Change of Numeraire: Alternative Derivation of Black-Scholes

The heat equation and Feynman-Kac sections derived the Black-Scholes formula \(C = S\mathcal{N}(d_1) - Ke^{-r\tau}\mathcal{N}(d_2)\) by solving the PDE and computing a risk-neutral expectation. But a question remains: why does the formula split into exactly two terms, each involving a normal CDF? The PDE route gives no clear answer --- the split emerges from completing the square in a Gaussian integral.

The change of numeraire technique answers this question directly. It shows that each term is a probability under a different measure: \(\mathcal{N}(d_2)\) is the risk-neutral probability that \(S_T > K\), while \(\mathcal{N}(d_1)\) is the same event's probability under the stock measure. Unlike the PDE methods developed earlier, this approach is entirely probabilistic --- it uses measure changes rather than differential equations. While the standard risk-neutral approach uses the money market account as numeraire, changing to alternative numeraires (e.g., the stock itself) simplifies certain pricing problems and reveals the measure-theoretic structure hidden inside familiar formulas.

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Numeraire and Pricing Measures

1. Definition of Numeraire

A numeraire \(N_t\) is a strictly positive traded asset used as a unit of account. All asset prices are expressed relative to the numeraire:

\[ \text{Relative price} = \frac{S_t}{N_t} \]

Key property: In an arbitrage-free market, there exists a probability measure (called the numeraire measure or \(N\)-measure) under which all asset prices relative to \(N_t\) are martingales.

2. Standard Risk-Neutral Measure

In the Black-Scholes framework, the standard numeraire is the money market account:

\[ B_t = e^{rt} \]

Under the risk-neutral measure \(\mathbb{Q}\): - The relative price \(S_t/B_t = e^{-rt}S_t\) is a martingale - This gives the standard BS dynamics: \(dS_t = rS_t dt + \sigma S_t dW_t^{\mathbb{Q}}\)

Option pricing formula:

\[ V_0 = \mathbb{E}^{\mathbb{Q}}\left[\frac{V_T}{B_T}\right] = e^{-rT}\mathbb{E}^{\mathbb{Q}}[V_T] \]

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General Numeraire Change

1. Fundamental Theorem

Numeraire Change Theorem: Let \(N_t\) be any numeraire (strictly positive traded asset). There exists a unique equivalent probability measure \(\mathbb{Q}^N\) under which all asset prices relative to \(N_t\) are martingales.

Specifically, for any traded asset \(S_t\):

\[ \frac{S_t}{N_t} = \mathbb{E}^{\mathbb{Q}^N}\left[\frac{S_T}{N_T} \Big| \mathcal{F}_t\right] \]

2. Radon-Nikodym Derivative

The measure \(\mathbb{Q}^N\) is related to the standard risk-neutral measure \(\mathbb{Q}\) via:

\[ \frac{d\mathbb{Q}^N}{d\mathbb{Q}}\Big|_{\mathcal{F}_T} = \frac{N_T/B_T}{\mathbb{E}^{\mathbb{Q}}[N_T/B_T]} \]

Intuition: We reweight paths according to the terminal value of the numeraire (discounted).

3. Girsanov Connection

The change of numeraire induces a change of Brownian motion via Girsanov's theorem. If under \(\mathbb{Q}\):

\[ dN_t = \mu_N N_t dt + \sigma_N N_t dW_t^{\mathbb{Q}} \]

then under \(\mathbb{Q}^N\):

\[ dW_t^{\mathbb{Q}^N} = dW_t^{\mathbb{Q}} - \sigma_N dt \]

Equivalently, \(dW_t^{\mathbb{Q}} = dW_t^{\mathbb{Q}^N} + \sigma_N dt\). The drift adjustment reflects the covariance between the asset and the numeraire.

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Stock Numeraire and Forward Measure

1. Setup: Stock as Numeraire

Choose \(N_t = S_t\) (the underlying stock itself).

