A collar is two options netted into one quote. Value each leg with Black-76 and the premium that was never invoiced turns up on one side or the other.
No. A zero-cost collar has zero premium, not zero value: it is a bought option paid for by a sold one, and the two are rarely worth the same. On a one-year commodity hedge priced with Black-76, a cap bought at 2,680 is worth 182.84 a tonne and the floor sold at 2,300 is worth 122.18, so 60.66 a tonne, or 1,577,235 on 26,000 tonnes, changed hands on a trade quoted at nothing.
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The case is Kilmartin Industrial, the company in Derivatives that consumes a commodity and wants protection against a rise in its price. It buys a cap and, to avoid paying a premium, sells a floor: below the floor it gives up the benefit of a falling price. The bank quotes the pair at zero. All figures are illustrative.
| Input | Value |
|---|---|
| Consumption hedged, tonnes | 26,000 |
| Spot, per tonne | 2,450 |
| Carry rate to one year | 4.12% |
| One-year forward, spot × (1 + carry) | 2,550.94 |
| Volatility | 24% |
| Cap bought at | 2,680 |
| Floor sold at | 2,300 |
| Premium quoted | 0 |
For an option on a forward, Black-76 is the standard model. The cap is a call on the forward and the floor a put, both discounted at the rate.
d1 = [ln(F/K) + σ²T/2] / (σ√T), d2 = d1 − σ√T
Cap (call) = DF × [F × N(d1) − K × N(d2)]
Floor (put) = DF × [K × N(−d2) − F × N(−d1)]
Excel, cap: =EXP(-r*T)*(F*NORM.S.DIST(d1,TRUE)-K*NORM.S.DIST(d2,TRUE))
For the cap at 2,680, d1 is −0.0856 and d2 is −0.3256; its value is 182.84 a tonne. For the floor at 2,300, d1 is 0.5515 and d2 is 0.3115; its value is 122.18 a tonne.
| Leg | Strike | Distance from forward | Value per tonne | On 26,000 tonnes |
|---|---|---|---|---|
| Cap, bought | 2,680 | 129.06 above | 182.84 | 4,753,803 |
| Floor, sold | 2,300 | 250.94 below | −122.18 | −3,176,567 |
| Collar, net to Kilmartin | 60.66 | 1,577,235 |
The strikes are not symmetric around the forward: the cap sits 129.06 above it and the floor 250.94 below. The nearer option is worth more, so the company bought more value than it sold and received 1,577,235, about 2.38 per cent of the 66.3M forward value of the hedged tonnage. The cliché about zero-cost structures is that the bank hides its margin in the strikes. Here the transfer runs the other way. The point is not that one side won; it is that nobody priced the structure in either direction.
The strike that would have made it genuinely free comes from solving for the floor whose value equals the cap's 182.84. There is no closed form, so iterate with Newton's method, using the floor's sensitivity to its own strike, DF × N(−d2). It settles at 2,447.51. A truly zero-premium collar with a cap at 2,680 would have given away every fall below 2,447.51, only 2.49 below today's spot. The floor actually sold, at 2,300, kept 147.51 a tonne more of the downside for Kilmartin.
At expiry the collar is just a price band. If the market ends at 3,000 the cap pays 320 a tonne, 8,320,000 in all. If it ends at 2,000 the floor costs 300 a tonne, 7,800,000. Between 2,300 and 2,680 nothing settles. None of that tells you whether the zero premium was fair; only the valuation on day one does.
| Volatility | Cap | Floor at 2,300 | Net per tonne | Net on 26,000 tonnes | Premium-free floor |
|---|---|---|---|---|---|
| 16% | 105.65 | 57.95 | 47.70 | 1,240,145 | 2,440.84 |
| 20% | 144.04 | 89.20 | 54.84 | 1,425,740 | 2,444.22 |
| 24% | 182.84 | 122.18 | 60.66 | 1,577,235 | 2,447.51 |
| 28% | 221.83 | 156.14 | 65.69 | 1,707,990 | 2,450.71 |
| 32% | 260.88 | 190.68 | 70.20 | 1,825,242 | 2,453.82 |
Because the cap is nearer the money, it gains more from volatility than the floor, and the collar is long volatility for the buyer: the net value rises from 47.70 to 70.20 a tonne across the range. Moving the floor matters more. At a floor of 2,400 the transfer shrinks to 21.11 a tonne, 548,768 in all; at 2,200 it grows to 93.63, or 2,434,392.
The mistake is to treat the absence of a premium as evidence of fairness and therefore to stop pricing. A treasurer who would challenge a 1.6M option premium on an invoice accepts the same value embedded in a pair of strikes because no cash moves. The discipline is to value both legs at the moment of dealing, with an independent volatility, and record the net. If it is in the bank's favour, that is the price of the hedge and belongs in the cost line. If it is in yours, the strikes can probably be improved. The same applies to the second year of a rolling collar, when the floor is re-fixed against a price that has already moved.
For the cash consequences of hedges that are marked to market, see how much more margin a five-year hedge calls than a rolling one.
Black-76 written out in cells, the Newton solve for the free floor and the collar year by year are in the free workbook for this case.
Value the option bought and the option sold separately, with Black-76 on the forward for commodities, and compare. The collar is fair only if they are equal. Here a cap at 2,680 is worth 182.84 a tonne and a floor at 2,300 is worth 122.18, so a collar quoted at zero premium transferred 60.66 a tonne to the buyer of the cap.
Fix the strike you care about, here the cap at 2,680, and solve for the other strike at which the two option values are equal. There is no closed form, so iterate: Newton's method converges in a few passes. At 24 per cent volatility the premium-free floor is 2,447.51, only 2.49 below a spot of 2,450.
Yes, unless the strikes are roughly symmetric around the forward. Here the cap is 129.06 above the forward and the floor 250.94 below, so the cap gains more from volatility. At 16 per cent the buyer receives 47.70 a tonne; at 32 per cent, 70.20. A collar is a volatility position as well as a price hedge.
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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