Black–Scholes PDE via Change of Numéraire¶

This derivation removes the physical drift \(\mu\) by unit normalization: expressing prices in units of the stock shifts the drift under Girsanov's theorem, and the martingale condition in the new units produces the PDE.

Recall (see § Risk-Neutral Pricing): under \(\mathbb{Q}\) the discounted stock \(e^{-rt}S_t\) is already a \(\mathbb{Q}\)-martingale, so \(Z_t = S_te^{-rt}/S_0\) defines a valid change of measure to \(\mathbb{Q}^S\). The general change-of-numéraire framework is developed in § Numéraire; this page is its concrete application to Black–Scholes.

Mechanism: Changing the Unit of Account in Two States¶

Numéraire change is fundamentally probability reweighting. Consider a one-period market with stock \(S_1 \in \{S^u, S^d\}\) and bond \(B_1 = 1+r\), both starting at \(S_0\) and \(B_0 = 1\). Pricing in dollars uses the risk-neutral measure \(\mathbb{Q}\), under which \(S_t/B_t\) is a martingale. Pricing in shares uses a new measure \(\mathbb{Q}^S\) obtained by reweighting:

The state where \(S_1\) is high gets up-weighted, the state where \(S_1\) is low gets down-weighted, by exactly the factor that makes \(1/S_t \cdot B_t = B_t/S_t\) a \(\mathbb{Q}^S\)-martingale. Asset prices in units of stock — \(V/S\) — are then the new "discounted prices." The continuous derivation below is the same operation in differential form: \(Z_t = S_te^{-rt}/S_0\) tilts \(\mathbb{Q}\) toward high-\(S_T\) paths, the new martingale condition is \(V/S\) rather than \(e^{-rt}V\), and the resulting PDE for the option price \(V\) turns out to be identical to the one obtained from the bond numéraire — the price is invariant to the chosen unit.

Setup¶

We work in the standard Black–Scholes model on a filtered probability space \((\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \ge 0}, \mathbb{P})\) satisfying the usual conditions. Under the physical measure \(\mathbb{P}\):

with constant parameters \(\mu, \sigma > 0\), and a money market account \(B_t = e^{rt}\) with constant rate \(r\). Under the risk-neutral measure \(\mathbb{Q}\) (the \(B\)-martingale measure):

We take the stock \(S_t\) as numéraire. This is valid because \(S_t\) is strictly positive and pays no dividends, so its discounted price \(S_t e^{-rt}\) is a \(\mathbb{Q}\)-martingale. (When the stock pays dividends, the ex-dividend price alone is not a valid numéraire; one must use the total return process instead—see Exercise 5.) We derive the associated measure \(\mathbb{Q}^S\), the stock dynamics under \(\mathbb{Q}^S\), and ultimately the Black–Scholes PDE.

Step 1: Density Process and Girsanov Transformation¶

Radon–Nikodym Derivative¶

Recall (see § Numéraire): the density process from \(\mathbb{Q}\) to \(\mathbb{Q}^S\) is the ratio of normalized numéraires,

Dynamics of Z_t¶

Itô's product rule applied to \(Z_t = (e^{-rt}/S_0)S_t\) gives

after the \(r\,dt\) terms cancel. Thus \(Z_t = \exp(\sigma W^{\mathbb{Q}}_t - \tfrac{1}{2}\sigma^2 t)\) is a true \(\mathbb{Q}\)-martingale (constant integrand \(\sigma\) trivially satisfies Novikov).

Girsanov Transformation¶

Recall (see § Girsanov's Theorem): with density volatility \(\sigma\), the \(\mathbb{Q}^S\)-Brownian motion is

Step 2: Dynamics Under the Stock Measure¶

Stock Dynamics¶

Substitute \(dW^{\mathbb{Q}}_t = dW^S_t + \sigma\, dt\):

Under \(\mathbb{Q}^S\), the stock drift increases from \(r\) to \(r + \sigma^2\). This is natural: the density \(Z_T \propto S_T e^{-rT}\) up-weights high-\(S_T\) paths, increasing the drift under the new measure.

Verification: B_t / S_t Is a Q^S-Martingale¶

The defining property of \(\mathbb{Q}^S\) is that all asset prices normalized by \(S_t\) are martingales. We verify this for \(B_t / S_t\).