The associated measure \(\mathbb{Q}^S\) is called the stock measure or forward measure.

2. Relative Prices Under Q^S

Under the stock measure, all assets relative to \(S_t\) are martingales.

Money market account relative to stock:

\[ \frac{B_t}{S_t} = \mathbb{E}^{\mathbb{Q}^S}\left[\frac{B_T}{S_T} \Big| \mathcal{F}_t\right] \]

Strike relative to stock:

\[ \frac{K}{S_t} = \mathbb{E}^{\mathbb{Q}^S}\left[\frac{K}{S_T} \Big| \mathcal{F}_t\right] \]

3. Radon-Nikodym Derivative

The stock measure is related to \(\mathbb{Q}\) by:

\[ \frac{d\mathbb{Q}^S}{d\mathbb{Q}}\Big|_{\mathcal{F}_T} = \frac{S_T e^{-rT}}{S_0} \]

4. Brownian Motion Under Q^S

If \(dS_t = rS_t dt + \sigma S_t dW_t^{\mathbb{Q}}\) under \(\mathbb{Q}\), then by Girsanov:

\[ dW_t^{\mathbb{Q}^S} = dW_t^{\mathbb{Q}} - \sigma dt \]

Substituting \(dW_t^{\mathbb{Q}} = dW_t^{\mathbb{Q}^S} + \sigma dt\) into the stock dynamics:

\[ dS_t = rS_t dt + \sigma S_t(dW_t^{\mathbb{Q}^S} + \sigma dt) = (r + \sigma^2)S_t dt + \sigma S_t dW_t^{\mathbb{Q}^S} \]

Key property: The ratio \(e^{rt}/S_t\) is a martingale under \(\mathbb{Q}^S\), since its drift vanishes.

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Black-Scholes via Stock Numeraire

1. Call Option Valuation

We want to price a European call with payoff \((S_T - K)^+\).

Step 1: Express payoff in numeraire units

Divide by \(S_T\):

\[ \frac{(S_T - K)^+}{S_T} = \left(1 - \frac{K}{S_T}\right)^+ = \left(1 - \frac{K}{S_T}\right)\mathbf{1}_{\{S_T > K\}} \]

Step 2: Change to stock measure

By the numeraire change theorem:

\[ \frac{C_0}{S_0} = \mathbb{E}^{\mathbb{Q}^S}\left[\frac{C_T}{S_T}\right] = \mathbb{E}^{\mathbb{Q}^S}\left[\left(1 - \frac{K}{S_T}\right)\mathbf{1}_{\{S_T > K\}}\right] \]

Step 3: Decompose expectation

\[ \frac{C_0}{S_0} = \mathbb{E}^{\mathbb{Q}^S}[\mathbf{1}_{\{S_T > K\}}] - K\mathbb{E}^{\mathbb{Q}^S}\left[\frac{1}{S_T}\mathbf{1}_{\{S_T > K\}}\right] \]

2. First Term: Q^S(S_T > K)

Under \(\mathbb{Q}^S\), the stock dynamics have drift \((r + \sigma^2)\), so by Ito's formula:

\[ S_T = S_0 \exp\left(\left(r + \frac{1}{2}\sigma^2\right)T + \sigma W_T^{\mathbb{Q}^S}\right) \]

Therefore:

\[ S_T > K \iff \sigma W_T^{\mathbb{Q}^S} > \ln(K/S_0) - \left(r + \frac{1}{2}\sigma^2\right)T \]

Dividing by \(\sigma\sqrt{T}\) and using \(W_T^{\mathbb{Q}^S}/\sqrt{T} \sim \mathcal{N}(0,1)\):

\[ S_T > K \iff \frac{W_T^{\mathbb{Q}^S}}{\sqrt{T}} > -\frac{\ln(S_0/K) + \left(r + \frac{1}{2}\sigma^2\right)T}{\sigma\sqrt{T}} = -d_1 \]

where

\[ d_1 = \frac{\ln(S_0/K) + (r + \frac{1}{2}\sigma^2)T}{\sigma\sqrt{T}} \]

