The Option With No Strike Price: Margrabe's Exchange Option

Every option you have ever heard of has a strike price: a fixed number of dollars you pay to receive the asset. A call on Apple at 200 means you hand over 200 dollars and get one share. The 200 is certain. The share is risky. One risky thing, one certain thing.

William Margrabe asked a deceptively small question in 1978. What if both sides are risky? What is the value of the right to hand over one share of Company A and receive one share of Company B?

The answer is one of the most elegant results in derivatives, and it contains a genuine surprise.

The problem: these contracts are everywhere, once you look

Margrabe's opening move is to point out that exchange options are not an exotic curiosity. They are hiding inside a long list of ordinary financial arrangements:

• A performance fee. A fund manager who is paid a bonus for beating a benchmark holds, in effect, an option to exchange the benchmark's performance for the fund's. If the fund beats the index, the manager exercises, so to speak. If not, the option expires worthless. The bonus is an option on the relative performance of two risky things.
• A margin account. The right to walk away from a collateralised position is the right to exchange the collateral for the debt.
• An exchange offer in a takeover. Shareholders in a target company are offered shares in the acquirer instead of their own. Whether that offer is worth taking depends on the relative value of the two stocks, and the optionality embedded in the offer has value.
• A standby commitment in underwriting.

None of these had a pricing formula. Black-Scholes did not apply, because Black-Scholes assumes you pay a fixed number of dollars.

The key idea via analogy: change your currency

Here is Margrabe's trick, and once you see it you cannot unsee it.

The problem is that we have two risky assets and no fixed reference point. So invent one. Stop measuring value in dollars. Start measuring everything in units of Asset B.

In that world, Asset B is now worth exactly one, by definition, always. It has become the boring, certain thing. It is the new dollar. And Asset A, measured in units of B, is some fluctuating number: it might take 1.3 units of B to buy one unit of A today, and 1.1 tomorrow.

Look at what has happened. The option to exchange A for B is now, in B-units, an option to pay one and receive a risky asset whose price is "A measured in B." That is a plain vanilla call option with a strike of one. And we already know how to price those.

This move, changing the unit of account from cash to a traded asset, is called a change of numeraire, and Margrabe's paper is the reason it is now standard equipment in every quant's toolkit. It is the financial equivalent of picking a smarter coordinate system: the problem has not changed, but in the right coordinates it becomes something you have already solved.

The two surprises

The formula that emerges looks almost exactly like Black-Scholes, but with two remarkable differences.

Surprise one: the interest rate disappears. It is simply not in the formula. This is startling until you understand it, and then it is obvious. Interest rates matter in ordinary option pricing because you are deferring a cash payment: the strike is paid at expiry, so its present value depends on how much you can earn on that cash in the meantime. But in an exchange option you never pay cash. You pay an asset. There is nothing to park in a deposit account, so the deposit rate is irrelevant. The exchange option's value does not care what the central bank does.

Surprise two: what matters is not each asset's volatility but the volatility of the two relative to each other. The formula uses a single "exchange volatility" built from three ingredients: the volatility of A, the volatility of B, and the correlation between them.

This is where the economics is genuinely interesting. Two assets can each be individually wild and yet, if they move in lockstep, their ratio barely budges, and the option to swap them is nearly worthless. Two oil majors, both volatile, both driven by the same crude price, are highly correlated, so the right to exchange one for the other is cheap. Conversely, two calm-looking assets that tend to move in opposite directions produce a ratio that swings violently, and the exchange option is expensive.

So the exchange option is not really a bet on volatility. It is a bet on correlation, and its price falls as correlation rises. That makes it the ancestor of an entire class of instruments whose whole purpose is to trade the relationship between assets rather than the assets themselves.

Why it mattered

• It opened the door to multi-asset derivatives. Before Margrabe, option theory was about one risky thing. After, it was about relationships. Spread options, basket options, rainbow options, best-of and worst-of structures, quanto options: all descend from this insight.
• It made the change of numeraire a standard technique. This is the single most useful trick in derivatives mathematics. Pricing an option on a bond? Use the bond as the numeraire and the messy interest-rate discounting collapses. Pricing an FX option? Switch currencies. Whole branches of interest-rate modelling (the forward-measure machinery behind the LIBOR market model) are applications of the move Margrabe introduced.
• It gave correlation a price. Once you have a formula in which correlation is an input, correlation becomes something you can imply from market prices, hedge, and trade. Correlation desks exist because of this lineage.
• It priced things nobody realised were options. The performance-fee example is quietly profound. It says that a manager compensated for beating a benchmark holds a real, valuable option, and that the option becomes more valuable the more the manager's portfolio diverges from the benchmark. That is an incentive to take risk, and it falls straight out of Margrabe's formula. Compensation design has never fully recovered from noticing this.

The honest limitations

• Correlation is the least stable input in finance. The formula needs the correlation between the two assets over the life of the option. Correlation estimated from history is notoriously unreliable, it drifts, and worse, it tends to lurch toward one exactly during crises, when diversification is most needed. An exchange option priced on calm-period correlation can be badly mispriced when it matters.
• You cannot easily hedge correlation. Volatility can be hedged by trading options. Correlation, for most asset pairs, cannot: there is no liquid instrument that isolates it. So a desk holding correlation risk is often holding it, not hedging it.
• It inherits Black-Scholes' assumptions, twice over. Both assets are assumed to follow smooth lognormal motion with constant volatility, and their correlation is assumed constant. All three assumptions fail in the real world, and the failures compound when you have two assets rather than one.
• It assumes no dividends by default. The basic result needs adjusting for assets that pay out, and the adjustment is where much of the practical fiddliness lives.
• Relative-value bets can stay wrong for a long time. The formula prices the option correctly given the model. It says nothing about whether two assets that have decoupled will ever recouple.

The one-line takeaway

Margrabe showed that if you stop measuring value in dollars and start measuring it in units of one of the assets, an option to swap two risky things becomes an ordinary call option, and the resulting price has no interest rate in it at all and depends above all on correlation, making it the founding document of every derivative that trades the relationship between assets rather than the assets themselves.