April 15, 2008. 2. Overview. PG
1. Hedging and Calculus at PG&E: Diablo Valley College Field Trip Chuck Riedhauser
April 15, 2008
2. April 15, 2008 2 Overview PG&E procurement activities
Valuation of power plants
Hedging spread options
3. April 15, 2008 3 Pacific Gas and Electric Combined gas and electric utility in northern and central California
4 million customer accounts
850 bcf (billion cubic feet)
5 million customer accounts
86,000 GWh (gigawatt-hours)
20,000 MW peak load
4. April 15, 2008 4 Electricity Procurement PG&E owned facilities produce less than half the electricity consumed in our service territory
PG&E purchases most of its power from third parties
Calpine, PPM, etc.
Energy Procurement division negotiates contracts, takes delivery, and schedules power
Quantitative Analysis group values all potential contracts and assists in risk management (hedging) of contracts
5. April 15, 2008 5 Procurement Contracts The typical contract we (PG&E) purchase is a “tolling agreement” for a gas-fired power plant
We get the ability to call on power when we need it
We pay a fee and provide the gas
Costs of power are heavily tied to gas commodity prices
Power prices are also random
6. April 15, 2008 6 Random Gas and Power Prices
7. April 15, 2008 7 Price Parameters Parameters describe random prices
Expected value of random prices
Measure of much prices fluctuate
Related to standard deviation
Measures how likely prices are to move together
8. April 15, 2008 8 The Valuation Problem How do we value a contract which depends on 2 random prices?
Power plant (or toll) is viewed as a “spread option” between gas and power
We want to know the value of a spread option which expires in the future
9. April 15, 2008 9 Spread Option
10. April 15, 2008 10 Hedging a Power Plant What is hedging?
Reducing the unexpected changes in the value of a plant or contract
Why do we hedge?
Our customers are risk averse—they don’t like unexpected price increases
How do we hedge a power plant?
We enter into contracts which offset value changes in power plant
How much of these other contracts should we purchase?
Look at how power plant value changes as the forward prices of gas and power change
11. April 15, 2008 11 Hedging a Power Plant, continued How does the value of a power plant (i.e., spread option) change for small changes in the forward prices?
Taylor series expansion of spread option value
12. April 15, 2008 12 Hedging a Power Plant, continued Spread option is “hedged” by acquiring other positions which have the opposite dependence on price moves
First order hedge
Add positions with first order expansion
Second order hedge positions have expansion
13. April 15, 2008 13 Hedge Ratios The partial derivatives of the option value with respect to forward prices are called “hedge ratios”
Hedge ratios derived from the option value formula
Repeated application of the chain rule for derivatives gives …
14. April 15, 2008 14 Hedge Ratios, continued
15. April 15, 2008 15 Hedging Power Plants Power plants modeled as spread options
Comprehends random gas and power prices
Spread option value expressed in terms of cumulative distribution function
Taylor series forms basis for hedging
Hedge ratios calculated by chain rule