Hence:

\[ \mathbb{Q}^S(S_T > K) = \mathcal{N}(d_1) \]

3. Second Term: Change to Q

For the second term, we use:

\[ \mathbb{E}^{\mathbb{Q}^S}\left[\frac{1}{S_T}\mathbf{1}_{\{S_T > K\}}\right] = \mathbb{E}^{\mathbb{Q}}\left[\frac{d\mathbb{Q}^S}{d\mathbb{Q}} \cdot \frac{1}{S_T}\mathbf{1}_{\{S_T > K\}}\right] \]

\[ = \mathbb{E}^{\mathbb{Q}}\left[\frac{S_T e^{-rT}}{S_0} \cdot \frac{1}{S_T}\mathbf{1}_{\{S_T > K\}}\right] = \frac{e^{-rT}}{S_0}\mathbb{Q}(S_T > K) \]

Under \(\mathbb{Q}\):

\[ \mathbb{Q}(S_T > K) = \mathcal{N}(d_2) \]

where \(d_2 = d_1 - \sigma\sqrt{T}\).

4. Final Result

Combining:

\[ \frac{C_0}{S_0} = \mathcal{N}(d_1) - K \cdot \frac{e^{-rT}}{S_0}\mathcal{N}(d_2) \]

\[ \boxed{C_0 = S_0\mathcal{N}(d_1) - Ke^{-rT}\mathcal{N}(d_2)} \]

This is the Black-Scholes formula, derived via stock numeraire.

Key insight: The term \(\mathcal{N}(d_1)\) arises naturally as the probability under the stock measure, not the risk-neutral measure.

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Foreign Exchange Options

1. Setup: Quanto Options

Consider an option on foreign exchange rate \(X_t\) (domestic per foreign).

Two numeraires: - Domestic money market: \(B_t^d = e^{r_d t}\) - Foreign money market: \(B_t^f = e^{r_f t}\)

Exchange rate dynamics under domestic risk-neutral measure \(\mathbb{Q}^d\):

\[ dX_t = (r_d - r_f)X_t dt + \sigma X_t dW_t^{\mathbb{Q}^d} \]

2. Foreign Numeraire Measure

Choose numeraire \(N_t = X_t B_t^f = X_t e^{r_f t}\) (foreign money market converted to domestic).

The Radon-Nikodym derivative:

\[ \frac{d\mathbb{Q}^f}{d\mathbb{Q}^d}\Big|_{\mathcal{F}_T} = \frac{X_T e^{r_f T - r_d T}}{X_0} \]

Under \(\mathbb{Q}^f\):

\[ dW_t^{\mathbb{Q}^f} = dW_t^{\mathbb{Q}^d} - \sigma dt \]

Exchange rate becomes:

\[ dX_t = (r_d - r_f + \sigma^2)X_t dt + \sigma X_t dW_t^{\mathbb{Q}^f} \]

3. Call on FX Rate

Using the foreign measure:

\[ C_0 = X_0 e^{-r_f T}\mathcal{N}(d_1) - K e^{-r_d T}\mathcal{N}(d_2) \]

where

\[ d_1 = \frac{\ln(X_0/K) + (r_d - r_f + \frac{1}{2}\sigma^2)T}{\sigma\sqrt{T}}, \quad d_2 = d_1 - \sigma\sqrt{T} \]

This is the Garman-Kohlhagen formula for FX options.

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Summary: When to Use Each Numeraire

Numeraire	Measure	Application	Advantage
Money market \(B_t\)	Risk-neutral \(\mathbb{Q}\)	Standard options	Familiar, drift = \(r\)
Stock \(S_t\)	Stock measure \(\mathbb{Q}^S\)	Forward contracts	\(\mathcal{N}(d_1)\) as probability
Zero-coupon bond \(P(t,T)\)	Forward measure \(\mathbb{Q}^T\)	Interest rate options	Simplifies swap pricing
Foreign money market	Foreign measure \(\mathbb{Q}^f\)	FX options	Symmetry between currencies

General principle: Choose the numeraire that eliminates drift from the payoff-relevant dynamics.

