Margrabe Exchange Option Pricing Lab & Relative Valuation

Q: What is a Margrabe exchange option?

A Margrabe exchange option is a multi-asset derivative that grants the holder the right, but not the obligation, to surrender one risky asset (Asset 2) in exchange for another risky asset (Asset 1) at expiration, with terminal payoff max(S1(T) - S2(T), 0). Developed by William Margrabe in 1978, it generalizes Black-Scholes by allowing the strike price itself to be a stochastic asset.

Q: Why does the risk-free interest rate (r) not appear in Margrabe's formula?

The risk-free rate r cancels out because Asset 2 serves as the stochastic numeraire rather than a deterministic cash bank account. Under the equivalent martingale measure defined by numeraire S2, the relative price S1/S2 behaves like a standard Black-Scholes underlying with a fixed strike of 1.0, eliminating the need for cash discounting.

Q: How does cross-asset correlation (ρ) affect the value of an exchange option?

An exchange option's value decreases monotonically as correlation ρ increases. The combined spread volatility is given by sigma_hat = sqrt(sigma1^2 + sigma2^2 - 2*rho*sigma1*sigma2). When two assets are strongly correlated, they move together, compressing the variance of their spread and diminishing the probability that one dramatically outperforms the other.

Q: What real-world financial contracts are valued using Margrabe's formula?

Margrabe's formula is widely used to price energy spark spreads (converting natural gas into wholesale electricity), refinery crack spreads (crude oil into refined gasoline/diesel), M&A stock-for-stock takeover offers, dual-currency foreign exchange options, and equity relative performance or outperformance options.

Market Presets:

Exchange Option Value

$14.285

14.29% of benchmark asset

Combined Spread Vol (σ̂)

36.19%

Spread volatility √(σ₁² + σ₂² - 2ρσ₁σ₂)

Relative Ratio (S₁ / S₂)

1.0000

At-The-Money Parity

Correlation Greek (dC/dρ)

-$18.420

Price change per +1.00 correlation

Exchange Contract Parameters

Payoff: max(S₁ - S₂, 0)

Exchange Direction / Payoff

Receive S₁, Deliver S₂ [max(S₁ - S₂, 0)]

Receive S₂, Deliver S₁ [max(S₂ - S₁, 0)]

Asset 1 (Acquired Asset S₁)

Spot Price (S₁) $100.00

Volatility (σ₁) 25.0%

Yield / Carry (q₁) 0.0%

Asset 2 (Delivered Asset S₂ / Strike)

Spot Price (S₂) $100.00

Volatility (σ₂) 25.0%

Yield / Carry (q₂) 0.0%

Cross-Asset Joint Dynamics

Correlation Coefficient (ρ) +0.50

-1.0 (Opposite Movement) 0.0 (Uncorrelated) +1.0 (Comovement)

Time to Expiration (T) 1.00 yrs (365 days)

Analytical Formula Breakdown

Margrabe Option Value	$14.285
Intrinsic Spread max(S₁ - S₂, 0)	$0.0000
Time Value Component	$14.285
Combined Volatility (σ̂)	36.19%
d₁ / d₂ Coordinates	d₁: 0.1809 \| d₂: -0.1809
N(d₁) / N(d₂) Probabilities	N(d₁): 0.5718 \| N(d₂): 0.4282
Discounted Asset Values	S₁e⁻ᑫᵀ: $100.00 \| S₂e⁻ᑫᵀ: $100.00
Risk-Neutral ITM Probability	42.82%
Euler Identity: S₁Δ₁ + S₂Δ₂	$14.285 (Exact match)

Margrabe exchange option value as a function of Asset 1 price (S₁), compared to intrinsic payoff and Asset 2 deliverable level.

