Hedging and Calculus at PG&E: Diablo Valley College Field Trip Chuck Riedhauser April 15, 2008

2 Overview PG&E procurement activities Valuation of power plants Hedging spread options

April 15, Pacific Gas and Electric Combined gas and electric utility in northern and central California Natural Gas –4 million customer accounts –850 bcf (billion cubic feet) Electric –5 million customer accounts –86,000 GWh (gigawatt-hours) –20,000 MW peak load

April 15, 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

April 15, 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

April 15, Random Gas and Power Prices

April 15, Price Parameters Parameters describe random prices Forward prices –Expected value of random prices Volatilities –Measure of much prices fluctuate –Related to standard deviation Correlation –Measures how likely prices are to move together

April 15, The Valuation Problem How do we value a contract which depends on 2 random prices? Option theory! 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

April 15, Spread Option

April 15, 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

April 15, 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

April 15, 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

April 15, 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 …

April 15, Hedge Ratios, continued

April 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

Questions?