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Connection to Other Solution Methods

In the context of solving the Black-Scholes PDE, change of numeraire is:

Not a PDE technique (like Fourier or separation of variables), but rather a probabilistic reinterpretation that: - Avoids solving PDEs entirely - Uses martingale representation instead - Provides intuition for why certain probabilities appear in formulas

Relation to Feynman-Kac: Both express PDE solutions as expectations, but: - Feynman-Kac uses the original measure and discounting - Numeraire change uses different measures without explicit discounting

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Summary

The change of numeraire technique provides an elegant alternative to standard risk-neutral pricing:

1. Key idea: Price assets relative to any traded numeraire, not just the money market

2. Mathematical tool: Radon-Nikodym derivative relating different numeraire measures

3. Girsanov connection: Numeraire change induces Brownian motion drift change

4. Black-Scholes derivation: Stock numeraire gives \(\mathcal{N}(d_1)\) direct probabilistic interpretation

5. Advantages:

6. Conceptual clarity (e.g., \(d_1\) as stock-measure probability)
7. Simplifies certain exotic options

8. Natural framework for FX and interest rate derivatives

9. Limitation: Requires understanding of measure theory; not always simpler than PDE methods

In the operator framework of the introduction, the change of numeraire expresses the pricing semigroup \(\mathcal{P}_\tau\) as an expectation under a different probability measure, revealing the structural meaning of each term in the Black--Scholes formula.

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Exercises

Exercise 1. Let \(N_t = e^{rt}\) be the money market account and \(\mathbb{Q}\) the associated risk-neutral measure. Verify the Radon-Nikodym derivative formula by showing that \(\frac{d\mathbb{Q}^S}{d\mathbb{Q}}\big|_{\mathcal{F}_T} = \frac{S_T e^{-rT}}{S_0}\) has expectation 1 under \(\mathbb{Q}\).

Solution to Exercise 1

We need to show that \(\mathbb{E}^{\mathbb{Q}}\left[\frac{S_T e^{-rT}}{S_0}\right] = 1\).

Under \(\mathbb{Q}\), the discounted stock price \(e^{-rt}S_t\) is a martingale, so:

\[ \mathbb{E}^{\mathbb{Q}}[e^{-rT}S_T] = e^{-r \cdot 0}S_0 = S_0 \]

Dividing both sides by \(S_0\):

\[ \mathbb{E}^{\mathbb{Q}}\left[\frac{S_T e^{-rT}}{S_0}\right] = 1 \]

Alternatively, using the explicit formula \(S_T = S_0 \exp\left((r - \frac{1}{2}\sigma^2)T + \sigma W_T^{\mathbb{Q}}\right)\):

\[ \frac{S_T e^{-rT}}{S_0} = \exp\left(-\frac{1}{2}\sigma^2 T + \sigma W_T^{\mathbb{Q}}\right) \]

This is the stochastic exponential \(\mathcal{E}(\sigma W^{\mathbb{Q}})_T\). Using the moment generating function of the normal distribution with \(W_T^{\mathbb{Q}} \sim \mathcal{N}(0, T)\):

\[ \mathbb{E}^{\mathbb{Q}}\left[\exp\left(-\frac{1}{2}\sigma^2 T + \sigma W_T^{\mathbb{Q}}\right)\right] = \exp\left(-\frac{1}{2}\sigma^2 T + \frac{1}{2}\sigma^2 T\right) = 1 \]

This confirms that the Radon-Nikodym derivative has unit expectation, as required for a valid change of measure.

