crack spread options

Crack Spread Options

Iris Mack, MBA/PhDOptions Trading Deskx3-6711

Outline of Crack Spread Options ModuleIntroduction

Classification of Crack Spread Options

What's in a Name? - Why the name "Crack Spread Options"?

Heat Rate

Payoffs of Crack Spread Options

Pricing of Crack Spread Options

Hedging of Crack Spread Options

Mini Case Study #1: Spark Spreads

Mini Case Study #2: ERCOT Example from Megawatt Daily

Mini Case Study #3: Spark Spread Call Option

Exotica Pricing Model: HEAT

Crack Spread Option Example

Pricing Models

References

Introduction

Many power generators may be exposed to the difference in the prices of two related commodities, rather than the price of an individual commodity. This type of exposure may be generated when one commmodity is used as an input to a process, which in turn produces another commodity. The difference between the input/output prices isreferred to as a "crack spread".

If the future contracts underlying an option are witten on two separate energy commodities (say natural gas

the same date or different dates. and electricity), then the option is referred to as a crack spread option. The maturity of the futures contracts can be on

Classification of Crack Spread Options

Common crack spreads are as follows:

isobutane, natural gasoline) and natural gas

* heat spreads - the difference beween the prices of No 2 heating oil and crude

* gasoline spreads - the difference between the prices of unleaded gasoline and crude

* resid spreads - the difference between the prices of No 6 fuel oil and crude

* frac spreads - the difference between the prices of gas liquids (propane, ethane, butane,

* spark spreads - the differences between the price of electricity and natural gas.

What's in a name? - Why the name "Crack Spread Options'?

The term "crack" refers to the technological process used in petroleum refineries - the application of vacuum, heat and catalysts tobreak down larger, heavier molecules of hydrocarbons into lighter ones, with higher economic value. Power traders have coined the term

electricity and natural gas reflects the costs or anticipated costs of producing power from a specific facility. It can be used as a method ofconverting millions of Btus (British thermal units) to megawatt hours and vice versa.

The usefulness of the spark spread evaluation is dependent on the market for power which reflects the relationship of supply and demand for power, not the efficiencies of the generating units. Other costs affecting the price of power using the spark spread evaluation includethose of gas transportation, power transmission, plant operations and maintenance, and fixed costs.

The spark spread may be used as a tool for managing the risk of owning a power plant. It is a primary metric for investors to track.The spark spread measures the difference between power prices (revenues) and natural gas prices (cost). This is the critical metric fortracking earnings of generating companies. The gross margin of power producers in the spot market is determined by the spark spread, andwhen these companies hedge, they'll have to hedge using the forward spark spread available at that point in time. Thus, a declining sparkspread affects the earnings power of generators, even those that are well hedged.

spark spread to denote the spread between power prices and the price of fuel burned to generate power. The spark spread between

Heat Rate

In the case of the US power industry, the obvious candidate for the development of the OTC spread option contract is the difference between power prices and natural gas prices. The main reason why natural gas was an obvious candidate is that among the alternative fuels used in the power industry, the market for natural gas is the most developed, both in terms of trading volatility and completeness (that is, the existence of forward markets). Also, in some regions of the US gas-fired power plants are the marginal units to be dispatched into the power grid for most time periods. As a result, power prices are well correlated (for some periods) with natural gas prices.

heat rate is a measure of the thermal efficiency of a power generating unit. It is calculated as follows:

Heat rate =: (Btu content of fuel consumed in electricity production)/(KWh generation of the facility).

10,000 Btu/KWh

Efficient unit: 7,500 Btu/KWh

Inefficient unit: 15,000 Btu/KWh

Spark spreads are often quoted using a convention based on the assumption of a heat rate of 10,000. The

Typical measures of heat rate are as follows:Average unit:

Payoffs of Crack Spread Options

Suppose

represented by the following:

Payoffs of the Crack Spread Options

Name Payoff

Crack Spread Call Option

Crack Spread Put Option

F1(t, s1) = price of an s1-maturity futures contract on commodity 1 (say natural gas)

F2(t, s2) = price of an s2-maturity futures contract on commodity 2 (say electricity)

The payoff to European crack spread options with maturity T and strike K on the spread between these two forward contracts can be

max(F1(T, s1) - F2(T, s2) - K, 0)

max (K - F1(T, s1) + F2(T, s2), 0)

Pricing of Crack Spread Options

Multi-commodity options have payoffs that depend on the prices of two or more underlying instruments. Modeling such optionsmay be a bit complicated, as the premiums depend on multidimensional joint probability distribution of prices.