Apply Itô's formula to \(f(B_t, S_t) = B_t / S_t\), using \(dB_t = rB_t\, dt\) and the \(\mathbb{Q}^S\)-dynamics of \(S_t\):

Computing each term:

Collecting the drift:

The drift vanishes identically. Therefore

This driftless SDE shows \(B_t/S_t\) is a local \(\mathbb{Q}^S\)-martingale. To confirm it is a true martingale, solve explicitly: \(B_t/S_t = (B_0/S_0)\exp(-\sigma W_t^S - \frac{1}{2}\sigma^2 t)\), which is a Doléans-Dade exponential with constant integrand \(-\sigma\). Since \(\sigma\) is constant, \(\mathbb{E}^{\mathbb{Q}^S}[B_t/S_t] = B_0/S_0\) for all \(t\) (by the moment generating function of a Gaussian), so the process is a true \(\mathbb{Q}^S\)-martingale. \(\checkmark\)

Step 3: The Martingale Condition for the Normalized Price¶

Under \(\mathbb{Q}^S\), any traded asset price normalized by \(S_t\) is a martingale.

Assume \(V \in C^{1,2}([0,T) \times (0,\infty))\) with polynomial growth (sufficient for Itô's formula and for the local martingale below to be a true martingale, by the standard GBM moment bounds). Under \(\mathbb{Q}^S\), the normalized price

must be a \(\mathbb{Q}^S\)-martingale — this is no-arbitrage in the chosen numéraire. The martingale condition requires zero drift:

where the generator \(\mathcal{L}^S\) is:

This gives the PDE for the normalized price:

Step 4: Transform Back to V¶

The PDE \((\star)\) governs the normalized price \(u = V/S\) under the stock measure. Now we translate this back into a PDE for the option price \(V\) itself, by expressing the derivatives of \(u\) in terms of \(V\) and substituting.

Since \(V = S \cdot u\), express the derivatives of \(u\) in terms of \(V\):

Substitute into \((\star)\) and multiply through by \(S\). Collecting terms:

The \(\sigma^2\) terms from the stock-measure drift and the Itô correction cancel perfectly, leaving:

This is the Black–Scholes PDE—derived without explicitly constructing a replicating portfolio or hedging argument. (The underlying arbitrage-free pricing theory still relies on completeness and replication in the background; the point is that the derivation method avoids them.)

Byproduct: Black–Scholes Formula as Two Probabilities¶

The stock-numéraire approach also yields an elegant decomposition of the Black–Scholes call price.

Derivation¶

For a European call with payoff \((S_T - K)^+\), the stock-numéraire pricing formula gives

Splitting the expectation:

For the second term, apply the abstract Bayes formula. Since \(Z_t = S_t e^{-rt}/S_0\) and \(Z_T = S_T e^{-rT}/S_0\):

Substituting \(S_0 Z_t = S_t e^{-rt}\):

Therefore:

Identification with N(d_1) and N(d_2)¶

Under \(\mathbb{Q}\), the stock drift is \(r\), so \(\ln S_T \mid \mathcal{F}_t \sim \mathcal{N}\!\left(\ln S_t + (r - \tfrac{1}{2}\sigma^2)\tau,\; \sigma^2\tau\right)\) where \(\tau = T - t\). Therefore

Under \(\mathbb{Q}^S\), the stock drift is \(r + \sigma^2\), so \(\ln S_T \mid \mathcal{F}_t \sim \mathcal{N}\!\left(\ln S_t + (r + \tfrac{1}{2}\sigma^2)\tau,\; \sigma^2\tau\right)\) (the Itô correction gives mean \(r + \sigma^2 - \frac{1}{2}\sigma^2 = r + \frac{1}{2}\sigma^2\)). Therefore

Combining:

The Black–Scholes formula emerges with a transparent probabilistic interpretation: \(N(d_1)\) is the probability of exercise under the stock measure \(\mathbb{Q}^S\), and \(N(d_2)\) is the probability of exercise under the risk-neutral measure \(\mathbb{Q}\).