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Exercise 2. Under the stock measure \(\mathbb{Q}^S\), the discounted bond price \(B_t / S_t = e^{rt}/S_t\) is a martingale. Verify this explicitly by computing \(d(e^{rt}/S_t)\) using Ito's lemma and the stock dynamics under \(\mathbb{Q}^S\), and confirming that the drift vanishes.

Solution to Exercise 2

Under \(\mathbb{Q}^S\), the stock dynamics are \(dS_t = (r + \sigma^2)S_t \, dt + \sigma S_t \, dW_t^{\mathbb{Q}^S}\).

Define \(Y_t = e^{rt}/S_t\). Apply Ito's lemma to \(f(t, S) = e^{rt}/S\).

For \(g(S) = 1/S\), we have \(g'(S) = -1/S^2\) and \(g''(S) = 2/S^3\). By Ito's lemma:

\[ d(1/S_t) = -\frac{1}{S_t^2} \, dS_t + \frac{1}{2} \cdot \frac{2}{S_t^3}(dS_t)^2 \]

Substituting \(dS_t = (r + \sigma^2)S_t \, dt + \sigma S_t \, dW_t^{\mathbb{Q}^S}\) and \((dS_t)^2 = \sigma^2 S_t^2 \, dt\):

\[ d(1/S_t) = \frac{-(r + \sigma^2) + \sigma^2}{S_t} \, dt - \frac{\sigma}{S_t} \, dW_t^{\mathbb{Q}^S} = \frac{-r}{S_t} \, dt - \frac{\sigma}{S_t} \, dW_t^{\mathbb{Q}^S} \]

Therefore:

\[ dY_t = \frac{re^{rt}}{S_t} \, dt + e^{rt}\left(\frac{-r}{S_t} \, dt - \frac{\sigma}{S_t} \, dW_t^{\mathbb{Q}^S}\right) = -\sigma Y_t \, dW_t^{\mathbb{Q}^S} \]

The drift vanishes, confirming that \(Y_t = e^{rt}/S_t\) is a martingale under \(\mathbb{Q}^S\). \(\square\)

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Exercise 3. Derive the Garman-Kohlhagen formula for a European call on a foreign exchange rate. Starting from the FX dynamics \(dX_t = (r_d - r_f)X_t \, dt + \sigma X_t \, dW_t^{\mathbb{Q}^d}\), use the change of numeraire to the foreign money market and show that the call price is \(C_0 = X_0 e^{-r_f T}\mathcal{N}(d_1) - Ke^{-r_d T}\mathcal{N}(d_2)\), where \(d_1 = \frac{\ln(X_0/K) + (r_d - r_f + \frac{1}{2}\sigma^2)T}{\sigma\sqrt{T}}\).

Solution to Exercise 3

Under \(\mathbb{Q}^d\), the FX rate satisfies \(dX_t = (r_d - r_f)X_t \, dt + \sigma X_t \, dW_t^{\mathbb{Q}^d}\).

Step 1: Choose numeraire. Take the foreign money market account converted to domestic currency: \(N_t = X_t e^{r_f t}\). The Radon-Nikodym derivative is:

\[ \frac{d\mathbb{Q}^f}{d\mathbb{Q}^d}\Big|_{\mathcal{F}_T} = \frac{X_T e^{(r_f - r_d)T}}{X_0} \]

Step 2: Girsanov shift. Under \(\mathbb{Q}^f\): \(dW_t^{\mathbb{Q}^f} = dW_t^{\mathbb{Q}^d} - \sigma \, dt\), so \(dW_t^{\mathbb{Q}^d} = dW_t^{\mathbb{Q}^f} + \sigma \, dt\).

