Two Currencies, Two Interest Rates: Garman and Kohlhagen Price FX Options

In the early 1980s, exchange rates had been floating for about a decade and the market for currency options was starting to grow up. Banks needed to quote prices. The obvious move was to reach for Black-Scholes.

It did not work. The prices it produced were visibly wrong, and put-call parity, the iron relationship between the price of a call and the price of a put, refused to hold. Something about currencies broke the model.

Mark Garman and Steven Kohlhagen diagnosed the problem in 1983 and fixed it. The fix is small, and once you understand why it is needed, it is completely obvious. But the diagnosis is the valuable part, because it reveals something important about what an option price really is.

The problem: a currency is not a stock

Black-Scholes was built for a non-dividend-paying stock. Hold the stock, and it just sits there. Its whole return comes from price appreciation.

A currency is not like that. If you hold euros, you do not stuff them in a mattress. You deposit them, and they earn the euro interest rate. Holding the asset pays you something, continuously, on top of any move in the exchange rate.

That single fact wrecks the standard model, and it does so in a specific place. Recall how Black-Scholes works: you replicate the option by holding a hedge in the underlying, financed by borrowing. The model assumes that holding the underlying costs you the financing rate and yields you nothing. But a hedged FX position yields the foreign interest rate while costing the domestic one. The cost of carrying the hedge is not the domestic rate, it is the difference between the two rates. Ignore that and every number you produce is wrong.

The key idea via analogy: a stock that pays a continuous dividend

Here is the reframing that makes it all fall into place.

Treat the foreign currency as a stock that pays a continuous dividend yield, and let that dividend yield be the foreign interest rate.

That is the whole idea. Merton had already worked out how to price options on dividend-paying stocks, and the adjustment is intuitive: a stock that pays out cash to its holders will, all else equal, drift lower than one that does not, because value is leaking out of it. So in the pricing formula, you discount the current price by the dividend yield before feeding it in.

Garman and Kohlhagen saw that a currency is exactly that case, with the "dividend" being the foreign deposit rate. The resulting formula, now universally called Garman-Kohlhagen, is Black-Scholes with the spot exchange rate discounted at the foreign interest rate and the strike discounted at the domestic interest rate. Two rates, two discounts, one symmetric formula.

The pleasing symmetry

The deepest thing about the Garman-Kohlhagen framework is a property that has no analogue in equity options, and it is worth pausing on.

A call on euros, struck at 1.10 dollars per euro, gives you the right to give up dollars and receive euros. But look at the same contract from the other side: it is also the right to give up dollars in exchange for euros, which, expressed in the other currency, is a put on dollars struck at the reciprocal rate. The very same contract is simultaneously a call on one currency and a put on the other. There is no "underlying" and "cash," there are just two currencies, and which one you call money is a matter of perspective.

The Garman-Kohlhagen formula respects this symmetry perfectly: price the contract in dollars-per-euro terms, then price it in euros-per-dollar terms, convert, and you get the same number. That is not an accident, it is a consistency check that the model has been set up correctly, and it is the reason the two interest rates appear in mirror-image positions in the formula.

This symmetry has a practical consequence that shapes the whole FX options market: because a call is a put, FX traders do not quote options by strike in the way equity traders do. They quote by delta, and they express the market's skew as a risk reversal (the price difference between an out-of-the-money call and the equivalent out-of-the-money put). The market's conventions are downstream of the model's symmetry.

Why it mattered

• It made a market possible. FX options are one of the largest derivatives markets on earth, and Garman-Kohlhagen is the language it is quoted in. Every bank's FX options desk runs on this formula or a smile-adjusted descendant of it.
• It connected options to carry. The interest rate differential in the formula is precisely the forward points, the premium or discount at which a currency trades for future delivery, and it is the same differential that drives the famous carry trade. So the FX option market and the FX forward market are two views of one quantity, and Garman-Kohlhagen is the bridge. A currency with a high interest rate trades at a forward discount, and its options price accordingly.
• It generalised beyond currencies. The trick, model a yield-paying asset as a stock with a continuous dividend, is now the standard way to price options on stock indices (which pay dividends), on commodities (which have storage costs and convenience yields, an effective negative dividend), and on anything else that throws off or consumes cash while you hold it. Garman-Kohlhagen is the template.
• It restored put-call parity. With both rates in the right place, the fundamental relationship between calls, puts, forwards and cash holds exactly, which means the market can be arbitraged into consistency. Without that, an options market cannot function.

The honest limitations

• It is Black-Scholes underneath, so it inherits every Black-Scholes flaw. Constant volatility, no jumps, lognormal returns. Real exchange rates do all the things this rules out.
• The FX smile is real, pronounced and structured. Currency options exhibit a clear smile (both out-of-the-money calls and puts trade at higher implied volatilities than at-the-money options) plus a skew that reflects which direction the market fears. Emerging-market currencies show violent skews reflecting devaluation risk. Garman-Kohlhagen, taken literally, says the smile should not exist. In practice it is used as a quoting convention, a way of converting a price into a volatility number, with the smile handled by a separate model layered on top (SABR and its relatives).
• Currencies jump, and they jump asymmetrically. Pegs break. Central banks intervene overnight. The Swiss franc moved roughly 20 percent in minutes when its floor was abandoned in 2015. A diffusion model with constant volatility cannot price the risk of a peg snapping, and any FX desk that relied on it would have been destroyed by exactly those events.
• Interest rates are not constant either. For short-dated options this hardly matters. For long-dated currency options, where the rate differential can move a great deal over the option's life, treating both rates as fixed is a meaningful simplification.
• It assumes you can borrow and lend freely in both currencies. Capital controls, funding squeezes and the cross-currency basis (the persistent, arbitrage-resistant gap between what interest-rate parity says funding should cost and what it actually costs) all violate this, and became much more visible after 2008.

The one-line takeaway

Garman and Kohlhagen fixed Black-Scholes for currencies by noticing that a currency is a stock that pays a dividend equal to its own interest rate, producing a symmetric two-rate formula in which a call on one currency is literally a put on the other, and which remains the quoting language of one of the world's largest derivatives markets, smile adjustments notwithstanding.