(1) Treat the spread between the two prices as if it were traded separately from the two underlying commodities. The nextlogical step is to offer options on this new commodity. The pricing of these options is complicated by the fact that the spread can have negativevalues. The Black-Scholes formula can't be used since it's based on the assumption of a lognormal distribution of prices at the horizon definedonly for positive values. However, one may assume that the spread has a normal distribution and use the pricing formula developed by Brennan (1979) and Wilcox (1991). A shortcoming of this approach is that the distribution of the spread is not necessarily normal or evensymmetric. The advantage of this method is the ease of computer implementation, because it produces a formula for option valuation.

(2) Another, and perhaps better, method is to assume that the two prices defining the spread follow a joint lognormaldistribution of the two forward prices and to price the option as the discounted, expected value of the payoff in a risk-neutral world. The value of the call option at time t can be written generally as:

where P(t,T) discounts the future cashflows back to today. The options prices are computed via Monte Carlo simulation. In order to simulate the joint behavior of the energy prices on which the portfolio depends one must compute the volatility functions of the two commodities from the covariance matrix of all the relevant forward prices.

which contain models for the pricing of crack spread options

There are several approaches to pricing spread options:

Crack_Spread(t; K, F1, F2, s1, s2) = P(t,T)*Et[max(0, F1(T, s1) - F2(T, s2) - K)]

The options trading desk has at its disposal two software libraries (Exotica and Financial Engineering Associates (FEA), Inc.)

Hedging of Crack Spread Options

A speculator may want to bet that operating a power plant with a specific heat rate during some period would be more profitablethan others, so that he might buy the spark spread in the futures market. Market risks result from the volatile nature of both inputs and outputs, and from the imperfect correlation between the input and output commodity prices. The high volatility of some of these spreads may be explained by considering that it's influenced by the volatilities of both underlying prices, moderated by their correlation.

Correlation is a measure of how the change in one commodity price is reflected in the other comodity price. The implied correlationin the market price of the spread option is more important than the individual implied volatility of either commodity. A trader will want to buyhigh correlation and sell weak correlation. As the correlation decreases, the price of the spread option increases.

The main problem in valuation of the spark spread options is the unstable nature of the correlation coefficient between powerand natural gas prices. Valuation of the spread option requires making an assumption about the correlation coefficient between the twocommodity prices.

The main risk that the writer and the holder of the option has is that this correlation will change due to market factors or other circumstances. For example, power plant outages or changing hydrological conditions may alter the composition of the supply stack. Inparticular, an abundant water supply in the northwest can reduce reliance on the gas-fired power plants and lower the correlation between natural gas prices and power prices in this region.

Mini Case Study #1: Spark Spreads

Recall that the heat rate corresponds to the number of Btus required to produce one kilowatt of electricity.

Output price =: Power price =: $25/MWh = $25/1,000kWh = $0.025/kWh

Input price =: (Heat rate)*(Gas price) =:10,000 Btu/kWh * $2.00/MMBtu = $0.02/kWh

This spark spread compares the cost of generating power at a certain heating efficiency with the cost of buying peaking from the grid. A positive spread indicates it's economical to buy gas, while a negative spread indicates it's economical to buy power from the grid.

Suppose that the price of natural gas is $2.00/MMBtu and the price of power is $25/MWh. What's the spark spread?

Spark spread = Output price - Input price = $0.025/kWh - $0.02/kWh = $0.005/kWh

Mini Case Study #2: ERCOT Example from Megawatt Daily

Let's look at a sample spark spread situation, courtesy of the 11/29/00 issue of Megawatt Daily Here we look at the electricity prices at ERCOT (Electric Reliability Council of Texas) and at gas prices at the Houston Ship Channel - as indicated in the following table.

$/MMBtu $/MWh 7000 8000 10,000 12,000

5.83 54.58 13.77 7.94 -3.72 -15.38

On 11/29/00, gas trading at the Ship Channel averaged $5.83/MMBtu, and peak power was selling at $52.00-56.00 perMWh in ERCOT (the average was $54.58). In the above table we calculate the cost of generating electricity using gas turbines that operateat various efficiencies, beginning with a highly efficient turbine which only requires 7 MMBtu to generate a megawatt hour, down to aturbine that requires 12 MMBtu to get the same unit of power.

