Change Of Numeraire

Pricing an option that pays off in a foreign currency, or an interest-rate option whose payoff depends on a bond price, using the standard risk-free-bank-account measure turns into a swamp of coupled stochastic discounting terms. Much of that mess is not really about the option — it's an artifact of which "ruler" you chose to measure value with. Pick a different ruler and the same problem can become almost trivially easy.

Measuring height in floors, not feet

If you want to know whether you're taller than a doorway, you don't need to know your height in feet and the doorway's height in feet separately — you just need the ratio. Express both in "doorway-heights" and the comparison becomes "are you more than 1?" A numeraire is that choice of measuring unit for money. Normally we price things in dollars, discounted by a bank account growing at the risk-free rate. But nothing forces that choice: you can instead price everything as a ratio to some other tradable asset — a bond, a foreign currency's bank account, an entire stock index — and as long as you're consistent, no-arbitrage pricing still works. Choose the numeraire that makes your payoff's ratio behave simply, and the pricing problem often collapses.

The formula

$$\frac{V_t}{N_t} = \mathbb{E}^{\mathbb{Q}^N}\!\left[\frac{V_T}{N_T} \;\middle|\; \mathcal{F}_t\right]$$

In plain English: the value of anything, $V_t$, divided by the chosen numeraire $N_t$, is a fair game (a martingale) under the probability measure $\mathbb{Q}^N$ associated with that numeraire — meaning its expected future value, still measured in numeraire units, equals its value today. Switch the numeraire from the bank account $B_t$ to something else, say a bond $P(t,T)$, and the probabilities $\mathbb{Q}^N$ themselves must change to keep the martingale property true; that reweighting is Girsanov's Theorem at work. The payoff for the effort: under the right numeraire, an ugly conditional expectation involving random interest rates can turn into a plain Black-Scholes-style formula.

Worked example 1: pricing a bond forward the easy way

You want the fair forward price for a bond, agreed today, for delivery at time $T$. Under the usual bank-account measure this requires the joint distribution of the discount factor and the bond price — messy, since both are random. Switch numeraire to the $T$-maturity zero-coupon bond $P(t,T)$ itself. Under this "T-forward measure," the forward price of any asset is simply its current price divided by the numeraire bond's current price:

$$F_0 = \frac{P_0(\text{bond price today})}{P(0,T)} \quad\Rightarrow\quad \text{no expectation needed at all}$$

If today's underlying bond trades at 98 and the discount bond $P(0,T)$ (paying $1 at $T$) trades at 0.95, the forward price is $98/0.95 = 103.16$. What would have required modeling the joint randomness of two curves became one division, because the T-forward measure was built precisely to make this ratio deterministic.

Worked example 2: a quanto option, numerically

A US investor wants a call on a Japanese stock, but wants the payoff settled in dollars at a fixed exchange rate agreed today (a "quanto") rather than the fluctuating spot rate — this removes currency risk from the payoff but couples the stock's volatility to the correlation between the stock and the yen. Under the domestic risk-neutral measure directly, you'd need the joint dynamics of the stock and the exchange rate. Switch to the foreign numeraire (the yen bank account) to price the option in yen terms first — an ordinary Black-Scholes problem, since currency has been numeraired away — then convert the answer using a single, well-known adjustment to the stock's drift, $\mu_{\text{quanto}} = \mu - \rho\sigma_S\sigma_{FX}$, where $\rho$ is the correlation between the stock and the exchange rate. If $\sigma_S = 25\%$, $\sigma_{FX} = 10\%$, and $\rho = -0.3$, the drift adjustment is $-(-0.3)(0.25)(0.10) = 0.0075$, or 0.75% a year added to the stock's growth rate before you price it as a plain call. One line of correlation algebra replaces a two-dimensional joint distribution.

What this means in practice

Every interest-rate derivatives desk lives inside numeraire changes: the T-forward measure prices bond options and caps, the swap-annuity measure prices Swaptions, and the "spot LIBOR" measure prices exotic path-dependent rate products. Cross-currency and quanto structuring desks change numeraire routinely to strip out correlation terms one at a time instead of solving a tangled multi-factor PDE from scratch.