Summary of the Derivation¶

The following diagram summarizes the logical flow:

flowchart LR
    A["Money-market frame
(numéraire B, measure Q)
drift = r"] -->|"Radon–Nikodym
Z = Se⁻ʳᵗ/S₀"| B["Stock frame
(numéraire S, measure Q^S)
drift = r + σ²"]
    B --> C["Martingale condition
u = V/S has zero drift"]
    C --> D["PDE for u"]
    D -->|"transform back (× S)
σ² terms cancel"| E["Black–Scholes PDE
(invariant)"]

The logical structure is:

1. Choose numéraire: take \(N_t = S_t\) (the stock).
2. Construct the density process: \(Z_t = S_t e^{-rt}/S_0\), verify \(dZ_t/Z_t = \sigma\, dW^{\mathbb{Q}}_t\).
3. Apply Girsanov: \(W^S_t = W^{\mathbb{Q}}_t - \sigma t\) is a \(\mathbb{Q}^S\)-Brownian motion.
4. Derive stock dynamics under \(\mathbb{Q}^S\): drift becomes \(r + \sigma^2\).
5. Impose the martingale condition: \(u = V/S\) satisfies \(\partial_t u + \mathcal{L}^S u = 0\).
6. Transform back to \(V\): the \(\sigma^2\) terms cancel, yielding the Black–Scholes PDE.

Recall (see § Self-Financing Replication): replication makes the strategy \((\alpha_t,\beta_t)\) explicit; here the same strategy is implicit, and the PDE is read off from the martingale condition for \(V/S\) rather than from \(\alpha_t = V_S\). The two methods rest on the same FTAP foundation — see § FTAP.

References¶

• Geman, H., El Karoui, N., and Rochet, J.-C. (1995). Changes of numéraire, changes of probability measure, and option pricing. Journal of Applied Probability, 32(2), 443–458.

• Shreve, S. E. (2004). Stochastic Calculus for Finance II: Continuous-Time Models. Springer.

• Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus. 2nd edition, Springer.

• Björk, T. (2009). Arbitrage Theory in Continuous Time. 3rd edition, Oxford University Press.

Exercises¶

Exercise 1. Verify that the density process \(Z_t = S_t e^{-rt}/S_0\) is a \(\mathbb{Q}\)-martingale by showing that \(dZ_t / Z_t = \sigma \, dW_t^{\mathbb{Q}}\) (i.e., the drift vanishes under \(\mathbb{Q}\)). Compute \(\mathbb{E}^{\mathbb{Q}}[Z_T]\) and confirm it equals 1.

Exercise 2. Under the stock measure \(\mathbb{Q}^S\), the process \(u_t = V(t, S_t)/S_t\) is a martingale. Apply Ito's lemma to \(u_t\) using the stock dynamics under \(\mathbb{Q}^S\) and show that setting the drift of \(u_t\) to zero yields the Black-Scholes PDE for \(V\).

Exercise 3. Explain why \(d_1 \neq d_2\) in the Black-Scholes formula from the change-of-numeraire perspective. Under \(\mathbb{Q}^S\), what is the distribution of \(\ln S_T\), and how does it differ from the distribution under \(\mathbb{Q}\)?

Exercise 4. The change-of-numeraire approach does not require constructing a self-financing portfolio. Compare the logical structure of this derivation with the delta-hedging derivation. In particular, identify which assumption (no-arbitrage, completeness, or Girsanov's theorem) plays the role that the hedging argument plays in the classical approach.

Exercise 5. Suppose the stock pays a continuous dividend yield \(q\). The ex-dividend stock price alone is not a valid numéraire (its discounted price is not a \(\mathbb{Q}\)-martingale). Instead, use the total return process \(\tilde{S}_t = e^{qt}S_t\) as numéraire, normalize by \(\tilde{S}_t\), and show that the Black–Scholes PDE becomes \(\frac{\partial V}{\partial t} + (r-q)S\frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} - rV = 0\).

Exercise 6. Price a European call option directly using the stock measure \(\mathbb{Q}^S\). Show that the call price can be written as \(C = S\,\mathbb{Q}^S(S_T > K) - Ke^{-r\tau}\,\mathbb{Q}(S_T > K)\), evaluate both probabilities using the distributions of \(\ln S_T\) under each measure, and verify that the result is \(C = S\,\mathcal{N}(d_1) - Ke^{-r\tau}\,\mathcal{N}(d_2)\).