The Garman-Kohlhagen formula worked on one exporter's dollar receipt, the option set against the forward at five settlement rates, and what volatility and strike do to the premium.
An FX option is priced with Garman-Kohlhagen: Black-Scholes with the foreign interest rate in place of a dividend yield. A UK exporter buying a one-year sterling call on USD 10,000,000, struck at the 1.2761 forward with 9 per cent volatility, pays 0.04383 dollars per pound: £270,449, or 3.45 per cent of the sterling it will receive. It does better than the forward only if sterling ends the year below 1.2319.
Worked in full in Derivatives by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
The forward locks a rate and costs no premium. The option fixes a worst case and leaves the upside open, for a price that has to be paid in cash on day one. Choosing between them means putting both on the same page, which starts with pricing the option properly.
The market is the same as in calculating an FX forward rate: one-year GBP/USD, quoted in dollars per pound. The volatility is illustrative.
| Input | Value |
|---|---|
| Dollars to be received in one year | USD 10,000,000 |
| Spot GBP/USD | 1.2700 |
| One-year USD rate, annual (domestic, the quote currency) | 4.5% |
| One-year GBP rate, annual (foreign, the base currency) | 4.0% |
| Implied volatility | 9% |
| Option bought | GBP call / USD put, struck at the forward |
The exporter loses if the dollar weakens, which is a rise in GBP/USD. The protection is therefore the right to buy pounds, a GBP call, at a fixed rate: the mirror image of a USD put.
Continuous rates: rd = ln(1.045) = 4.4017%; rf = ln(1.040) = 3.9221%
Forward = 1.2700 × 1.045 ÷ 1.040 = 1.2761 (1.27611), 61.1 points above spot
Strike K = 1.2761; sterling notional = 10,000,000 ÷ 1.2761 = 7,836,341
d1 = [ln(S ÷ K) + (rd − rf + σ² ÷ 2) T] ÷ (σ√T); d2 = d1 − σ√T
At the forward the first two terms cancel: d1 = σ√T ÷ 2 = 0.0450; d2 = −0.0450
N(d1) = 0.5179; N(d2) = 0.4821
Call = S e−rfT N(d1) − K e−rdT N(d2)
= 1.22115 × 0.5179 − 1.22115 × 0.4821 = 0.04383 dollars per pound (438.3 pips)
Excel: =S*EXP(-rf*T)*NORM.S.DIST(d1,TRUE)-K*EXP(-rd*T)*NORM.S.DIST(d2,TRUE)
At the forward strike, S e−rfT and K e−rdT are both 1.22115, the present value of the forward, which is why the call is simply that number times N(d1) − N(d2).
Premium in dollars: 0.04383 × 7,836,341 = USD 343,471
In sterling at spot: 343,471 ÷ 1.2700 = £270,449, 3.45 per cent of the sterling notional
Carried to the settlement date at the GBP rate: 270,449 × 1.04 = £281,267
| GBP/USD in a year | Unhedged | Forward | Option, net | Option less forward |
|---|---|---|---|---|
| 1.1500 | 8,695,652 | 7,836,341 | 8,414,385 | 578,044 |
| 1.2000 | 8,333,333 | 7,836,341 | 8,052,066 | 215,725 |
| 1.2761 | 7,836,341 | 7,836,341 | 7,555,074 | −281,267 |
| 1.3500 | 7,407,407 | 7,836,341 | 7,555,074 | −281,267 |
| 1.4000 | 7,142,857 | 7,836,341 | 7,555,074 | −281,267 |
Above the strike the exporter exercises and receives 7,836,341, less the premium: a worst case of 7,555,074. Below it the option lapses and the dollars are sold at spot. The option beats the forward once 10,000,000 ÷ spot exceeds 7,836,341 + 281,267, which happens at a spot of 1.2319. Sterling has to fall 3.46 per cent below the forward before the premium is earned back.
| Case | Premium | % of notional | Worst case, net |
|---|---|---|---|
| At the forward, 7% vol | 210,378 | 2.68% | |
| At the forward, 9% vol | 270,449 | 3.45% | 7,555,074 |
| At the forward, 11% vol | 330,494 | 4.22% | |
| Strike 1.2500, 9% vol | 359,126 | 4.49% | 7,626,509 |
| Strike 1.3000, 9% vol | 204,386 | 2.66% | 7,479,747 |
Near the money the premium is almost proportional to volatility: each two points add about £60,000. Moving the strike away from the forward to 1.3000 saves £66,064 of premium but lowers the guaranteed sterling by more. Selling an option on the other side to pay for it is the collar, whose hidden transfer is worked in is a zero-cost collar really zero cost.
Garman-Kohlhagen is Black-Scholes with the foreign rate as the yield: 0.04383 dollars per pound, £270,449 or 3.45 per cent, for a one-year at-the-forward sterling call at 9 per cent. Price it, carry the premium forward, and compare with the forward at several outcomes, not at one. What a hedge costs, what it saves and what it does to cash are the three questions of the book; its workbooks are on the free companion page for this book.
Garman-Kohlhagen is Black-Scholes for currencies: the foreign interest rate plays the part of a dividend yield. The spot is discounted at the foreign rate and the strike at the domestic rate. Pricing on the forward with the Black model gives the same answer: here 0.04383 dollars per pound for a one-year at-the-forward GBP call at 9 per cent volatility.
A forward has no premium; its points are the interest differential. An at-the-forward option costs its premium, here £270,449 on USD 10,000,000, or 3.45 per cent at 9 per cent volatility for a year. In exchange the exporter keeps the gain if the dollar strengthens. If it does not, the option leaves it £281,267 behind the forward, the premium plus a year's interest.
Yes. An at-the-forward option is worth about 0.4 x volatility x root T x forward, discounted. At 9 per cent for a year on a 1.2761 forward that gives 0.04396 dollars per pound against the exact 0.04383. The approximation is good near the money and short of a few years.
This article is one calculation from Derivatives. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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