Book Review: Chapter 6 ’Spot Price Models and Pricing Standard Instruments’ Anatoliy Swishchuk Dept of Math & Stat, U of C ‘Lunch at the Lab’ Talk January 31 st, 2007

Outline Intro Single Factor Models Two Factor Models Three Factor Models Choosing a Spot Price Model

Intro Models for Pricing Energy Derivatives Formulated in Terms of the Spot Energy Price Derivatives: Futures/Forwards, European Options Range: from 1 factor Black (1976) model to a three- factor model with stochastic convenience yield and stochastic term structure of interest rate Comments: seasonal factors Volatility Smile + Numerical Techniques: Chapter 7 (Lance’s Talk)

Single Factor Models Futures and Forward Pricing Option Pricing The Schwartz Single Factor Model (Futures/Forward, Option Pricing)

Single Factor Models: SDE vs PDE

Futures and Forward Pricing (Black, 1976)

Volatilities (FP)=Volatility SP

Option Pricing (Black, 1976): European Futures Option

The Schwartz Single Factor Model (1997) Mean-Reverting Positive S Alpha-mean reverting rate Mu-long term level Lambda-market price of energy risk

The Schwartz Single Factor Model (x=ln S): SDE and PDE

Futures and Forward Pricing (Schwartz SF Model)

Futures and Forward Pricing (Schwartz SF Model): long maturity level and volatility

Futures Prices and Their Volatility (Schwartz SF Model):

European Call Option Pricing (Schwartz SFM): Clewlow, Strickland (1999)

European Call Option Pricing (Schwartz SFM): Clewlow, Strickland (1999): s=T

European Call Option Pricing (Schwartz SFM)

Comparison: Option Prices in the Black and Schwartz SFM

Two Factor Models (Stochastic Convenience Yield) Gibson & Schwartz (1990) Schwartz (1997) Pilipovich (1997) Hillard & Reis (1998)

Two Factor Models (Stochastic Convenience Yield): PDE

Two Factor Models (Stochastic Convenience Yield): Futures/Forward Pricing (Schwartz(1997))

Two Factor Models (Stochastic Convenience Yield): Futures/Forward Pricing ( HR (1998))

Two Factor Models (Stochastic Convenience Yield): Futures Pricing (Schwartz(1997))

Two Factor Models (Stochastic Convenience Yield): Volatility of Futures Pricing (Schwartz(1997)&HR(1998))

Two Factor Models (Stochastic Convenience Yield): Volatility (Schwartz(1997))

Two Factor Models: Option Pricing (Clewlow & Strickland (1999))

The Schwartz 1 Factor Approximation: Rate of Change in the Futures Prices (Two Factor vs One Factor)

The Schwartz 1 Factor Approximation: Rate of Change in the Futures Prices (Two Factor vs One Factor, convenience yield)

The Schwartz 1 Factor Approximation: Rate of Change in the Futures Prices (Two Factor vs One Factor): Shadow Spot Price vs Futures Price

Three Factor Models Schwartz (1997): extension of his TFM (Vasicek short term rate r) Hillard & Reis (1998): interest rate follows HJM (1992) model

Three Factor Models: Schwartz (1997)

Three Factor Models: HR (1998)

Three Factor Models PDE: Schwartz (1997) &HR (1998)

Futures/Forward Pricing: (Three Factor Models, Schwartz (1997))

Futures/Forward Pricing: (Three Factor Models, HR (1998))

Volatility of the Futures Prices (S&HR)

TFM: Option Pricing (Milstein & Schwartz (1998))

Choosing a Spot Price Model For Short Maturity Options on Long Maturity Forward Contract: Black Model could be used For Short Maturity Options on Short Maturity Forward Contract: Schwartz One Factor Model could be used Large and Diverse Portfolio of Energy Contracts: Two Factor Stochastic Convenience Yield Model is good

Choosing a Spot Price Model II Not Necessary to Use Three Factor Model: Stochastic Interest Rate has a relatively minor impact on Energy Derivatives Prices Jumps?-loss of the simple analytical solutions and numerical techniques HR-presented a quasi-analytical solution for standard options under 3FM with jumps: but it’s not consistent with the attenuation of the jumps in the case of simple mean reversion

Summary

The End Thank You for Your Attention!