\) and quantos.
Change of numeraire and measure transformation provide an elegant way to price these payoffs quickly.
Suppose, we want to price the following payoff:
\[ V(T) = \max( S(T)(S(T) - K), 0 ) \]
By the risk-neutral pricing formula,
\[ V(t) = M(t) \mathbf{E}^{Q}\left[\frac{S(T)}{M(T)}(S(T) - K)1_{S_T > K}|\mathcal{F}_t\right] \]
The Radon-Nikodym derivative \(\frac{dQ}{dQ^S}\) is given by:
\[ L(t) = \frac{dQ}{dQ^S} = \frac{M(T)/M(t)}{S(T)/S(t)} = \frac{M(T)}{S(T)}\cdot \frac{S(t)}{M(t)} \]
Consequently, we can write:
\[ \begin{align*} V(t) &= M(t) \mathbf{E}^{Q^S}\left[\frac{dQ}{dQ^S}\cdot\frac{S(T)}{M(T)}(S(T) - K)1_{S_T > K}|\mathcal{F}_t\right]\\ &= M(t) \mathbf{E}^{Q^S}\left[\frac{M(T)}{S(T)}\cdot \frac{S(t)}{M(t)}\cdot\frac{S(T)}{M(T)}(S(T) - K)1_{S_T > K}|\mathcal{F}_t\right]\\ &= \mathbf{E}^{Q^S}\left[S(t)(S(T) - K)1_{S_T > K}|\mathcal{F}_t\right]\\ &= S(t)\mathbf{E}^{Q^S}\left[(S(T) - K)1_{S_T > K}|\mathcal{F}_t\right] \end{align*} \tag{14}\]
This looks much like the familiar European vanilla call option payoff, except that, the conditional expectation needs to be taken under the probability measure induced by the stock numeraire \((S,Q^S)\).
To find the dynamics of the stock price \(S(t)\) under the probability measure \(Q^S\), we use the intuitive fact, that, all normalized asset prices - the asset price \(X(t)\) deflated by the numeraire price \(S(t)\), \(\frac{X(t)}{S(t)}\) must be martingales under \(Q^S\). Thus, the price process \(M(t)/S(t)\) must be a martingale.
Let’s find the dynamics of the process \(\left(\frac{M(t)}{S(t)}\right)\).
Consider \(f(x,y) = \frac{x}{y}\). We have:
\[ \begin{align*} f_x = \frac{1}{y}, \quad f_y = -\frac{x}{y^2}\\ f_{xx} = 0, \quad f_{xy} = -\frac{1}{y^2}, f_{yy} = \frac{2x}{y^3} \end{align*} \]
We have:
\[ \begin{aligned} d\left(\frac{M_{t}}{S_{t}}\right) & =\frac{1}{S_{t}} dM( t) -\frac{M_{t}}{S_{t}^{2}} dS( t)\\ & -\frac{1}{S_{t}^{2}} dM( t) \cdotp dS( t) +\frac{1}{2} \cdot \frac{2M_{t}}{S_{t}^{3}}( dS_{t})^{2}\\ & =\frac{1}{S_{t}} rM( t) dt-\frac{M_{t}}{S_{t}^{2}}\left( rdt+\sigma dW^{\boxed{Q}}( t)\right)\\ & +\frac{M_{t}}{S_{t}^{3}} \sigma ^{2} S_{t}^{2} dt\\ & =\left(\frac{M_{t}}{S_{t}}\right)\left( rdt-rdt-\sigma dW^{\boxed{Q}}( t) +\sigma ^{2} dt\right)\\ & =-\sigma \left(\frac{M_{t}}{S_{t}}\right)\left( -\sigma dt+dW^{\boxed{Q}}( t)\right) \end{aligned} \]
But, we know that the process \((M_t/S_t)\) is a \(Q^S\)-martingale and should be driftless. Thus, we should have:
\[ d\left(\frac{M_t}{S_t}\right) = -\sigma \left(\frac{M_{t}}{S_{t}}\right)(0 \cdot dt + dW^{\boxed{Q^S}}( t)) \]
So, we perform the measure transformation:
\[ dW^{\boxed{Q^S}}(t) = -\sigma dt+dW^{\boxed{Q}}(t) \tag{15}\]
Consequently, \(Q^S\)-dynamics of the asset \(S(t)\) can be expressed as:
\[ \begin{align*} dS(t) &= rSdt + \sigma S dW^{\boxed{Q}}(t)\\ &=rSdt + \sigma S (dW^{\boxed{Q^S}}(t) + \sigma dt)\\ &=(r+\sigma^2)Sdt + \sigma S dW^{\boxed{Q^S}}(t) \end{align*} \]
