# Some Theory on Price and Volatility Modeling

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1 Some Theory on Price and Volatility Modeling by Frank Graves and Julia Litvinova The Brattle Group 44 Brattle Street Cambridge, Massachusetts Voice Fax January, 2009 Copyright 2009 The Brattle Group, Inc.

2 Introduction and Overview To get the benefits of a risk analysis tool, managers and regulators must have a basic understanding of its analytic foundations. Users also must know how design choices affect results. Key elements of electric or gas price risk management modeling: Typical price distribution: lognormal Term structure of volatility: the two-factor model Mean reversion Seasonality Simulating forward curve dynamics: Monte Carlo Calculating future option premiums Calculating and summarizing total cost distributions Assessing cash and credit risk to the utility 2

3 Price Mean and Uncertainty Forward price of a commodity F t,t will change over procurement horizon in relation to expected spot price P T Small t subscript refers to current transaction date (trading day) Big T subscript refers to future delivery or settlement date F price changes randomly with new information up to settlement date F at delivery date is spot price, i.e. F T,T = P T F price treated in model (as is common practice) as an unbiased estimate of future spot, i.e. no trend Process of F evolving over time is modeled as variation on a random walk in a risk-neutral environment 3

4 Probability Distribution of Prices Commodity prices are typically asymmetric, in the sense that they are non-negative and have more upside potential than down-side. We model percentage changes in forward prices F t,t over short time intervals having a normal distribution Then future spot prices P T will have a lognormal distribution Empirically found to describe distribution of many commodities, including natural gas 4

5 Properties of Lognormal Distribution Lognormal Distribution: Has the right shape Bounded on the lower end by zero Skewed to the right (no theoretical upper limit on prices) Up or down percentage price movements (returns) are equally likely Completely described by two parameters: mean and variance Can be enhanced to capture other time patterns (see next) Probability Volatility 25% Price Volatility 50% 5

6 PJM Price Histories and Distributions Historical LMP Daily Prices Distributions, % 18% 16% Historical LMP Daily Prices and Returns, Summer % LMP Price, \$/MWh Daily Prices 100% 80% 60% 40% 20% 0% -20% Frequency, % 12% 10% 8% 6% 4% 2% 0% LMP Price, \$/MWh Historical LMP Daily Returns Distributions, Daily Changes in Prices (Returns) -40% -60% 14% 12% 6/1/07 6/8/07 6/15/07 6/22/07 6/29/07 7/6/07 7/13/07 7/20/07 Date 7/27/07 8/3/07 8/10/07 8/17/07 8/24/07 8/31/07 Frequency, % 10% 8% 6% 4% 2% 6 0% -60% -50% -40% -30% -20% -10% 0% 10% 20% 30% 40% 50% 60% Return, %

7 Volatility Term Structure Volatility term structure refers to the relation between volatility and delivery time T. 70% Typically, term structure of annualized volatilities is declining (but not smoothly) High near-term volatility that decays to a steady, lower long-term level Annualized Volatility 60% 50% 40% 30% 20% May exhibit seasonality (some months usually more volatile than others) Seasonality can vary by trading date as well as by delivery date 10% 0% Mar-07 May-07 Jul-07 Sep-07 Nov-07 Jan-08 Mar-08 May-08 Delivery Month Jul-08 Sep-08 Nov-08 Quoted volatilities implied by contracts for natural gas call options (available from brokers) as of February 07 Jan-09 Mar-09 7

8 Volatility Model: Two-Factor Model Volatility term structure can be modeled as a meanreverting, two-factor model. Forward curve response to new information is subject to mean reversion (declining influence into future) represented as σ ( t, T ) = β η ( T t) s e where β s is the short volatility η is the mean reversion rate t is current date T is settlement date Long volatility for delivery in distant months tends to level off at a positive level, β L,reflecting uncertainty in long-run marginal costs Overall volatility is a combination of both; these information types are independent, so their variances add: σ 2 2η ( T t) 2 ( t, T ) = βs e + β L 8

9 Fitting the Two-Factor Model The parameters of the two factor model can be chosen to best-fit quoted volatilities: σ ( T ) = = 1 T T 0 2 σ ( t, T ) dt 2 1 e βs 2η ( T 2η ( T t) t) + β 2 L Annualized Volatility 70% 60% 50% 40% 30% 20% 10% 0% Mar-07 Fitted values: Delivery Month short volatility = β S = 57% / year mean reversion = η = 200% / year has a half-life of four months, i.e. it takes four months for the volatility shock to die out to half of its original value long volatility = β L = 26% / year May-07 Fitted with two-factor model volatility, but no seasonality Quoted volatility Jul-07 Sep-07 Nov-07 Jan-08 Mar-08 May-08 Jul-08 Sep-08 Nov-08 Jan-09 Mar-09 9

10 Understanding the Impact of the Two-Factor Model Parameters Higher longfactor volatility (e.g., twice the original value) causes higher long term volatility level Lower shortfactor mean reversion (e.g., one third of the original value) slows the decay of volatility to its long-term level Annualized Volatility 70% 60% 50% 40% 30% 20% 10% 0% Mar-07 May-07 Fitted with two-factor model volatility, no seasonality (mr = 200% per year, half-life = 4.2 month, lv = 26%) Jul-07 Sep-07 Nov-07 High Long Volatility (lv = 52% per year) Jan-08 Mar-08 May-08 Delivery Month Low Mean Reversion (mr = 67% per year, half-life = 12.5 month) Jul-08 Sep-08 Nov-08 Jan-09 Mar-09 10

