A gas plant's dispatch right priced as a call on the spread, step by step, and the correlation that decides it.
Kirk's approximation prices a spark spread option like a Black-Scholes call on power, with the gas cost plus the fixed costs as the strike and a blended volatility that nets out the correlation between power and gas. For an illustrative month six months out, with power at €110.0/MWh, a fuel cost of €76.0, €28.9 of carbon and variable cost, 45 and 40 per cent volatilities and a correlation of 0.70, the option is worth €12.32/MWh, against an intrinsic value of €5.02: 59.2 per cent of the value is time value.
Worked in full in Energy Trading by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
A gas-fired plant converts gas into power whenever the clean spark spread is positive, and stays off when it is not. Its owner therefore holds a call on the spread between power and fuel, struck at the plant's carbon and variable cost. The spread on today's forwards gives the intrinsic value, the margin locked in if both legs were hedged now; the option value adds what the right to wait and dispatch later is worth. That is the number a trader marks, and the one a buyer of a tolling agreement or a plant should not leave out. The spread itself is covered in how to calculate a clean spark spread.
| Input | Value |
|---|---|
| Power forward, F1 | €110.0/MWh |
| Gas forward | €38.0/MWh thermal |
| Heat rate (50% efficiency) | 2.0 |
| Fuel cost per MWh of power, F2 | €76.0 |
| Carbon: 0.37 t/MWh at €70/t | €25.9 |
| Variable operating cost | €3.0 |
| Fixed strike, K | €28.9 |
| Volatility of power / gas | 45% / 40% |
| Correlation | 0.70 |
| Time to expiry | 0.5 years |
| Interest rate | 3% |
The clean spark spread on the forwards is 110.0 − 76.0 − 25.9 = €8.1/MWh, and €5.1 after variable cost. Carbon is treated as fixed in the strike to keep the option two-legged; with a volatile carbon price the spread is a three-asset option and Kirk's formula no longer applies directly.
w = F2 / (F2 + K) = 76.0 / 104.9 = 0.7245
σ = √(σ1² − 2ρσ1σ2w + σ2²w²) = √(0.45² − 2 × 0.70 × 0.45 × 0.40 × 0.7245 + (0.40 × 0.7245)²) = 32.2%
d1 = [ln(F1 / (F2 + K)) + σ²T/2] / (σ√T) = 0.3222; d2 = d1 − σ√T = 0.0943
Value = e−rT [F1 N(d1) − (F2 + K) N(d2)] = 0.9851 × (110.0 × 0.6264 − 104.9 × 0.5376) = €12.32/MWh
Excel: =EXP(-r*T)*(F1*NORM.S.DIST(d1,TRUE)-(F2+K)*NORM.S.DIST(d2,TRUE))
The blended volatility is the heart of it. Power and gas move together, so the spread is far less volatile than power alone: 32.2 per cent rather than 45. The weight w scales the gas volatility down because only part of the strike, the fuel, moves; the carbon and variable cost do not. With no fixed strike, w is 1 and Kirk's formula collapses to Margrabe's exact formula for an exchange option.
The option is worth €12.32/MWh. Its intrinsic value, the €5.1 spread discounted, is €5.02, so €7.30, or 59.2 per cent, is extrinsic. On a 400 MW plant over a 744-hour month, 297,600 MWh, that is €3.67 million of option value against €1.50 million locked in on the forwards, a difference of €2.17 million that exists only if the plant keeps the flexibility to dispatch on the day. A Monte Carlo simulation of the two correlated forwards gives €12.36, so the approximation is close here; it is least accurate when the fixed strike is large relative to the fuel cost.
This is one option on the monthly average spread. A real plant holds an option in every hour, on prices far more volatile than a monthly forward, so its true flexibility value is higher than this, and it pays start costs and minimum run times that reduce it. The hourly approach is in how to price a power plant tolling agreement.
| Case | Spread volatility | Value | Extrinsic |
|---|---|---|---|
| Correlation 0.5 | 39.5% | 14.44 | 9.41 |
| Correlation 0.7, base | 32.2% | 12.32 | 7.30 |
| Correlation 0.9 | 22.7% | 9.59 | 4.57 |
| One month to expiry | 32.2% | 7.03 | 1.94 |
| One year to expiry | 32.2% | 15.97 | 11.02 |
| Power at €95.0, out of the money | 4.89 | 4.89 | |
| Power at €125.0, deep in the money | 23.05 | 3.25 |
Correlation is the input that moves the answer most and is the hardest to observe. Moving it from 0.7 to 0.9 takes a fifth off the value. Time value decays as delivery approaches, from €7.30 at six months to €1.94 at one month. And extrinsic value is largest near the money: a plant that is out of the money on the forwards, with power at €95.0, is still worth €4.89/MWh, all of it option value, while one deep in the money is mostly intrinsic.
Kirk's formula values a spark spread option in one line once the strike is split into a moving fuel leg and a fixed carbon and cost leg: here €12.32/MWh against €5.02 of intrinsic value, with correlation the assumption to defend. Why the hours a plant actually runs matter more than the annual forward, and what that does to its value, is worked in the free workbooks for Energy Trading.
It is a closed-form approximation for an option on the difference between two prices with a fixed strike. It treats the second asset plus the strike as one lognormal asset and uses the volatility sqrt(s1^2 - 2 rho s1 s2 w + (s2 w)^2), with w equal to F2 / (F2 + K). In the worked case w is 0.7245 and the spread volatility 32.2 per cent.
Higher correlation between power and gas makes the spread less volatile and the option cheaper. In the illustrative month the value is 14.44 EUR/MWh at a correlation of 0.5, 12.32 at 0.7 and 9.59 at 0.9, so a move of 0.2 in an input that is hard to observe changes the value by about a fifth.
Intrinsic value is the spread locked in on today's forwards; extrinsic value is what the right to dispatch later adds. For the illustrative 400 MW plant over a 744-hour month, the option is worth 3.67 million euros, of which 1.50 million is intrinsic and 2.17 million extrinsic.
This article is one calculation from Energy Trading. 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.
Get the book on Amazon →Free companion files
Also on Amazon UK · Amazon Germany · Amazon France · Amazon Canada
Reading guide: infrastructure, data centres and energy → · All 1501 articles →
If this book helped, or didn’t, a few lines on Amazon are worth more than they look: they are what the next reader goes on. Write a review. The workbook stays free either way.