Exchange Options: Margrabe Pricing and Relative-Value Risk

Value an exchange option with explicit asset quantities, carry, volatility, and correlation, then control settlement, execution, and counterparty risk.

For educational purposes only; not investment advice. Investing may result in loss.

Direct answer

An exchange option gives its holder the right, but not the obligation, to surrender a stated quantity of one asset and receive a stated quantity of another. If the claim delivers a units of asset 1 against b units of asset 2 at T, its payoff in a common currency is max(a x S1(T) - b x S2(T), 0). The exchange ratio, asset identity, currency, fixing, and deliverable are contractual inputs, not a multiplier to infer later.

The Margrabe formula prices a narrow European, no-cash-strike claim under jointly lognormal assets, deterministic proportional carry, constant volatilities and correlation, continuous trading, and frictionless replication. It provides a model benchmark. It does not establish an executable quote, a hedge guarantee, a standardized listed contract, or a bilateral close-out amount.

Build the claim and model

Write the exact payoff, quantities, common valuation currency, exercise style, fixing, settlement, and corporate-action rules. Do not rely on the label “exchange option.”
For the basic model, define X1 = a x S1 and X2 = b x S2. Lock synchronized spots, continuous payout yields q1 and q2, annualized volatilities sigma1 and sigma2, return correlation rho, and year fraction T.
Calculate relative volatility as sigmaR = sqrt(sigma1^2 + sigma2^2 - 2 x rho x sigma1 x sigma2). Check the covariance inputs, units, timestamp, and positive-semidefinite consistency before using the result.
Price the European claim with V = X1 x exp(-q1 x T) x N(d1) - X2 x exp(-q2 x T) x N(d2), where d1 = [ln(X1 / X2) + (q2 - q1 + 0.5 x sigmaR^2) x T] / (sigmaR x sqrt(T)) and d2 = d1 - sigmaR x sqrt(T).
Reprice across both volatility surfaces, correlation, carry, jumps, and liquidity. Lower rho normally raises sigmaR and this vanilla exchange-option value, but a single historical correlation is not a tradable input or a full joint distribution.
Map the model claim to the real instrument. Listed, FLEX, and bilateral OTC contracts can differ in multiplier, exercise, physical or cash settlement, fixing source, collateral, close-out, netting, dispute, and counterparty terms.
Record executable entry and exit sides, fees, hedges, funding, collateral, payouts, settlement cash or inventory, and model-to-market residuals. Reconcile the legal confirmation and cash ledger rather than treating the model mark as realized P/L.

The absence of an explicit risk-free-rate term in the elementary formula is a numeraire result, not proof that financing is irrelevant. Forward levels, payout yields, stock borrow, collateral remuneration, FX conversion, funding spreads, and close-out terms still affect a real trade. If the assets use different currencies, first state the settlement currency and FX conversion or quanto rule; subtracting raw prices in different currencies is meaningless.

Four worked examples

One-for-one model value. Let S1=$110, S2=$100, a=b=1, T=0.5, sigma1=30%, sigma2=20%, rho=0.50, and q1=q2=0. Then sigmaR=sqrt(0.30^2+0.20^2-2x0.50x0.30x0.20)=26.4575131106%, d1=0.6029957751, and d2=0.4159129058. The model value is $13.815554 per right, versus current intrinsic value max(110-100,0)=$10. These are model outputs, not an offer to trade.
Quantities belong inside the model. One right exchanges b=200 shares of asset 2 for a=150 shares of asset 1. With S1=$80, S2=$55, T=0.75, sigma1=28%, sigma2=22%, rho=0.35, q1=1%, and q2=3%, the current asset values are X1=$12,000 and X2=$11,000. Relative volatility is 28.9274955708%; d1=0.5324591424, d2=0.2819396820, and model value is about $1,799.02 per right. Replacing the contractual quantities with a presumed 100-share multiplier changes the claim.
Correlation is a model exposure. Return to the first example. At rho=0.90, sigmaR=14.8323969742% and model value is about $11.086226; at rho=-0.50, sigmaR=43.5889894354% and value is about $18.464726. The difference is not a forecast that realized correlation must follow either input. Surface moves, jumps, carry, and hedge costs can dominate the isolated comparison.
Settlement and execution are separate ledgers. Three rights each exchange 80 shares of asset 2 for 50 shares of asset 1. At contractual fixings S1=$140 and S2=$80, each payoff is max(50x140-80x80,0)=$600, so cash settlement is $1,800; physical settlement instead requires surrendering 240 asset-2 shares and receiving 150 asset-1 shares. If a different valid fixing clause produces S1=$138.50 and S2=$80.40, cash settlement is 3xmax(50x138.50-80x80.40,0)=$1,479, or $321 less. A dealer model mark cannot replace the contractual fixing.

Seven-step workflow and controls

Wrong asset, share class, index, currency, or legal issuer.
Wrong exchange direction, sign, quantity, ratio, multiplier, or deliverable.
Payoff written in incompatible currencies without an FX or quanto rule.
Exercise style, maturity, fixing time, time zone, or settlement convention mismatch.
Corporate action, payout, special dividend, borrow, or withholding-tax error.
Stale or asynchronous spots, forwards, curves, surfaces, or FX inputs.
Volatility units, day count, return convention, or covariance scaling error.
Correlation matrix inconsistency or unstable historical estimation.
One scalar correlation hiding strike, tenor, regime, and tail dependence.
Constant-volatility, constant-correlation, or lognormal assumptions failing.
Common or idiosyncratic jumps, defaults, suspensions, and market closures.
Relative Delta, cross-Gamma, Vega, correlation, and carry changing together.
Continuous replication assumed through a gap, halt, or illiquid session.
Model price mistaken for a synchronized executable bid or offer.
FLEX or listed specifications generalized to a bespoke two-asset claim.
OTC confirmation, collateral, netting, close-out, or valuation-dispute mismatch.
Counterparty default, wrong-way risk, collateral gap, or recovery uncertainty.
Partial hedge fills, basis risk, market impact, fees, and funding asymmetry.
Physical inventory, cash fixing, FX conversion, and tax not reconciled.
Model version, input vintage, confirmation, collateral, and final P/L not reconciled.

Common misconceptions

“Exchange option means exchange-traded option.” The first describes a payoff; the second describes a venue and market structure.
“The exchange ratio can be applied after pricing.” Quantities change the current relative value and belong inside the logarithm and both discounted asset terms.
“The risk-free rate disappears, so funding does not matter.” Numeraire cancellation does not remove carry, collateral, borrow, FX, or dealer funding.
“Correlation is directly observable and stable.” It is estimated, state-dependent, and only one part of the joint surface and jump risk.
“A closed-form value is a tradable close-out price.” Execution, credit, legal terms, liquidity, hedging costs, and valuation disputes can produce a different result.