A zero-premium participating forward on a USD 10 million receivable, priced with Garman-Kohlhagen and compared with the plain forward at every settlement rate.
A participating forward buys an option on the whole exposure at a protected rate worse than the forward, and sells the opposite option at the same strike on part of it, sized so that the two premiums cancel. For an illustrative sterling exporter selling USD 10 million in one year, with the forward at 1.2637, a protected rate of 1.2900 buys 47.7 per cent participation in a falling GBP/USD for zero premium. The price is the 263 pips given up against the forward: GBP 161,546 if sterling ends at or above 1.2900.
Worked in full in Derivatives by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
The company reports in sterling and will receive USD 10,000,000 in twelve months. It loses if GBP/USD rises, because each dollar then buys less sterling. All market levels are illustrative, and the dealer's margin is left out so that the mechanics are visible; in practice it is taken out of the participation rate.
| Input | Value |
|---|---|
| Spot GBP/USD | 1.2700 |
| USD rate, continuous | 4.00% |
| GBP rate, continuous | 4.50% |
| One-year forward | 1.2637 |
| Volatility | 8.0% |
| Protected rate (strike) | 1.2900 |
Forward = 1.2700 × e(4.00% − 4.50%) × 1 = 1.2637, so forward points are minus 63 pips
Bought: GBP call / USD put, strike 1.2900, on the full amount. Garman-Kohlhagen price 0.02778 USD per GBP
Sold: GBP put / USD call, strike 1.2900, on (1 − participation) of the amount. Price 0.05309 USD per GBP
Zero premium: 0.02778 = (1 − p) × 0.05309, so p = 1 − 0.02778 / 0.05309 = 47.7%
Option notionals: USD 10,000,000 / 1.2900 = GBP 7,751,938 bought; GBP 4,057,255 sold
Premium check: USD 215,384 paid and USD 215,384 received
The sold option is in the money: its strike is 263 pips above the forward, so it is worth almost twice the bought option. That is why selling it on only about half the amount pays for full protection.
If GBP/USD settles at or above 1.2900, the bought option is exercised and all USD 10 million is sold at 1.2900. If it settles below 1.2900, the bank exercises the sold put: USD 5,233,859 is sold at 1.2900 and the remaining USD 4,766,141, the participating share, is sold at spot. The blended rate is not a simple average of the two rates, because sterling received is dollars divided by rate:
GBP received below the strike = USD 10m × (47.7% / spot + 52.3% / 1.2900)
Break-even against the forward: 47.7% / S + 52.3% / 1.2900 = 1 / 1.2637, so S = 1.2360
| GBP/USD at maturity | Effective rate | Participating forward, GBP | Forward, GBP | Unhedged, GBP | Versus forward, GBP |
|---|---|---|---|---|---|
| 1.1800 | 1.2351 | 8,096,358 | 7,913,484 | 8,474,576 | 182,873 |
| 1.2200 | 1.2557 | 7,963,928 | 7,913,484 | 8,196,721 | 50,444 |
| 1.2360 (break-even) | 1.2637 | 7,913,484 | 7,913,484 | 8,090,884 | 0 |
| 1.2900 | 1.2900 | 7,751,938 | 7,913,484 | 7,751,938 | −161,546 |
| 1.3300 | 1.2900 | 7,751,938 | 7,913,484 | 7,518,797 | −161,546 |
The structure is a forward at a worse rate with a partial option to do better. It beats the forward only once sterling has fallen 277 pips below it, to 1.2360. Across the whole range from 1.2360 upwards it pays less than the forward, by up to GBP 161,546. Against doing nothing, it keeps 47.7 per cent of the gain below 1.2900, gives up the other 52.3 per cent, and removes all of the loss above it.
| Protected rate | Pips worse than forward | Participation | Worst case, GBP | Given up vs forward, GBP | Break-even spot |
|---|---|---|---|---|---|
| 1.2700 | 63 | 14.5% | 7,874,016 | 39,469 | 1.2276 |
| 1.2800 | 163 | 33.2% | 7,812,500 | 100,984 | 1.2320 |
| 1.2900 | 263 | 47.7% | 7,751,938 | 161,546 | 1.2360 |
| 1.3000 | 363 | 59.0% | 7,692,308 | 221,177 | 1.2396 |
| 1.3100 | 463 | 67.8% | 7,633,588 | 279,897 | 1.2428 |
Every extra 100 pips of protection given up buys between 9 and 19 points of participation, at a falling rate. The break-even barely moves: it sits between 1.2276 and 1.2428 whatever strike is chosen, because the better participation is paid for by a worse floor. Volatility matters too: at 1.2900 participation is 57.9 per cent with 6 per cent volatility and 40.4 per cent with 10, so the same structure quoted in a calm and a nervous market can look very different.
Calling it zero cost because no premium changes hands. The premium is zero; the cost is the protected rate set below the forward from the company's point of view. On this trade that is GBP 161,546 whenever sterling ends above the break-even, which, on the pricing distribution centred on the forward, is more likely than not. The second mistake is ignoring the sold option in the board paper: below 1.2900 the company is obliged to sell 52.3 per cent of the dollars at 1.2900 even if spot is far better. The same logic applied to a two-strike structure is in is a zero-cost collar really zero cost.
Price the two options at the protected rate, set participation as one minus their ratio, then compare sterling received with the forward at every settlement rate, not with spot. Here 47.7 per cent participation costs 263 pips of certainty and pays off only below 1.2360. The book's own forward and swap cost lines are in the free workbook for this case, and the option pricing in the Garman-Kohlhagen worked example.
Set the premium of the bought option equal to the premium of the sold option on the unprotected share: participation equals one minus the bought option price divided by the sold option price at the same strike. In the worked case the GBP call costs 0.02778 and the GBP put is worth 0.05309 dollars per pound at 1.2900, so participation is 47.7 per cent.
Only if the currency moves in the company's favour by more than a break-even amount. Here the exporter receives more sterling than under the 1.2637 forward only if GBP/USD settles below 1.2360. Anywhere above that it receives less, and at or above the 1.2900 protected rate it receives GBP 161,546 less than the forward would have paid.
It falls. A call and a put at the same strike have the same vega, so both premiums rise by the same amount, but that amount is a larger share of the cheaper, out-of-the-money bought option, so the ratio of the two rises, more of the sold option is needed to pay for the protection and participation drops. At the same 1.2900 protected rate, participation is 57.9 per cent at 6 per cent volatility, 47.7 per cent at 8 and 40.4 per cent at 10.
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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