Under \(\mathbb{Q}^f\), the FX dynamics become:

\[ dX_t = (r_d - r_f)X_t \, dt + \sigma X_t(dW_t^{\mathbb{Q}^f} + \sigma \, dt) = (r_d - r_f + \sigma^2)X_t \, dt + \sigma X_t \, dW_t^{\mathbb{Q}^f} \]

Step 3: Price the call. The call payoff is \((X_T - K)^+\). Using the numeraire \(N_t = X_t e^{r_f t}\):

\[ \frac{C_0}{N_0} = \mathbb{E}^{\mathbb{Q}^f}\left[\frac{(X_T - K)^+}{N_T}\right] = \mathbb{E}^{\mathbb{Q}^f}\left[\frac{(X_T - K)^+}{X_T e^{r_f T}}\right] \]

\[ C_0 = X_0 \mathbb{E}^{\mathbb{Q}^f}\left[\frac{(X_T - K)^+}{X_T e^{r_f T}} \cdot e^{r_f T} \cdot \frac{X_T}{X_0}\right] \]

Alternatively, use the standard risk-neutral approach under \(\mathbb{Q}^d\):

\[ C_0 = e^{-r_d T}\mathbb{E}^{\mathbb{Q}^d}[(X_T - K)^+] \]

Under \(\mathbb{Q}^d\), \(\ln X_T \sim \mathcal{N}(\ln X_0 + (r_d - r_f - \frac{1}{2}\sigma^2)T, \sigma^2 T)\). Following the standard Black-Scholes integral evaluation (splitting into two Gaussian integrals and completing the square):

\[ C_0 = X_0 e^{-r_f T}\mathcal{N}(d_1) - K e^{-r_d T}\mathcal{N}(d_2) \]

where:

\[ d_1 = \frac{\ln(X_0/K) + (r_d - r_f + \frac{1}{2}\sigma^2)T}{\sigma\sqrt{T}}, \quad d_2 = d_1 - \sigma\sqrt{T} \]

This is the Garman-Kohlhagen formula.

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Exercise 4. Consider the exchange option (Margrabe's formula) with payoff \((S_T^{(1)} - S_T^{(2)})^+\) where both assets follow GBM with correlation \(\rho\). Using \(S_t^{(2)}\) as numeraire, show that the option price is \(V_0 = S_0^{(1)}\mathcal{N}(d_1) - S_0^{(2)}\mathcal{N}(d_2)\) and determine the effective volatility \(\hat{\sigma}\) that appears in \(d_1\) and \(d_2\).

Solution to Exercise 4

Let \(S_t^{(1)}\) and \(S_t^{(2)}\) follow correlated GBMs under \(\mathbb{Q}\):

\[ dS_t^{(i)} = rS_t^{(i)} \, dt + \sigma_i S_t^{(i)} \, dW_t^{(i)}, \quad d\langle W^{(1)}, W^{(2)}\rangle_t = \rho \, dt \]

Step 1: Use \(S_t^{(2)}\) as numeraire. The Radon-Nikodym derivative is:

\[ \frac{d\mathbb{Q}^{S^{(2)}}}{d\mathbb{Q}}\Big|_{\mathcal{F}_T} = \frac{S_T^{(2)} e^{-rT}}{S_0^{(2)}} \]

Step 2: Pricing formula. The exchange option price is:

\[ V_0 = e^{-rT}\mathbb{E}^{\mathbb{Q}}[(S_T^{(1)} - S_T^{(2)})^+] \]

Using the numeraire \(S_t^{(2)}\):

\[ \frac{V_0}{S_0^{(2)}} = \mathbb{E}^{\mathbb{Q}^{S^{(2)}}}\left[\left(\frac{S_T^{(1)}}{S_T^{(2)}} - 1\right)^+\right] \]

Step 3: Dynamics of the ratio. Define \(R_t = S_t^{(1)}/S_t^{(2)}\). By Ito's lemma:

\[ \frac{dR_t}{R_t} = (\sigma_1 \, dW_t^{(1)} - \sigma_2 \, dW_t^{(2)}) + (\sigma_2^2 - \rho\sigma_1\sigma_2) \, dt \]

Under \(\mathbb{Q}^{S^{(2)}}\), the Girsanov shift removes the drift from \(R_t/1\) (since \(R_t = S_t^{(1)}/S_t^{(2)}\) must be a martingale under this measure). The volatility of \(R_t\) is:

\[ \hat{\sigma} = \sqrt{\sigma_1^2 - 2\rho\sigma_1\sigma_2 + \sigma_2^2} \]

Under \(\mathbb{Q}^{S^{(2)}}\), \(R_t\) is a driftless geometric Brownian motion with volatility \(\hat{\sigma}\).