11 Adding Seasonality to the Two Factor Model The volatility with respect to delivery time has peaks and valleys at particular delivery seasons. We capture delivery seasonality by scaling average volatility up or down in each month by a factor: Jan = 1.07, Feb = 1.11, Mar = 1.13, Apr = 0.94, May = 0.90, Jun = 0.88, Jul = 0.90, Aug = 0.94, Sep = 0.99, Oct = 1.05, Nov = 1.04, Dec = 1.04 Annualized Volatility 70% 60% 50% 40% 30% 20% 10% 0% Mar-07 May-07 Fitted without seasonality Fitted with seasonality Quoted volatility Jul-07 Sep-07 Nov-07 Jan-08 Mar-08 May-08 Delivery Month Jul-08 Sep-08 Nov-08 Jan-09 Mar-09 11

12 Simulating Future Volatility Using Two-Factor Model From the parameter estimates of the two-factor model, we can simulate how future average volatility will shift, in a fashion consistent with the original quotes. 70% New t causes nearmonth volatility to increase Also different T-t remaining, and must apply appropriate seasonality factors Annualized Volatility 60% 50% 40% 30% 20% Projected volatility as of 8/07 given 2/07 quotes Quoted and fitted volatility as of 2/07 Projected volatility as of 2/08 given 2/07 quotes 10% 0% Mar-07 May-07 Jul-07 Sep-07 Nov-07 Jan-08 Mar-08 May-08 Jul-08 Sep-08 Nov-08 Jan-09 Mar-09 Delivery Month 12

13 Simulating Forward Curve Dynamics The next step is to simulate many possible future forward curves created by sampling from the volatility equation and perturbing the original forward curve accordingly. In the two-factor model, forward price changes occur in response to two types of information: short-term and long-term % ΔF t, T df F t, T t, T T t S = βse η ( ) dzt + mean-reversion coefficient short and long volatility coefficients β dz L L t random variables in simulations 13

14 Simulating Forward Curve Dynamics Goal: simulate forward prices F T,T+i i=1,2, at trading date T for delivery at T+1,T+2 etc. (outlook as of the current time, t) 1. Simulate uncorrelated standard normal random variables dz ts and dz tl representing draws of new short-term and long-term information (factors) in period t. 2. Compute cumulative uncertainty for each factor from the current time, t, to delivery time, T. 3. Compute future spot price P T consistent with cumulative price uncertainty and prevailing forward curve. 4. Compute future forward price F T,T+1 perturbing the original forward curve F t,t+1 using the forward price dynamics: exponentially decay short factor at its mean reversion rate and do not decay long factor. 5. Repeat thousands of times, such that average of simulations recapitulates the original forward curve and volatility quotes. 14

15 Simulated Future Spot Prices Recall that if changes in forward prices are normally distributed, we can write forecasts of future spot prices in the standard lognormal form: P T Z ~ ) σ (T where is a standard normal w S, wl dw dw S, t L, t = F ( w w T T ) 1 2 L, T + S, T 0, T exp 2 σ ( ) is cumulative volatility from trading date t to delivery date T are short and long volatility factors following dynamic equations: = ηw = β dz L S, t L t dt + β dz S S t 15 Mean correction of lognormal distribution References: 1) Chapter 8 in Clewlow, L., and C. Strickland (2000): Energy Derivatives: Pricing and Risk Management. 2) Electric Power Research Institute (EPRI) Technical Brief WO3581

16 Evolution of Forward Curves The volatility function can be used to repeatedly perturb the forward price curve in order to create simulations of what forward curves might prevail in the future. \$20 \$18 \$16 \$14 Possible forward curves for 12/1/06 Possible Forward Curves on 6/1/07 Forward Price \$12 \$10 \$8 Forward Price on 6/1/06 \$6 \$4 \$2 Realized Forward Price on 12/1/06 \$0 Jun-06 Jul-06 Aug-06 Sep-06 Oct-06 Nov-06 Dec-06 Jan-07 Feb-07 Mar-07 Apr-07 May-07 Jun-07 Jul-07 Aug-07 Sep-07 Oct-07 Nov-07 Dec-07 Date 16

17 Updating Costs and Risks Over Time Prices and volatility both change over time. Periodically updating volatility reveals shifting market conditions that may affect desired hedging decisions. 90% Implied Volatility from 2005 NYMEX Natural Gas Options Implied Volatility 80% 70% 60% 50% 40% 30% Post Katrina Jul-05 Aug-05 20% Pre Katrina 10% 0% Nov-05 Dec-05 Jan-06 Feb-06 Mar-06 Apr-06 May-06 Jun-06 Jul-06 Aug-06 Sep-06 Oct-06 Nov-06 Dec-06 Delivery Month Notes: Calculations for Implied Volatility use the Black-Scholes Model. Implied Volatility Calculations are based on the last trading day of each calendar month, only including options with a positive volume. 17

18 Impact of the Two-Factor Model Parameters on Purchasing Strategy Changes in volatilities (and forward curves) will cause initial cost distribution results to shift. Annual Model Results No material impact on expected total costs from increase in volatility alone But procurement cost distribution becomes much wider And collars become more expensive (may raise expected total costs slightly) Cumulative Probability High Long Term Volatility Current Outlook Low Mean Reversion 0.0 \$3.00 \$5.00 \$7.00 \$9.00 \$11.00 \$13.00 \$/MMBtu 18

19 Potential Extensions In addition to the energy price volatility modeling described above, several other factors affecting portfolio risk and asset value are stochastic: Congestion Capacity prices RECs Correlations between uncertain variables may be important to certain resources (e.g., gas-fired power plants). And, long term volatility may change over time, e.g., when/if regulatory rules change (CO 2 ). These extensions, as well as mark-to-market exposure analysis, can be accommodated with similar techniques. 19

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