Cross-Asset Sensitivities & Greeks

Delta 1 (Δ₁) +0.5718

∂C/∂S₁ = e⁻ᑫ¹ᵀ N(d₁) [Long Asset 1]

Delta 2 (Δ₂) -0.4282

∂C/∂S₂ = -e⁻ᑫ²ᵀ N(d₂) [Short Asset 2]

Gamma 1 (Γ₁) +0.0108

∂²C/∂S₁² = Rate of Delta 1 shift

Gamma 2 (Γ₂) +0.0108

∂²C/∂S₂² = Rate of Delta 2 shift

Cross-Gamma (Γ₁₂) -0.0108

∂²C/(∂S₁∂S₂) [Cross curvature]

Corr Greek (∂C/∂ρ) -$0.1842

Dollar change per +0.01 shift in ρ

Vega 1 (𝒱₁) $0.1963

Per 1% shift in Asset 1 volatility

Vega 2 (𝒱₂) $0.1963

Per 1% shift in Asset 2 volatility

Daily Theta (Θ) -$0.0195/day

Calendar time decay per day

Sensitivity 1: S₁ vs S₂ on Exchange Value

Evaluates option premium across joint price movements of Asset 1 and Asset 2 around base spots.

Sensitivity 2: Correlation (ρ) vs Expiry (T)

Demonstrates how negative correlation and longer time horizons amplify the option value by increasing combined volatility σ̂.

Institutional Framework: William Margrabe's (1978) Model

1. Foundation: The Option to Exchange One Asset for Another

Published in the Journal of Finance (1978), William Margrabe solved the valuation of a contract granting the holder the right to surrender asset 2 to acquire asset 1 at expiration date $T$:

Payoff(T) = max(S₁(T) - S₂(T), 0)

Unlike the standard Black-Scholes model where the strike price $K$ is a deterministic, constant cash amount, Margrabe allows the strike itself to be a stochastic asset $S_2(t)$. Both asset prices follow correlated geometric Brownian motions:

dS₁/S₁ = (μ₁ - q₁)dt + σ₁ dW₁
dS₂/S₂ = (μ₂ - q₂)dt + σ₂ dW₂, with Corr(dW₁, dW₂) = ρ dt

2. Change of Numeraire & Closed-Form Formula

By using $S_2(t)$ as the stochastic numeraire instead of the money market account, the relative price $Z(t) = S_1(t) / S_2(t)$ becomes a martingale under the $S_2$-measure with volatility equal to the combined spread volatility:

σ̂ = √(σ₁² + σ₂² - 2ρσ₁σ₂)

Under this measure, the contract is identical to a standard European call on $Z(t)$ with a fixed strike of $1.0$, yielding the exact closed form:

C = S₁ e⁻ᑫ¹ᵀ N(d₁) - S₂ e⁻ᑫ²ᵀ N(d₂)
d₁ = [ln(S₁/S₂) + (q₂ - q₁ + ½σ̂²)T] / (σ̂√T)
d₂ = d₁ - σ̂√T

3. Euler's Homogeneity Identity & Cross-Greeks

Because the pricing function is linearly homogeneous of degree 1 with respect to $(S_1, S_2)$—i.e., $C(\lambda S_1, \lambda S_2) = \lambda C(S_1, S_2)$—Euler's homogeneous function theorem guarantees:

C = S₁ (∂C/∂S₁) + S₂ (∂C/∂S₂) = S₁ Δ₁ + S₂ Δ₂

Where the cross-asset Greeks exhibit remarkable symmetry:

Delta 1 (Δ₁): $e^{-q_1 T} N(d_1) > 0$ (long underlying asset 1).
Delta 2 (Δ₂): $-e^{-q_2 T} N(d_2) < 0$ (short deliverable asset 2).
Cross-Gamma (Γ₁₂): $\frac{\partial^2 C}{\partial S_1 \partial S_2} = -\frac{e^{-q_1 T} n(d_1)}{S_2 \hat{\sigma} \sqrt{T}} < 0$.
Correlation Greek: $\frac{\partial C}{\partial \rho} = -\frac{S_1 e^{-q_1 T} n(d_1) \sqrt{T} \sigma_1 \sigma_2}{\hat{\sigma}} < 0$.

4. Industry Applications: Spark Spreads & M&A

Merchant Power Plants (Spark Spread): The option to convert fuel into electricity is modeled as a Margrabe option where $S_1 = \text{Electricity Price}$ and $S_2 = \text{Heat Rate} \times \text{Natural Gas Price}$. A plant operates only when the spark spread is positive.

M&A Takeover Collars & Stock Swaps: Valuing stock-for-stock exchange ratios where an acquiring company's shares are swapped for target company shares at closing.

Exchange Option Mastery Quiz

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