Step 4: Apply Black-Scholes to the ratio. The problem reduces to pricing a call on \(R_T\) with strike 1 in a world with zero interest rate:

\[ \frac{V_0}{S_0^{(2)}} = R_0 \mathcal{N}(d_1) - \mathcal{N}(d_2) \]

where \(R_0 = S_0^{(1)}/S_0^{(2)}\), and:

\[ d_1 = \frac{\ln(S_0^{(1)}/S_0^{(2)}) + \frac{1}{2}\hat{\sigma}^2 T}{\hat{\sigma}\sqrt{T}}, \quad d_2 = d_1 - \hat{\sigma}\sqrt{T} \]

Multiplying by \(S_0^{(2)}\):

\[ V_0 = S_0^{(1)}\mathcal{N}(d_1) - S_0^{(2)}\mathcal{N}(d_2) \]

This is Margrabe's formula. The effective volatility \(\hat{\sigma} = \sqrt{\sigma_1^2 - 2\rho\sigma_1\sigma_2 + \sigma_2^2}\) is the volatility of the log-ratio \(\ln(S^{(1)}/S^{(2)})\).

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Exercise 5. Explain why the term \(\mathcal{N}(d_1)\) in the Black-Scholes call formula is both the delta of the option and the probability of exercise under the stock measure. Is this a coincidence, or does the change-of-numeraire framework make this relationship transparent? Justify your answer.

Solution to Exercise 5

The relationship between \(\mathcal{N}(d_1)\) being both the delta and the stock-measure probability of exercise is not a coincidence -- it is a direct consequence of the change-of-numeraire framework.

Delta as stock-measure probability. From the Black-Scholes formula \(C = S\mathcal{N}(d_1) - Ke^{-rT}\mathcal{N}(d_2)\), differentiating with respect to \(S\):

\[ \Delta = \frac{\partial C}{\partial S} = \mathcal{N}(d_1) + S\mathcal{N}'(d_1)\frac{\partial d_1}{\partial S} - Ke^{-rT}\mathcal{N}'(d_2)\frac{\partial d_2}{\partial S} \]

Since \(\frac{\partial d_1}{\partial S} = \frac{\partial d_2}{\partial S} = \frac{1}{S\sigma\sqrt{T}}\) and \(S\mathcal{N}'(d_1) = Ke^{-rT}\mathcal{N}'(d_2)\) (a standard identity), the last two terms cancel, giving \(\Delta = \mathcal{N}(d_1)\).

Stock-measure probability. Under the stock measure \(\mathbb{Q}^S\), the call price can be written as \(C = S_0\mathbb{Q}^S(S_T > K) - Ke^{-rT}\mathbb{Q}(S_T > K)\). Comparing with the Black-Scholes formula:

\[ \mathbb{Q}^S(S_T > K) = \mathcal{N}(d_1) \]

Why this is transparent from the numeraire framework. Under the numeraire change to the stock, \(C_0/S_0 = \mathbb{E}^{\mathbb{Q}^S}[(1 - K/S_T)^+]\). The derivative of \(C_0\) with respect to \(S_0\) equals the probability under \(\mathbb{Q}^S\) that the option expires in the money, because \(C_0 = S_0 \cdot \mathbb{Q}^S(S_T > K) - Ke^{-rT}\mathcal{N}(d_2)\) and the first term is linear in \(S_0\) (through the dependence of \(\mathbb{Q}^S(S_T > K)\) on \(S_0\), with the derivative simplifying to \(\mathcal{N}(d_1)\)). The change-of-numeraire framework makes this relationship structurally transparent: the delta is the hedge ratio, which equals the probability of exercise under the measure where the stock is the numeraire.