Looking at the most efficient turbine: 7*$5.83 = $40.81, the cost to generate one megawatt hour. If one comparesthat to the cost of buying power in ERCOT at $54.58/MWh, it turns out to be $13.77 cheaper per MWh to generate the power with thehigh-efficiency turbine than to buy it off the grid ($54.58 - $40.81 = $13.77).

However, the story changes if the turbine burns 8 MMBtu, 10 MMBtu or 12 MMBtu to yield a hypotheticalmegawatt hour. At 8,000, it's still economical to self-generate by $7.94/MMBtu. But at 10,000 the spread is -$3.72, and power is cheaperto buy than to generate with gas. At a heat rate of 12,000 it's cheaper to buy from the grid by $15.38. Recall that negative numbersindicate gas is more expensive than power.

Spark spreads look at a very simple world - one where the turbine sits next to the switchyard, and both sit on top of anatural gas pipleline. There are no transaction costs and no transmission or transportation costs. In the above example it is assumed thatthe peaking plant is natural gas-fired, and that gas turbines operate at only four standard heat rates. Too bad the world is not like that.At any rate the spark spread is still a useful way to view the energy marketplace and benchmark relative energy costs.

ERCOT/ Houston Ship Channel

Mini Case Study #3: Spark Spread Call Option

Note: This call option is exercised when the following condition holds:

Market power price > (Market gas price)*(Plant HR) + VAR O&M

Power Market Power Desk

e-

Market Power Price

(Plant HR)*(Market gas) + VAR O&M

(Plant HR)*(Market gas) + VAR O&M

Plant

e-

Exercise Call

NoNo

Yes

Profit = 0

Revenue =: Market power price = (Market gas)*(Market HR)

- Cost =: - (Market gas price)*(Plant HR) - (VAR O&M)

Profit =: (Market gas price)*(Market HR - Plant HR) - (VAR O&M)

Revenue =: Market power price = (Market gas)*(Market HR)

- Cost =: - (Market gas price)*(Plant HR) - (VAR O&M)

Profit =: (Market gas price)*(Market HR - Plant HR) - (VAR O&M)

Exotica Crack Spread Option: HEAT

The fundamental assumption is that the two prices evolved as correlated GBM.In this function the strike price K is zero.

Inputs:Price1 Forward price of commodity 1

Price2 Forward price of commodity 2

Heat Rate (Heat Rate)/1000

IntRt Interest rate

Vol1 Annualized volatility of commodity 1

Vol2 Annualized volatility of commodity 2

Correl Correlation coefficient of commodity 1 and commodity 2

ExpDays Time to option expiration (days

OptType 1 = Call, 0 = Put

Outputs:0 Option Premium

1 Delta with respect to commodity 1

2 Delta with respect to commodity 2

3 Gamma with respect to commodity 1

4 Gamma with respect to commodity 2

5 Vega with respect to Vol1

6 Vega with respect to Vol2

7 eta (premium sensitivity to change in correlation)

8 rho

9 theta (annualized)

10 charm for commodity 111 charm for commodity 212 Cross gamma

Description: The HEAT() function in Exotica Library finds the premium and risk parameters for an option on the spread between two underlyings.

Exotica Crack Spread Option: HEAT(ERCOT Example from Megawatt Daily in Case Study #2)

EffDt 11/30/2000

INPUTS RetType

Fwd Price 1 Fwd Price 2 (Heat Rate)/1000 Ann.IntRt Vol.1 Vol.2 Correlation ExpDt Days OptType

54.83 5.83 7 5.25% 90% 85% 50% 12/29/2000 29 154.83 5.83 8 5.25% 90% 85% 50% 12/29/2000 29 154.83 5.83 9 5.25% 90% 85% 50% 12/29/2000 29 154.83 5.83 10 5.25% 90% 85% 50% 12/29/2000 29 154.83 5.83 12 5.25% 90% 85% 50% 12/29/2000 29 154.83 5.83 15 5.25% 90% 85% 50% 12/29/2000 29 1

OUTPUTS0 1 2 3 4 5 6 7 8 9 10 11 12

Price Delta 1 Delta 2 Gamma 1 Gamma 2 Vega 1 Vega 2 Eta Rho Theta Charm 1 Charm 2 CrossGamma#ADDIN? #ADDIN? #ADDIN? #ADDIN? #ADDIN? #ADDIN? #ADDIN? #ADDIN? #ADDIN? #ADDIN? #ADDIN? #ADDIN? #ADDIN?

