What is the change of numeraire technique in derivatives pricing?

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AcadiFi · Accepted Answer

Change of numeraire is a Girsanov-theorem-powered technique that says: you can price derivatives using any tradeable asset as your reference currency, as long as you use the probability measure under which assets-divided-by-that-numeraire are martingales.

The generic formula: Price(0) = N(0) * E^N [Payoff / N(T)], where N is the numeraire and E^N is the expectation under the measure induced by N.

Why change numeraire? Because clever choices eliminate stochastic discounting. If the payoff depends on the numeraire itself, using N makes the discount factor trivial.

```mermaid
graph TD
    A[Complex Payoff] --> B[Choose Right Numeraire]
    B --> C[Discount Becomes Trivial]
    C --> D[Closed-Form Pricing]
```

Classic examples:
- Bond option: use T-bond as numeraire, forward price is martingale, Black-76 emerges
- Swaption: use swap annuity, forward swap rate is martingale, swaption Black formula emerges
- Quanto option: change to foreign-currency risk-neutral measure, adjust drift by quanto term
- LIBOR-in-arrears: change from T_i+1 to T_i forward measure, pick up convexity adjustment

The Radon-Nikodym derivative dQ^N / dQ^M = (N(T)/N(0)) * (M(0)/M(T)) handles the measure change mechanically. For any practitioner at a firm like Solstice Quant Labs, mastering change of numeraire is the difference between pricing exotic products cleanly and being stuck with slow Monte Carlo.

For CFA Level II, know the intuition: different numeraires simplify different products.

What is the change of numeraire technique in derivatives pricing?

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