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Exercise 6. A zero-coupon bond maturing at time \(T\) with price \(P(t,T) = e^{-r(T-t)}\) can serve as a numeraire, giving rise to the \(T\)-forward measure \(\mathbb{Q}^T\). Show that under \(\mathbb{Q}^T\), the forward price \(F(t,T) = S_t / P(t,T)\) is a martingale. Use this to re-derive the Black-Scholes call price starting from \(C_0 = P(0,T)\mathbb{E}^{\mathbb{Q}^T}[(F(T,T) - K)^+]\).

Solution to Exercise 6

Step 1: Show \(F(t,T)\) is a martingale under \(\mathbb{Q}^T\). The forward price is:

\[ F(t,T) = \frac{S_t}{P(t,T)} = S_t e^{r(T-t)} \]

Under \(\mathbb{Q}\), \(dS_t = rS_t \, dt + \sigma S_t \, dW_t^{\mathbb{Q}}\). The numeraire is \(P(t,T) = e^{-r(T-t)}\) with \(dP = rP \, dt\) (deterministic). The Radon-Nikodym derivative:

\[ \frac{d\mathbb{Q}^T}{d\mathbb{Q}}\Big|_{\mathcal{F}_T} = \frac{P(T,T)/B_T}{P(0,T)/B_0} = \frac{1/e^{rT}}{e^{-rT}/1} = 1 \]

Since the bond is deterministic, \(\mathbb{Q}^T = \mathbb{Q}\) and there is no Girsanov shift. Now compute \(dF\):

\[ dF = d(S_t e^{r(T-t)}) = e^{r(T-t)} \, dS_t + S_t \, d(e^{r(T-t)}) \]

\[ = e^{r(T-t)}(rS_t \, dt + \sigma S_t \, dW_t) - rS_t e^{r(T-t)} \, dt = \sigma F_t \, dW_t^{\mathbb{Q}^T} \]

The drift vanishes, so \(F(t,T)\) is a martingale under \(\mathbb{Q}^T\).

Step 2: Re-derive the call price. Starting from:

\[ C_0 = P(0,T)\mathbb{E}^{\mathbb{Q}^T}[(F(T,T) - K)^+] \]

Note that \(F(T,T) = S_T\) and \(P(0,T) = e^{-rT}\). Under \(\mathbb{Q}^T = \mathbb{Q}\), \(F(t,T) = F(0,T)\exp(-\frac{1}{2}\sigma^2 t + \sigma W_t)\) with \(F(0,T) = S_0 e^{rT}\).

So:

\[ F(T,T) = S_0 e^{rT} \exp\left(-\frac{1}{2}\sigma^2 T + \sigma W_T\right) \]

The problem is now a standard Black-Scholes pricing with the forward \(F_0 = S_0 e^{rT}\) replacing \(S_0\), zero interest rate (since discounting is already handled by \(P(0,T)\)), and the same volatility \(\sigma\). By the Black formula:

\[ C_0 = e^{-rT}[F_0 \mathcal{N}(d_1) - K\mathcal{N}(d_2)] \]

where:

\[ d_1 = \frac{\ln(F_0/K) + \frac{1}{2}\sigma^2 T}{\sigma\sqrt{T}} = \frac{\ln(S_0/K) + rT + \frac{1}{2}\sigma^2 T}{\sigma\sqrt{T}} = \frac{\ln(S_0/K) + (r + \frac{1}{2}\sigma^2)T}{\sigma\sqrt{T}} \]

\[ d_2 = d_1 - \sigma\sqrt{T} \]

Therefore:

\[ C_0 = e^{-rT}[S_0 e^{rT}\mathcal{N}(d_1) - K\mathcal{N}(d_2)] = S_0\mathcal{N}(d_1) - Ke^{-rT}\mathcal{N}(d_2) \]

This recovers the Black-Scholes formula.