Book Review: Chapter 6 ’Spot Price Models and Pricing Standard

Instruments’Anatoliy SwishchukDept of Math & Stat, U of C‘Lunch at the Lab’ TalkJanuary 31st, 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 PricingOption PricingThe 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

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!