So, \(S(t)\) is still a lognormal random variable and evolves according to:
\[ S(T) = S(t)\exp\left[\left(r+\frac{\sigma^2}{2}\right)(T-t) + \sigma (W^{\boxed{Q^S}}(T) - W^{\boxed{Q^S}}(t))\right] \tag{16}\]
The price of the option would be given by:
\[ V(t) = S(t)[S(t)\Phi(d_{+}) - K\Phi(d_{-})] \]
where
\[ \begin{align*} d_{+} &= \frac{\ln\left(\frac{S(t)}{K}\right) + (r + \frac{3\sigma^2}{2})(T-t)}{\sigma\sqrt{T - t}}\\ d_{-} &= \frac{\ln\left(\frac{S(t)}{K}\right) + (r + \frac{\sigma^2}{2})(T-t)}{\sigma\sqrt{T - t}} \end{align*} \]
Pricing the payoff \(V(T) = S_1(T)1_{S_2(T) > K}\)
Suppose the dynamics of two assets \(S_1(t)\) and \(S_2(t)\) are given by:
\[ \begin{align*} dS_1(t) &= rS_1(t)dt + \sigma_1 S_1 W_1^{Q}(t) \\ dS_2(t) &= rS_2(t)dt + \sigma_2 S_2 W_2^{Q}(t) \end{align*} \]
Assume that the two driving brownian motions are correlated and their instantaneous correlation is given by:
\[ dW_1^{Q}(t) dW_2^{Q}(t) = \rho dt \]
By the risk-neutral pricing formula, we have:
\[ V(t) = M(t)\mathbb{E}^{\boxed{Q}}\left[\frac{S_1(T)}{M(T)}1_{S_2(T) > K}\right] \]
By change of numeraire, we have:
\[ \begin{align*} V(t) &= M(t)\mathbb{E}^{\boxed{Q}}\left[\frac{S_1(T)}{M(T)}1_{S_2(T) > K}\right]\\ &= M(t) \mathbb{E}^{\boxed{Q^{S_1}}}\left[\frac{dQ}{dQ^{S_1}}\cdot \frac{S_1(T)}{M(T)}1_{S_2(T) > K}\right]\\ &= M(t) \mathbb{E}^{\boxed{Q^{S_1}}}\left[\frac{M(T)}{S_1(T)}\cdot \frac{S_1(t)}{M(t)}\cdot \frac{S_1(T)}{M(T)}1_{S_2(T) > K}\right]\\ &= M(t) \mathbb{E}^{\boxed{Q^{S_1}}}\left[\frac{M(T)}{S_1(T)}\cdot \frac{S_1(t)}{M(t)}\cdot \frac{S_1(T)}{M(T)}1_{S_2(T) > K}\right]\\ &=S_1(t)\mathbb{E}^{\boxed{Q^{S_1}}}[1_{S_2(T) > K}]\\ &=S_1(t)Q^{S_1}(S_2(T) > K) \end{align*} \]
Thus, we need to derive the \(Q^{S_1}\)-dynamics of the asset \(S_2\). Under the \(Q^{S_1}\)-measure, \(\left(\frac{S_2(t)}{S_1(t)}\right)\) must be a martingale.
Let \(f(x,y) = \frac{y}{x}\). We have:
\[ \begin{align*} f_x = -\frac{y}{x^2}, \quad f_y = \frac{1}{x} \\ f_{xx} = \frac{2y}{x^3}, \quad f_{xy} = -\frac{1}{x^2}, \quad f_{yy} = 0 \end{align*} \]
We have:
\[ \begin{aligned} d\left(\frac{S_{2}( t)}{S_{1}( t)}\right) & =-\frac{S_{2}}{S_{1}^{2}} dS_{1}( t) +\frac{1}{S_{1}} dS_{2}( t) +\frac{1}{2}\left(\frac{2S_{2}}{S_{1}^{3}}\right)( dS_{1}( t))^{2} -\frac{1}{S_{1}^{2}} dS_{1}( t) \cdot dS_{2}( t)\\ & =-\frac{S_{2}}{S_{1}^{2}}\left( rS_{1}( t) dt+\sigma _{1} S_{1} dW_{1}^{\boxed{Q}}( t)\right) +\frac{1}{S_{1}}\left( rS_{2}( t) dt+\sigma _{2} S_{2} dW_{2}^{\boxed{Q}}( t)\right)\\ & +\frac{S_{2}}{S_{1}^{3}} \sigma _{1}^{2} S_{1}^{2} dt-\frac{1}{S_{1}^{2}} S_{1} S_{2} \sigma _{1} \sigma _{2} \rho dt\\ & =\left(\frac{S_{2}}{S_{1}}\right)\left( -rdt-\sigma _{1} dW_{1}^{\boxed{Q}}( t) +rdt+\sigma _{2} dW_{2}^{\boxed{Q}}( t) +\sigma _{1}^{2} dt-\rho \sigma _{1} \sigma _{2} dt\right)\\ & =\left(\frac{S_{2}}{S_{1}}\right)\left( \sigma _{1}^{2} dt-\rho \sigma _{1} \sigma _{2} dt-\sigma _{1} dW_{1}^{\boxed{Q}}( t) +\sigma _{2} dW_{2}^{\boxed{Q}}( t)\right)\\ & \quad \left\{\text{ Applying the measure transformation } dW_{1}^{\boxed{Q}}( t) =dW_{1}^{\boxed{Q^{S_{1}}}}( t) +\sigma _{1} dt\right\}\\ & =\left(\frac{S_{2}}{S_{1}}\right)\left( \sigma _{1}^{2} dt-\rho \sigma _{1} \sigma _{2} dt-\sigma _{1}\left( dW_{1}^{\boxed{Q^{S_{1}}}}( t) +\sigma _{1} dt\right) +\sigma _{2} dW_{2}^{\boxed{Q}}( t)\right)\\ & =\left(\frac{S_{2}}{S_{1}}\right)\left( -\rho \sigma _{1} \sigma _{2} dt-\sigma _{1} dW_{1}^{\boxed{Q^{S_{1}}}}( t) +\sigma _{2} dW_{2}^{\boxed{Q}}( t)\right)\\ & =\left(\frac{S_{2}}{S_{1}}\right)\left[\left( -\rho \sigma _{1} \sigma _{2} dt+\sigma _{2} dW_{2}^{\boxed{Q}}( t)\right) -\sigma _{1} dW_{1}^{\boxed{Q^{S_{1}}}}( t)\right]\\ & =\sigma _{2}\left(\frac{S_{2}}{S_{1}}\right)\left[\left( -\rho \sigma _{1} dt+dW_{2}^{\boxed{Q}}( t)\right) -\frac{\sigma _{1}}{\sigma _{2}} dW_{1}^{\boxed{Q^{S_{1}}}}( t)\right] \end{aligned} \]
But, we must have:
\[ d\left(\frac{S_2}{S_1}\right) = \sigma_2 \left(\frac{S_2}{S_1}\right) \left[0 \cdot dt + dW_{2}^{\boxed{Q^{S_1}}}(t) -\frac{\sigma_{1}}{\sigma _{2}} dW_{1}^{\boxed{Q^{S_{1}}}}( t)\right] \]
This suggests the measure transformation:
\[ dW_2^{\boxed{Q^{S_1}}}(t) = -\rho \sigma _{1} dt + dW_{2}^{\boxed{Q}}( t) \]
So, finally, the model under the stock measure \(Q^{S_1}\) is given by:
\[ \begin{aligned} dS_1(t) &= (r + \sigma_1^2) S_1 dt + \sigma_1 S_1 dW_1^{\boxed{Q^{S_1}}}(t)\\ dS_2(t) &= (r +\rho \sigma_1) S_2 dt + \sigma_2 S_2 dW_2^{\boxed{Q^{S_1}}}(t)\\ dM(t) &= r M(t) dt \end{aligned} \]
The evolution of the second asset can be expressed as:
\[ \begin{aligned} S_2(T) &= S_2(t) \exp\left[\left(r + \rho \sigma_1 - \frac{\sigma_2^2}{2}\right)(T-t) + \sigma_2 (W_2^{\boxed{Q^{S_1}}}(T) - W_2^{\boxed{Q^{S_1}}}(t))\right] \end{aligned} \]
The option payoff can be simplified as:
\[ \begin{aligned} V( t) & =S_{1}( t) Q^{\boxed{S_{1}}}[ S_{2}( T) >K]\\ & =S_{1}( t) Q^{\boxed{S_{1}}}\left[ S_{2}( t)\exp\left\{\left( r+\rho \sigma _{1} -\frac{\sigma _{2}^{2}}{2}\right)( T-t) +\sigma _{2}( W_{2}( T) -W_{2}( t))\right\} >K\right]\\ & =S_{1}( t) Q^{\boxed{S_{1}}}\left[ S_{2}( t)\exp\left\{\left( r+\rho \sigma _{1} -\frac{\sigma _{2}^{2}}{2}\right)( T-t) -\sigma _{2}\sqrt{T-t} \cdot Z\right\} >K\right]\\ & =S_{1}( t) Q^{\boxed{S_{1}}}\left[\ln\frac{S_{2}( t)}{K} +\left( r+\rho \sigma _{1} -\frac{\sigma _{2}^{2}}{2}\right)( T-t) >\sigma _{2}\sqrt{T-t} \cdot Z\right]\\ & =S_{1}( t) Q^{\boxed{S_{1}}}\left[ Z< \frac{\ln\frac{S_{2}( t)}{K} +\left( r+\rho \sigma _{1} -\frac{\sigma _{2}^{2}}{2}\right)( T-t)}{\sigma _{2}\sqrt{T-t}}\right]\\ & =S_{1}( t) \Phi \left[\frac{\ln\frac{S_{2}( t)}{K} +\left( r+\rho \sigma _{1} -\frac{\sigma _{2}^{2}}{2}\right)( T-t)}{\sigma _{2}\sqrt{T-t}}\right] \end{aligned} \]
Pricing an assymetric power option \(V_T = \frac{1}{K^2}(K^3 - S_T^3)^{+}\)
References
- Girsanov, Numeraires and all that, Andrew Lesniewski.
- Interest-rate Models - Theory and Practice, Damiano Brigo and Fabio Mercurio.