#ADDIN? #ADDIN? #ADDIN? #ADDIN? #ADDIN? #ADDIN? #ADDIN? #ADDIN? #ADDIN? #ADDIN? #ADDIN? #ADDIN? #ADDIN?





Pricing Models

In the previous worksheets we gave case studies and examples of how to price spark spread options. Other crack spread pricing models are as follows:

spread between two average prices.)

Example worksheets:SPRDOPT - O:\research\exotica\xll\xll_templates\sprdopt.xls and M:\exotica\xll\xll_templates\sprdopt.xlsAsnSprd - O:\research\exotica\xll\xll_templates\asnsprd.xls and M:\exotica\xll\xll_templates\asnsprd.xlsAsnSprd2 - O:\research\exotica\xll\xll_templates\asnsprd2.xls and M:\exotica\xll\xll_templates\asnsprd2.xlsOSTRIPSPRD - O:\research\exotica\xll\xll_templates\ostripspread.xls and M:\exotica\xll\xll_templates\ostripspread.xls

on three physical commodities.

physical commodities.

commodities.

on two physical commodities.

on two physical commodities.

on a strip of spread options.

Example worksheets:spread.xls (Advanced) - Examples of SPREADAPO, SPREADASO.

EXOTICA:SPRDOPT (Finds the premium and risk parameters for an option on the spread between two underlyings.)AsnSprd, AsnSprd2 - spread option on Asian spreads (Finds the premium and risk parameters for an option on the

OSTRIPSPRD (STRIP of daily fixed price spread options)

Financial Engineering Associates (FEA), Inc.:

CRACKAPO - Returns an array containing the price and risk measures of a European crack average-price option

CRACKPOT - Returns an array containing the price and risk measures of a European crack price option on three

SPREADOPT - Returns an array containing the price and risk measures of a European spread option on two physical

STRIPSPREADOPT - Returns an array containing the price and risk measures of a strip of European spread options

OPTSPREADOPT - Returns an array containing the price and risk measures of a European compound spread option

OPTSTRIPSPREADOPT - Returns an array containing the price and risk measures of a European compound option

SPREADAPO, SPREADASO - spread options on Asians

References

Financial and Quantitative Analysis, 26, (December), pp. 477-95.* Amin, K., 1991, "On the Computation of Continuous Time Option Prices Using Discrete Approximations," Journal of

* Barrett, J.G., Moore and P. Wilmott, 1992, "Inelegant Inefficiency", Risk, 5 (October), pp. 82-4.

* Boyle, P.P., 1988, "A Lattice Framework for Option Pricing with Two State Variables", Journal of Financial andQuantitative Analysis, 23, (March), pp. 1-12.

* Brennan, M. J., 1979, "The Pricing of Contingent Claims in Discrete Time Models", Journal of Finance, 34, (March), pp. 53-68.

* Clewlow, L. and C. Strickland, 2000, "Energy Derivatives: Pricing and Risk Management," Lacima Publications.

* El-Hawary, M. E., 1995, "Electrical Power Systems: Design and Analysis", IEEE Press, Power Systems Engineering Series.

* Enron's Houston Research Group, Exotica Options Library

* Enron Power Marketing, Product Descriptions

* Financial Engineering Associates (FEA), Inc., 2001, User Guide

* Hull, J., 2000, "Options, Futures, and Other Derivatives," Fourth Edition, Prentice Hall.

* Kaminski, V., 1999, "Managing Energy Price Risk," Second Edition, Risk Publications.

* Kirk, E., 1995, "Correlation in the Energy Markets," in Managing Energy Price Risk. London: Risk Publications.

* NYMEX , Energy Glossary (http://www.nymex.com)

* Platts Megawatt Daily, 2000, Spark Spreads, (November 29).

* Rubeinstein, M., 1991b, "Somewhere Over the Rainbow," Risk 4, (November), pp. 63-6.

* Wilcox, D., 1991, "Spread Options Enhance Risk Management Choices", Nymex Energy in the News, (Autumn), pp. 9-13.

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