Topic 4 – Measure / numeraire intuition

1. What are “measure” and “numeraire” in plain language?

Forget probability theory for a moment.

• Numeraire = the thing you measure value in.
  • In practice: a bank account or discount bond.
  • If your numeraire is a bank account $B_t$, then “1 unit of numeraire” is worth 1 by definition.
• Measure (here: probability measure) = which “lens” you use to assign probabilities to future scenarios.
  • Under the usual “real-world” lens (call it $\mathbb{P}$), expected drift of a risky asset includes risk premia.
  • Under a special “risk-neutral” lens (call it $\mathbb{Q}$), you tweak those probabilities so that discounted asset prices have no drift on average.

Key idea:

Choose a numeraire; switch to a measure under which every asset price divided by that numeraire is a fair game (martingale). Then prices today = discounted expectations under that measure.

2. Bank account as numeraire: why discounting is natural

Take a bank account:

$$B_t = e^{rt}$$

with constant short rate $r$.

Take a risky asset $S_t$ (stock, forward, whatever). Define the discounted price:

$$\tilde{S}_t = \frac{S_t}{B_t}$$

If markets are arbitrage-free, there exists a measure $\mathbb{Q}$ (risk-neutral measure) such that:

$$\mathbb{E}^{\mathbb{Q}}[\tilde{S}_T \mid \mathcal{F}_t] = \tilde{S}_t$$

for all $T \ge t$. This is the “fair game” property.

At time 0, this gives:

$$\tilde{S}_0 = \mathbb{E}^{\mathbb{Q}}[\tilde{S}_T] = \mathbb{E}^{\mathbb{Q}}\left[\frac{S_T}{B_T}\right]$$

Multiply by $B_0 = 1$:

$$S_0 = \mathbb{E}^{\mathbb{Q}}\left[ \frac{S_T}{B_T} \right]$$

For any cash flow $X_T$ at time $T$, same logic gives:

$$\text{Price at } 0 = \mathbb{E}^{\mathbb{Q}}\left[ \frac{X_T}{B_T} \right] = D(0,T)\, \mathbb{E}^{\mathbb{Q}}[X_T]$$

because $B_T = 1/D(0,T)$.

That is the risk-neutral pricing formula.

3. Risk-neutral measure vs real-world measure

In a simple diffusion world (GBM):

Real-world dynamics:

$$\frac{dS_t}{S_t} = \mu \, dt + \sigma\, dW_t^{\mathbb{P}}$$

• $\mu$ includes risk-free rate + risk premium.
• Under $\mathbb{P}$, discounted $S_t/B_t$ has drift (not a martingale).

Risk-neutral dynamics:

$$\frac{dS_t}{S_t} = r \, dt + \sigma\, dW_t^{\mathbb{Q}}$$

• Drift is exactly risk-free rate $r$.
• Discounted price $\tilde{S}_t = S_t/B_t$ has zero drift, only noise. It’s a martingale under $\mathbb{Q}$.

Practitioner summary:

“Under real-world measure, assets drift at risk-free + risk premia. Under risk-neutral measure, assets drift at risk-free only, so discounted prices are martingales. We price under risk-neutral, we backtest / risk-manage under real-world.”

4. Why this gives the usual pricing formulae

Take a payoff $X_T = f(S_T)$, e.g. a call:

$$X_T = (S_T - K)^+$$

Risk-neutral pricing formula with bank account numeraire:

$$\text{Option price at } 0 = \mathbb{E}^{\mathbb{Q}}\left[ \frac{X_T}{B_T} \right] = D(0,T)\, \mathbb{E}^{\mathbb{Q}}[(S_T - K)^+]$$

This is the conceptual backbone of:

• Black–Scholes / Black-76,
• Generic Monte Carlo pricing,
• Lattice models.

You always see the pattern:

Price = discount factor × risk-neutral expected payoff.

In commodities, same story:

• Choose money market as numeraire → discounting.
• Or choose a forward as numeraire → “forward measure” for that maturity. Under the $T$-forward measure, that forward’s price is a martingale.

5. Why practitioners actually care

1. Algorithm design:
   • When coding Monte Carlo, you simulate under risk-neutral dynamics: drift = r (plus carry, dividend, convenience yield).
   • Then you take discounted averages of payoffs.
2. Curve consistency:
   • Given a discount curve $D(0,T)$ and a forward curve $F(0,T)$, risk-neutral measure ensures: $$F(0,T) = \mathbb{E}^{\mathbb{Q}}[S_T]$$ once you’ve adjusted for carry.
3. Separation of duties:
   • Modelling real-world drifts ($\mu$) is about forecasting, risk and PnL.
   • Pricing often ignores $\mu$ and uses the risk-neutral world (drift = r + carry), because that is what no-arb arguments pin down.
4. Change of numeraire:
   • In interest-rate models, you often switch numeraires (money market, swap annuity) to simplify dynamics.
   • Measure changes track that switch and preserve the “discounted martingale” property.

6. Short interview soundbites

• On why discounting works:

  “If I discount by the bank account and work under the risk-neutral measure, each tradable asset divided by the bank account is a martingale. That’s why the present price is the discounted expected payoff under that measure.”

• On risk-neutral vs real-world:

  “Risk-neutral is a pricing device: it strips out risk premia and forces discounted prices to be fair games. Real-world is for risk and forecasting.”

• On numeraires:

  “The choice of numeraire is a choice of units. For each numeraire there’s a measure where numeraire-denominated prices are martingales. The money-market numeraire gives the usual risk-neutral measure; forward numeraires give forward measures, which are very convenient for interest-rate products.”